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1
TAMBORINI, CAROLINA. « On totally geodesic subvarieties in the Torelli locus and their uniformizing symmetric spaces ». Doctoral thesis, Università degli Studi di Milano-Bicocca, 2022. http://hdl.handle.net/10281/371476.
Texte intégralRésumé :
Oggetto di questa tesi sono le sottovarietà totalmente geodetiche dello spazio dei moduli A_g di varietà abeliane principalmente polarizzate e la loro relazione con il luogo di Torelli. Questo è definito come la chiusura in A_g dell'immagine dello spazio dei moduli M_g di curve algebriche complesse lisce di genere g tramite la mappa di Torelli j: M_g-->A_g. Lo spazio dei moduli A_g è un quoziente dello spazio di Siegel, che è uno spazio simmetrico. Una sottovarietà algebrica di A_g è totalmente geodetica se è l'immagine, tramite la naturale mappa di proiezione, di una qualche sottovarietà totalmente geodetica dello spazio di Siegel. Ci si aspetta che j(M_g) contenga poche sottovarietà totalmente geodetiche di A_g. Questo è anche in accordo con la congettura di Coleman-Oort. La geometria differenziale degli spazi simmetrici si può descrivere attraverso la teoria di gruppi e algebre di Lie. In particolare, le sottovarietà totalmente geodetiche di spazi simmetrici possono essere caratterizzate in termini di algebre di Lie. Queste considerazioni sono alla base della trattazione svolta in questa tesi, in cui utilizziamo alcuni strumenti della teoria di Lie per indagare alcuni aspetti geometrici dell'inclusione di j(M_g) in A_g. I principali risultati presentati sono i seguenti. Nel Capitolo 2, consideriamo il pull-back dell'operazione di Lie-bracket sullo spazio tangente ad A_g tramite la mappa di Torelli e lo caratterizziamo in termini della geometria della curva. Per farlo usiamo il nucleo di Bergman associato alla curva. Inoltre, colleghiamo il nucleo di Bergman alla seconda forma fondamentale della mappa Torelli. Nel Capitolo 3, determiniamo quale spazio simmetrico uniforma ciascuno dei controesempi noti alla congettura di Coleman-Oort attraverso il calcolo della decomposizione dell'algebra di Lie associata. Questi esempi noti erano stati ottenuti studiando famiglie di rivestimenti di Galois. Nel capitolo 4 ci concentriamo sullo studio di queste famiglie e descriviamo una nuova costruzione topologica di famiglie di G-rivestimenti di P^1.This thesis deals with totally geodesic subvarieties of the moduli space A_g of principally polarized abelian varieties and their relation with the Torelli locus. This is the closure in A_g of the image of the moduli space M_g of smooth, complex algebraic curves of genus g via the Torelli map j: M_g-->A_g. The moduli space A_g is a quotient of the Siegel space, which is a Riemannian symmetric space. An algebraic subvariety of A_g is totally geodesic if it is the image, under the natural projection map, of some totally geodesic submanifold of the Siegel space. Geometric considerations lead to the expectation that j(M_g) should contain very few totally geodesic subvarieties of A_g. This expectation also agrees with the Coleman-Oort conjecture. The differential geometry of symmetric spaces is described through Lie theory. In particular, totally geodesic submanifolds can be characterized via Lie algebras. This motivates the discussion carried out in this thesis, in which we use some Lie-theoretic tools to investigate geometric aspects of the inclusion of j(M_g) in A_g. The main results presented are the following. In Chapter 2, we consider the pull-back of the Lie bracket operation on the tangent space of A_g via the Torelli map, and we characterize it in terms of the geometry of the curve. We use the Bergman kernel form associated with the curve. Also, we link the Bergman kernel form to the second fundamental form of the Torelli map. In Chapter 3, we determine which symmetric space uniformizes each of the known counterexamples to the Coleman-Oort conjecture via the computation of the associated Lie algebra decomposition. These known examples were obtained studying families of Galois coverings of curves. Chapter 4 focuses on these families for their own sake, and we describe a new topological construction of families of G-coverings of the line.
2
歐偉民 et Wai-man Au. « Families of polarized abelian varieties ». Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1997. http://hub.hku.hk/bib/B31214897.
Texte intégral
3
Lemos, Pedro. « Residual representations of Abelian varieties ». Thesis, University of Warwick, 2017. http://wrap.warwick.ac.uk/94788/.
Texte intégralRésumé :
This thesis is divided in two parts, corresponding to two papers in which I collaborated during the course of my PhD studies. Both of these parts are concerned with the question of surjectivity of residual Galois representations arising from abelian varieties defined over Q. At the start of each chapter, a full introduction to the topic covered is provided.
4
Fhlathuin, Brid ni. « Mahler's measure on Abelian varieties ». Thesis, University of East Anglia, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.296951.
Texte intégralRésumé :
This thesis is a study of the integration of proximity functions over certain compact groups. Mean values are found of the ultrametric valuation of certain rational functions associated with a divisor on an abelian variety, and it is shown how these may be expressed in terms of an integral, thus finding the analogue, for an abelian variety, of Mahler's definition of the measure of a polynomial. These integrals are shown to arise in a manner which mimics classical Riemann sums, and their relation with the global canonical height is investigated. It is shown that the measure is a rational multiple of log p. Similar results are given for elliptic curves, taking the divisor to be the identity of the group law, and somewhat stronger mean value theorems proven in this more specific case by working directly with local canonical heights rather than approaching them through related functions. Effective asymptotic formulae for the local height are derived, first for the kernel of reduction of a curve and then, via a detailed analysis of the local reduction of the curve, for the group of rational points. The theory of uniform distribution is used to show that the mean value also takes an integral form in the case of an archimedean valuations, and recent inequalities for elliptic forms in logarithms are used to give error terms for the convergence towards the measure. This is undertaken first for the local height on an elliptic curve, and then, in terms of general theta-functions, on an abelian variety. We then seek to exploit these generalisations of the Mahler measure to yield an alternative method to that of Silverman and Tate for the determining of the global height. The integration over a cyclic group of the laws satisfied locally by the height allows us to reformulate our theorems in a manner conducive to practical application. It is demonstrated how our asymptotic formulae may be used together with an appropriate computer software package, PARI in our case, to calculate the mean value of heights, and, more generally, of rational functions, on an elliptic curve and on abehan varieties of higher genus. Some such calculations are displayed, with comments on their efficacy and their possible future development.
5
Au, Wai-man. « Families of polarized abelian varieties / ». Hong Kong : University of Hong Kong, 1997. http://sunzi.lib.hku.hk/hkuto/record.jsp?B19471117.
Texte intégral
6
Young, Ian David. « Symmetric squares of modular Abelian varieties ». Thesis, University of Sheffield, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.500087.
Texte intégral
7
Giangreco, Maidana Alejandro José. « Cyclic abelian varieties over finite fields ». Thesis, Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0316.
Texte intégralRésumé :
L'ensemble A(k) des points rationnels d'une variété abélienne A définie sur un corps fini k forme un groupe abélien fini. Ce groupe convient pour des multiples applications, et sa structure est très importante. Connaître les possibles structures de groupe des A(k) et quelques statistiques est donc fondamental. Dans cette thèse, on s'intéresse aux "variétés cycliques", i.e. variétés abéliennes définies sur des corps finis avec groupe des points rationnels cyclique.Les isogénies nous donnent une classification plus grossière que celle donnée par les classes d'isomorphisme des variétés abéliennes, mais elles offrent un outil très puissant en géométrie algébrique. Chaque classe d'isogénie est déterminée par son polynôme de Weil. On donne un critère pour caractériser les "classes d'isogénies cycliques", i.e. classes d'isogénies de variétés abéliennes définies sur des corps finis qui contiennent seulement des variétés cycliques. Ce critère est basé sur le polynôme de Weil de la classe d'isogénie.À partir de cela, on donne des bornes de la proportion de classes d'isogénies cycliques parmi certaines familles de classes d'isogénies paramétrées par ses polynômes de Weil.On donne aussi la proportion de classes d'isogénies cycliques "locaux" parmi les classes d'isogénie définies sur des corps finis mathbb{F}_q avec q éléments, quand q tend à l'infiniThe set A(k) of rational points of an abelian variety A defined over a finite field k forms a finite abelian group. This group is suitable for multiple applications, and its structure is very important. Knowing the possible group structures of A(k) and some statistics is then fundamental. In this thesis, we focus our interest in "cyclic varieties", i.e. abelian varieties defined over finite fields with cyclic group of rational points. Isogenies give us a coarser classification than that given by the isomorphism classes of abelian varieties, but they provide a powerful tool in algebraic geometry. Every isogeny class is determined by its Weil polynomial. We give a criterion to characterize "cyclic isogeny classes", i.e. isogeny classes of abelian varieties defined over finite fields containing only cyclic varieties. This criterion is based on the Weil polynomial of the isogeny class.From this, we give bounds on the fractions of cyclic isogeny classes among certain families of isogeny classes parameterized by their Weil polynomials.Also we give the proportion of "local"-cyclic isogeny classes among the isogeny classes defined over the finite field mathbb{F}_q with q elements, when q tends to infinity
8
Joyce, Adam Jack. « The Manin constants of modular abelian varieties ». Thesis, Imperial College London, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.440468.
Texte intégral
9
Lahoz, Vilalta Marti. « Theta-duality in abelian varieties and the bicanonical map of irregular varieties ». Doctoral thesis, Universitat Politècnica de Catalunya, 2010. http://hdl.handle.net/10803/77898.
Texte intégralRésumé :
The first goal of this Thesis is to contribute to the study of principally polarized abelian varieties (ppav), especially to the
Schottky and the Torelli problems. Ppav admit a duality theory analogous to that of projective spaces, where the role
played by hyperplanes in projective spaces is played by divisors representing the principal polarization. Thus, given a
subvariety Y of a ppav, we can define its thetadual
T(Y) as the set of divisors representing the principal polarization
that contain this subvariety. This set admits a natural schematic structure (as defined by Pareschi and Popa).
Jacobian and Prym varieties are classical examples of ppav constructed from curves. Besides, they are interesting
because some properties of the curves involved in their construction are reflected in their geometry or in the geometry
of some special subvarieties. For example, in the case of Jacobians we have the BrillNoether
loci Wd ( W1 corresponds
to the AbelJacobi
curve) and in the case of Pryms we have the AbelPrym
curve C.
In chapter III, we study the schematic structure of the thetadual
of the BrillNoether
loci Wd and the AbelPrym
curve.
In the first case, we obtain with different methods, the result of Pareschi and Popa T(Wd)= Wgd1.
In the case of the
AbelPrym
curve C, we get that T(C)=V², where V² is the second PrymBrillNoether
locus with the schematic structure
defined by Welters.
Pareschi and Popa have proved a result for ppavs analogous to the Castelnuovo Lemma for projective spaces. That is,
if (A,Θ) is a ppav of dimension g, then g+2 distinct points in general position with respect to Θ, but in special position
with respect to 2Θ, have to be contained in a curve of minimal degree in A, i.e. an AbelJacobi
curve. In particular,
they obtain a Schottky result because A has to be a Jacobian variety and a Torelli result, because the curve is the
intersection of all the divisors in |2Θ| that contain the g+2 points. In chapter IV, as Eisenbud and Harris have done in
the projective Castelnuovo Lemma, we extend this result to possibly nonreduced
finite schemes.
The second goal of this thesis is the study of varieties of general type. Almost by definition, pluricanonical maps are the
essential tool to study them. One of the main problems in this area is to find geometric or numerical conditions to
guarantee that the mth
pluricanonical map (for low m) induces a birational equivalence with its image.
The classification of surfaces whose bicanonical map is nonbirational
has attracted considerable interest among
algebraic geometers. In chapter V, we give a sufficient numerical condition for the birationality of the bicanonical map of
irregular varieties of arbitrary dimension.
We also prove that, if X is a primitive variety, then it only admits very special fibrations to other irregular varieties. For
primitive varieties we get that the following are equivalent:
X
is birational to a divisor Θ in an indecomposable ppav,
the
irregularity q(X) > dim X and the bicanonical map is nonbirational.
When X is a primitive variety of general type and q(X) = dim X we prove, under certain conditions over the Stein
factorization of the Albanese map, that the only possibility for the bicanonical map being nonbirational
is that X is a
double cover branched along a divisor in |2Θ|. These results extend to arbitrary dimension, wellknown
theorems in the
case of surfaces and curves.El primer objectiu d'aquesta tesi és contribuir a l'estudi de les varietats abelianes principalment polaritzades (vapp), especialment als problemes de Schottky i Torelli. Les vapp admeten una teoria de dualitat anàloga a la dualitat dels espais projectius, on el paper que juguen els hiperplans de l'espai projectiu és substituït pels divisors que representen la polarització principal. Així doncs, donada una subvarietat Y d'una vapp, podem definir el seu thetadual T(Y) com el conjunt dels divisors que representen la polarització principal i contenen aquesta subvarietat. Aquest conjunt admet una estructura esquemàtica natural (tal i com la defineixen Pareschi i Popa). Les varietats Jacobianes i de Prym són exemples clàssics de vapp construïdes a partir de corbes. A més, són interessants perquè certes propietats de les corbes involucrades es veuen reflectides en elles o en algunes subvarietats especials. Per exemple, en el cas de les Jacobianes tenim els llocs de BrillNoether Wd ( W1 correspon a la corba d'AbelJacobi) i en el cas de les Pryms tenim la corba d'AbelPrym C. Al capítol III de la tesi s'estudia l'estructura esquemàtica del thetadual dels llocs de BrillNoether Wd i de la corba d'AbelPrym. En el primer cas, es reobté amb uns altres mètodes, el resultat de Pareschi i Popa T(Wd)= Wgd1. En el cas de la corba d'AbelPrym C, s'obté que T(C)=V², onV² és el segon lloc de PrymBrillNoether amb l'estructura esquemàtica definida per Welters. Pareschi i Popa han demostrat un resultat anàleg per les vapp al Lemma de Castelnuovo pels espais projectius. És a dir, si (A,Θ) és una vapp de dimensió g, aleshores g+2 punts en posició general respecte Θ, però en posició especial respecte 2Θ, han d'estar continguts en una corba de grau minimal a A, i.e. una corba d'AbelJacobi. En particular, s'obté un resultat de Schottky ja que A ha de ser una Jacobiana i un resultat de Torelli, ja que la corba és la intersecció de tots els divisors de |2Θ| que contenen els g+2 punts. Al capítol IV, tal i com Eisenbud i Harris van fer en el cas projectiu, s'estén aquest resultat a esquemes finits possiblement no reduïts. El segon objectiu d'aquesta tesi és contribuir a l'estudi de les varietats de tipus general. Pràcticament per definició, les aplicacions pluricanòniques són essencials pel seu estudi. Un dels problemes principals de l'àrea és donar condicions geomètriques o numèriques per assegurar que la mèsima aplicació pluricanònica (per m baix) indueix una equivalència biracional amb la imatge. La classificació de les superfícies que tenen l'aplicació bicanònica no biracional ha atret l'atenció de molts geòmetres algebraics. Al capítol V, es dóna un criteri numèric suficient per assegurar la biracionalitat de l'aplicació bicanònica de les varietats irregulars de dimensió arbitrària. També es demostra que si X és una varietat primitiva, aleshores només admet fibracions molt especials a altres varietats irregulars. Per aquestes varietats s'obté que és equivalent que X sigui biracional a un divisor Θ en una vapp indescomponible, a què la irregularitat q(X) > dim X i l'aplicació bicanònica sigui no biracional. Quan X és una varietat primitiva de tipus general i q(X) = dim X es demostra sota certes condicions de la descomposició de Stein del morfisme d'Albanese, que l'única possibilitat per tal que l'aplicació bicanònica sigui no biracional és que X sigui un recobriment doble sobre una vapp ramificat al llarg d'un divisor a |2Θ|. Aquest resultats estenen a dimensió arbitrària, teoremes ben coneguts en el cas de superfícies i corbes.
10
Borowka, Pawel. « Non-simple abelian varieties and (1,3) Theta divisors ». Thesis, University of Bath, 2012. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.564009.
Texte intégralRésumé :
This thesis studies non-simple Jacobians and non-simple abelian varieties. The moti- vation of the study is a construction which gives a distinguished genus 4 curve in the linear system of a (1, 3)-polarised surface. The main theorem characterises such curves as hyperelliptic genus 4 curves whose Jacobian contains a (1, 3)-polarised surface. This leads to investigating the locus of non-simple principally polarised abelian g- folds. The main theorem of this part shows that the irreducible components of this locus are Is~, defined as the locus of principally polarised g-folds having an abelian subvariety with induced polarisation of type d. = (d1, ... , dk), where k ≤ g/2 Moreover, there are theorems which characterise the Jacobians of curves that are etale double covers or double covers branched in two points. There is also a detailed computation showing that, for p > 1 an odd number, the hyperelliptic locus meets IS4(l,p) transversely in the Siegel upper half space
11
Marseglia, Stefano. « Isomorphism classes of abelian varieties over finite fields ». Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-130316.
Texte intégralRésumé :
Deligne and Howe described polarized abelian varieties over finite fields in terms of finitely generated free Z-modules satisfying a list of easy to state axioms. In this thesis we address the problem of developing an effective algorithm to compute isomorphism classes of (principally) polarized abelian varieties over a finite field, together with their automorphism groups. Performing such computations requires the knowledge of the ideal classes (both invertible and non-invertible) of certain orders in number fields. Hence we describe a method to compute the ideal class monoid of an order and we produce concrete computations in dimension 2, 3 and 4.
12
Reid, Fergus. « Varieties for modules of small dimension ». Thesis, University of Aberdeen, 2013. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=203509.
Texte intégralRésumé :
This thesis focuses on the subject of varieties for modules for elementary abelian p-groups. Given a homogeneous polynomial over an algebraically closed field of char- acteristic 2 we will give constructions for modules of small dimension having that polynomial as variety. This is similar to an earlier construction given by Jon Carlson but our modules will in general be of considerably smaller dimension. We also investigate the connection between the variety of a module and its Loewy length. We show that working over an algebraically closed field of characteristic 2 with modules of Loewy length 2 allows us to find modules with any hypersurface as their variety. On the other hand we also demonstrate that in odd characteristic p, with modules of Loewy length p, the only possible varieties are finite unions of linear hypersurfaces.
13
Yan, Fengsheng. « Tate property and isogeny estimate for semi-abelian varieties / ». [S.l.] : [s.n.], 1994. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=10798.
Texte intégral
14
Bisatt, Matthew David. « Root numbers of abelian varieties and their Galois representations ». Thesis, King's College London (University of London), 2018. https://kclpure.kcl.ac.uk/portal/en/theses/root-numbers-of-abelian-varieties-and-their-galois-representations(c5072bf1-5719-46aa-9177-bb7e33bf6d40).html.
Texte intégralRésumé :
The Birch–Swinnerton-Dyer conjecture is one of the most famous open problems in modern number theory; this is reflected by its inclusion in the Clay Mathematics Insti-tute million-dollar problems. The conjecture asserts that the rank of an abelian variety can be recovered from its L-function. In this thesis, we examine some of the conse-quences that are predicted by the Birch–Swinnerton-Dyer conjecture with the aid of Galois representations. The first consequence is the parity conjecture: this states that the expected sign of the functional equation, known as the root number, should control the rank modulo 2; i.e. whether it is odd or even. We derive explicit formulae for the root number in terms of Jacobi symbols, as well as their generalisation to twisted root numbers. This is a very useful tool for numerically verifying the Birch–Swinnerton-Dyer conjecture and we give worked examples of computing the root number associated to the Jacobian of a hyperelliptic curve. As an application, we give sufficient criteria for an abelian variety such that every quadratic twist has infinitely many rational points, assuming the parity conjecture. If one combines the Birch–Swinnerton-Dyer conjecture with a conjecture of Deligne–Gross, then one can obtain a generalised version concerning twisted L-functions. One can then use tools from representation theory to give predictions about: orders of van-ishing of the twisted L-functions; the corank of the ∞-Selmer group; and the existence of certain extensions where high orders of vanishing of the (untwisted) L-function al-ways occur, independently of the abelian variety. Finally, we investigate the classical problem of distinguishing conjugacy classes of Frobenius elements in images of Galois representations. Using elliptic curves as the source of our Galois representations, we present two algorithms to distinguish between conjugacy classes of matrix groups in a small number of situations.
15
Banwait, Barinder S. « On some local to global phenomena for abelian varieties ». Thesis, University of Warwick, 2013. http://wrap.warwick.ac.uk/58400/.
Texte intégral
16
Seveso, M. A. « Stark-Heegner points and Selmer groups of abelian varieties ». Doctoral thesis, Università degli Studi di Milano, 2009. http://hdl.handle.net/2434/151050.
Texte intégral
17
Malabre, François. « Eigenvalue varieties of abelian trees of groups and link-manifolds ». Doctoral thesis, Universitat Autònoma de Barcelona, 2015. http://hdl.handle.net/10803/308323.
Texte intégralRésumé :
L’A-polinomi d’un nus en S3 és un poliomi de dues variables obtingut projectant la
varietat de SL2C-caràcters de l’exterior del nus sobre la varietat de caràcters del grup perifèric.
Distingeix el nus trivial i detecta alguns pendents a la vora de superfícies essencials
dels exteriors de nus.
El concepte de A-polinomi va ser generalitzat a les 3-varietats amb vores tòriques no
connexes; una 3-varietat M amb n tors de vora produeix un sub-espai algebraic E(M) de
C2n anomenat varietat de valors propis de M. Té dimensió maximal n i E(M) també
detecta sistemes de pendents a les vores de superfícies essencials en M.
La varietat de valors propis de M sempre conté una part Ered(M), de dimensió maximal,
produïda pels caràcters reductibles. Si M és hiperbòlica, E(M) conté una altra
component de dimensió maximal; saber quines altres 3-varietats compleixen això encara
és una pregunta oberta.
En aquesta tesi, estudiem aquest assumpte per dues famílies de 3-varietats amb vores
tòriques i, amb dues tècniques diferents, aportem una resposta positiva en ambdós casos.
Primerament, estudiem els enllaços Brunnians en S3, enllaços per els quals tot subenllaç
estricte és trivial. Algunes propietats d’aquests enllaços i llur estabilitat sota alguns
ompliments de Dehn permet mostrar que, si M és l’exterior d’un enllaç Brunnià no trivial
i diferent de l’enllaç de Hopf, E(M) conté una component de dimensió maximal diferent
de Ered(M). Aquest resultat s’obté generalitzant la tècnica prèviament utilitzada per els
nusos en S3 fent servir el teorema de Kronheimer-Mrowka.
Per altre banda, considerem una família de varietats-enllaç, varietats obtingudes com
exteriors d’enllaços en esferes d’homologia entera. Les varietats-enllaç tenen sistemes
perifèrics estàndards de meridans i longituds i són estables per splicing, l’enganxament de
dues varietats-enllaç al llarg de tors perifèrics, identificant el meridià de cada costat amb la
longitud oposada. El splicing indueix una noció de descomposició tòrica per les varietatsenllaç
i anomenem grafejades les varietats-enllaç que admeten una descomposició tòrica
per la qual totes les peces són fibrades de Seifert. Mostrem que, excloent els casos trivials,
totes les varietats-enllaç grafejades produeixen una altre component de dimensió maximal
en les seves varietats de valors propis.
Per aquesta segona demostració, presentem una nova generalització de la varietat de
valors propis, que també té en compte tors interns, i que presentem en el context més general
d’arbres abelians de grups. Un arbre de grup és abelià quan tots els grups de arestes
són commutatius; en aquest cas, definim la varietat de valors propis d’un arbre abelià de
grup, una varietat algebraica compatible amb dues operacions naturales sobre els arbres: la
fusió i la contracció. Això permet estudiar la varietat de valors propis d’una varietat-enllaç
mitjançant les varietats de valors propis de les seves descomposicions tòriques. Combinant resultats generals sobre varietats de valors propis d’arbres abelians de grup i les descripcions
combinatòries de les varietats-enllaç grafejades, construïm components de dimensió
maximal en les seves varietats de valors propis.Le A-polynôme d’un noeud dans S3 est un polynôme à deux variables obtenu en projetant la variété des SL2C-caractères de l’extérieur du noeud sur la variété de caractères du groupe périphérique. Il distingue le noeud trivial et détecte certaines pentes aux bords de surfaces essentielles des extérieurs de noeud. La notion de A-polynôme a été généralisée aux 3-variétés à bord torique non connexe ; une 3-variétéM bordée par n tores produit un sous-espace algebrique E(M) de C2n appelé variété des valeurs propres deM. Sa dimension est inférieure ou égale à n et E(M) détecte également des systèmes de pentes aux bords de surfaces essentielles dans M. La variété des valeurs propres de M contient toujours un sous-ensemble Ered(M) produit par les caractères réductibles, et de dimension maximale. Si M est hyperbolique, E(M) contient une autre composante de dimension maximale ; pour quelles autres 3- variétes est-ce le cas reste une question ouverte. Dans cette thèse, nous étudions cette question pour deux familles de 3-variétés à bords toriques et, via deux techniques distinctes, apportons une réponse positive dans ces deux cas. Dans un premier temps, nous étudions les entrelacs Brunniens dans S3, entrelacs pour lesquels tout sous-entrelacs strict est trivial. Certaines propriétés de ces entrelacs, et leur stabilité par certains remplissages de Dehn nous permettent de prouver que, siM est l’extérieur d’un entrelacs Brunnien non trivial et différent de l’entrelacs de Hopf, E(M) contient une composante de dimension maximale différente de Ered(M). Ce résultat est obtenu en généralisant la technique préalablement utilisée pour les noeuds dans S3 grâce au théorème de Kronheimer-Mrowka. D’autre part, nous considérons une famille de variétés-entrelacs, variétés obtenues comme extérieurs d’entrelacs dans des sphères d’homologie entière. Les variétés-entrelacs possèdent des systèmes périphériques standard de méridiens et longitudes et sont stables par splicing, le recollement de deux variétés-entrelacs le long de tores périphériques en identifiant le méridien de chaque coté avec la longitude opposée. Ceci induit une notion de décomposition torique de variété-entrelacs et une telle variété est dite graphée si elle admet une décomposition torique où toutes les pièces sont fibrées de Seifert. Nous montrons que, mis-à-part les cas triviaux, toutes les variétés-entrelacs graphées produisent une autre composante de dimension maximale dans leur variétés des valeurs propres. Pour cette seconde preuve, nous présentons une nouvelle généralisation de la variété des valeurs propres, qui prend également en compte les tores intérieurs, que nous introduisons dans le contexte plus général des arbres abéliens de groupes. Un arbre de groupe est appelé abélien si tous les groupes d’arête sont commutatifs ; dans ce cas, nous définissions la variété des valeurs propres d’un arbre abélien de groupe, une variété algébrique compatible avec deux opérations naturelles sur les arbres : la fusion et la contraction. Ceci permet d’étudier la variété des valeurs propres d’une variété-entrelacs à travers les variétés des valeurs propres de ses décompositions toriques. En combinant des résultats généraux sur les variétés des valeurs propres d’arbres abéliens de groupe et les descriptions combinatoires des variétés-entrelacs graphées, nous contruisons des composantes de dimension maximale dans leur variétés des valeur propres.
The A-polynomial of a knot in S3 is a two variable polynomial obtained by projecting the SL2C-character variety of the knot-group to the character variety of its peripheral subgroup. It distinguishes the unknot and detects some boundary slopes of essential surfaces in knot exteriors. The notion of A-polynomial has been generalized to 3-manifolds with non-connected toric boundaries; ifM is a 3-manifold bounded by n tori, this produces an algebraic subset E(M) of C2n called the eigenvalue variety of M. It has dimension at most n and still detects systems of boundary slopes of surfaces in M. The eigenvalue variety of M always contains a part Ered(M) arising from reducible characters and with maximal dimension. If M is hyperbolic, E(M) contains another topdimensional component; for which 3-manifolds is this true remains an open question. In this thesis, this matter is studied for two families of 3-manifolds with toric boundaries and, via two very different technics, we provide a positive answer for both cases. On the one hand, we study Brunnian links in S3, links in the standard 3-sphere for which any strict sublink is trivial. Using special properties of these links and stability under certain Dehn fillings we prove that, if M is the exterior of a Brunnian link different from the trivial link or the Hopf link, then E(M) admits a top-dimensional component different from Ered(M). This is achieved generalizing the technic applied to knots in S3, using Kronheimer-Mrowka theorem. On the other hand, we consider a family of link-manifolds, exteriors of links in integerhomology spheres. Link-manifolds are equipped with standard peripheral systems of meridians and longitudes and are stable under splicing, gluing two link-manifolds along respective boundary components, identifying the meridian of each side to the longitude of the other. This yields a well-defined notion of torus decomposition and a link-manifold is called a graph link-manifold if there exists such a decomposition for which each piece is Seifert-fibred. Discarding trivial cases, we prove that all graph link-manifolds produce another top-dimensional component in their eigenvalue variety. For this second proof, we propose a further generalization of the eigenvalue variety that also takes into account internal tori and this is introduced in the broader context of abelian trees of groups. A tree of group is called abelian if all its edge groups are commutative; in that case, we define the eigenvalue variety of an abelian tree of groups, an algebraic variety compatible with two natural operations on trees: merging and contraction. This enables to study the eigenvalue variety of a link-manifold through the eigenvalue varieties of its torus splittings. Combining general results on eigenvalue varieties of abelian trees of groups with combinatorial descriptions of graph link-manifolds, we construct top-dimensional components in their eigenvalue varieties.
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Bradford, Jeremy. « Commutative endomorphism rings of simple abelian varieties over finite fields ». Thesis, University of Maryland, College Park, 2013. http://pqdtopen.proquest.com/#viewpdf?dispub=3557641.
Texte intégralRésumé :
In this thesis we look at simple abelian varieties defined over a finite field k = [special characters omitted]pn with Endk( A) commutative. We derive a formula that connects the p -rank r(A) with the splitting behavior of p in E = [special characters omitted](π), where π is a root of the characteristic polynomial of the Frobenius endomorphism. We show how this formula can be used to explicitly list all possible splitting behaviors of p in [special characters omitted]E, and we do so for abelian varieties of dimension less than or equal to four defined over [special characters omitted]p. We then look for when p divides [[special characters omitted]E : [special characters omitted][π, π]]. This allows us to prove that the endomorphism ring of an absolutely simple abelian surface is maximal at p when p ≥ 3. We also derive a condition that guarantees that p divides [[special characters omitted]E: [special characters omitted][π, π]]. Last, we explicitly describe the structure of some intermediate subrings of p-power index between [special characters omitted][π, π] and [special characters omitted]E when A is an abelian 3-fold with r(A) = 1.
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Laface, Roberto [Verfasser]. « Picard numbers of abelian varieties and related arithmetic / Roberto Laface ». Hannover : Technische Informationsbibliothek (TIB), 2017. http://d-nb.info/1131186710/34.
Texte intégral
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Nicole, Marc-Hubert. « Superspecial abelian varieties, theta series and the Jacquet-Langlands correspondence ». Thesis, McGill University, 2005. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=85946.
Texte intégralRésumé :
Let Ei, i = 1, ..., n, be all the supersingular elliptic curves over Fp up to isomorphism. The modules of isogenies Hom(Ei, Ej), equipped with the degree map, are quadratic modules that give rise to theta series of level p. The space of modular forms of weight two for Gamma0(p) is thus spanned by the theta series coming from supersingular elliptic curves in this fashion. We generalize this classical result to Hilbert modular forms by showing that for totally real fields L of narrow class number one, the space of Hilbert modular newforms of parallel weight 2 for Gamma0( p), p unramified, is spanned by theta series coming from quadratic modules HomOL , (Ai, Aj), where Ai, Aj range across all superspecial abelian varieties with real multiplication by OL . We also provide a version of this theorem in the more delicate case where p is totally ramified in OL , building on the classification of superspecial crystals following from the generalization of Manin's Habilitationschrift that we present in the first Chapter.
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Lombardo, Davide. « Galois representations and Mumford-Tate groups attached to abelian varieties ». Thesis, Université Paris-Saclay (ComUE), 2015. http://www.theses.fr/2015SACLS196/document.
Texte intégralRésumé :
Soient $K$ un corps de nombres et $A$ une variété abélienne sur $K$ dont nous notons $g$ la dimension. Pour tout premier $ell$, le module de Tate $ell$-adique de $A$ nous fournit une représentation $ell$-adique du groupe de Galois absolu de $K$, et c'est à l'image de ces représentations galoisiennes que l'on s'intéresse dans cette thèse.Pour de nombreuses classes de variétés abéliennes on possède une description de ces images à une erreur finie près : le premier but de ce travail est de quantifier explicitement cette erreur dans plusieurs cas différents. On parvient à résoudre complètement le problème pour une courbe elliptique sans multiplication complexe, ou plus généralement pour un produit de telles courbes elliptiques, et pour toute variété abélienne géométriquement simple admettant multiplication complexe. Pour d'autres classes de variétés abéliennes $A/K$ on obtient seulement une description de l'image de Galois pour tout premier $ell$ plus grand qu'une certaine borne (que l'on calcule explicitement, et qui est polynomiale en le degré de $K$ et en la hauteur de Faltings de $A$) : nous prouvons de tels résultats pour toute surface abélienne semistable et géométriquement simple et pour les variétés dites "de type $operatorname{GL}_2$''. On montre également un résultat semblable, mais un peu affaibli, pour de nombreuses variétés abéliennes de dimension impaire dont l'anneau des endomorphismes est réduit à $mathbb{Z}$.On s'intéresse ensuite à l'action de Galois sur des variétés abéliennes non simples, et on donne des conditions suffisantes pour que les représentations galoisiennes qui leur sont associées se décomposent elles-mêmes en produit. Finalement on étudie l'intersection entre les extensions cyclotomiques d'un corps de nombres $K$ et les corps engendrés par les points de torsion d'une variété abélienne sur $K$, et on établit des propriétés d'uniformité des degrés de ces intersectionsLet $K$ be a number field and $A$ be a $g$-dimensional abelian variety over $K$. For every prime $ell$, the $ell$-adic Tate module of $A$ gives rise to an $ell$-adic representation of the absolute Galois group of $K$; in this thesis we set out to study the images of the Galois representations arising in this way.For various classes of abelian varieties a description of these images is known up to finite error, and the first aim of this work is to explicitly quantify this error for a number of different cases. We provide a complete solution for the case of elliptic curves without complex multiplication (and more generally for products thereof) and for geometrically simple abelian varieties of CM type. For other classes of abelian varieties we can only describe the Galois image when the prime $ell$ is above a certain bound (which we compute explicitly in terms of $A$, and which is polynomial in $[K:mathbb{Q}]$ and in the Faltings height of $A$): we obtain such results for geometrically simple, semistable abelian surfaces and for "$operatorname{GL}_2$-type" varieties. We also prove similar (but slightly weaker) results for many abelian varieties of odd dimension with trivial endomorphism algebra.We then consider the Galois action on non-simple abelian varieties, and we give sufficient conditions for the associated Galois representations to decompose as a product.Finally, we investigate the structure of the intersection between the cyclotomic extensions of a number field $K$ and the fields generated by the torsion points of an abelian variety over $K$, proving a uniformity property for the degrees of such intersections
22
Tommasi, Orsola [Verfasser]. « Cohomological aspects of moduli of curves and Abelian varieties / Orsola Tommasi ». Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2013. http://d-nb.info/1041652380/34.
Texte intégral
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Hao, Yun [Verfasser]. « A Simpson Correspondence for Abelian Varieties in Positive Characteristic / Yun Hao ». Berlin : Freie Universität Berlin, 2019. http://d-nb.info/1192755626/34.
Texte intégral
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Grant, David R. MD FRCSC. « Theta functions and division points on Abelian varieties of dimension two ». Thesis, Massachusetts Institute of Technology, 1985. http://hdl.handle.net/1721.1/115469.
Texte intégralRésumé :
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1985.MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE.
Bibliography: leaves 123-124.
by David R. Grant.
Ph.D.
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Kadets, Borys. « Arboreal representations, sectional monodromy groups, and abelian varieties over finite fields ». Thesis, Massachusetts Institute of Technology, 2020. https://hdl.handle.net/1721.1/126927.
Texte intégralRésumé :
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 93-97).
This thesis consists of three independent parts. The first part studies arboreal representations of Galois groups - an arithmetic dynamics analogue of Tate modules - and proves some large image results, in particular confirming a conjecture of Odoni. Given a field K, a separable polynomial [mathematical expression], and an element [mathematical expression], the full backward orbit [mathematical expression] has a natural action of the Galois group [mathematical expression]. For a fixed [mathematical expression] with [mathematical expression] and for most choices of t, the orbit [mathematical expression] has the structure of complete rooted [mathematical expression]. The Galois action on [mathematical expression] thus defines a homomorphism [mathematical expression]. The map [mathematical expression] is the arboreal representation attached to f and t.
In analogy with Serre's open image theorem, one expects [mathematical expression] to hold for most f, t, but until very recently for most degrees d not a single example of a degree d polynomial [mathematical expression] with surjective [mathematical expression],t was known. Among other results, we construct such examples in all sufficiently large even degrees. The second part concerns monodromy of hyperplane section of curves. Given a geometrically integral proper curve [mathematical expression], consider the generic hyperplane [mathematical expression]. The intersection [mathematical expression] is the spectrum of a finite separable field extension [mathematical expression] of degree [mathematical expression]. The Galois group [mathematical expression] is known as the sectional monodromy group of X. When char K = 0, the group [mathematical expression] equals [mathematical expression] for all curves X.
This result has numerous applications in algebraic geometry, in particular to the degree-genus problem. However, when char K > 0, the sectional monodromy groups can be smaller. We classify all nonstrange nondegenerate curves [mathematical expression], for [mathematical expression] such that [mathematical expression]. Using similar methods we also completely classify Galois group of generic trinomials, a problem studied previously by Abhyankar, Cohen, Smith, and Uchida. In part three of the thesis we derive bounds for the number of [mathematical expression]-points on simple abelian varieties over finite fields; these improve upon the Weil bounds. For example, when q = 3, 4 the Weil bound gives [ .. ] for all abelian varieties A. We prove that [mathematical expression], [mathematical expression] hold for all but finitely many simple abelian varieties A (with an explicit list of exceptions).
by Borys Kadets.
Ph. D.
Ph.D. Massachusetts Institute of Technology, Department of Mathematics
26
Frejlich, Pedro. « Transformada de Nahm de fibrados de Higgs sobre superficies de Riemann de genero ao menos dois ». [s.n.], 2006. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306023.
Texte intégralRésumé :
Orientador: Marcos Benevuto JardimDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
Made available in DSpace on 2018-08-08T04:40:41Z (GMT). No. of bitstreams: 1 Frejlich_Pedro_M.pdf: 852390 bytes, checksum: bcde0cd89c0bcfe3b2515b5cfe48e528 (MD5) Previous issue date: 2007
Resumo: Construímos a transformada de Nahm de um fibrado de Higgs estável de grau nulo sobre uma superfície de Riemann de gênero pelo menos 2. Para tanto, empregamos a Teoria do Índice de Atiyah-Singer e um vanishing theorem que segue da hipótese de estabilidade do fibrado. O principal resultado é que o fibrado transformado é hiperholomorfo e sem fatores planos. Desse modo não só recuperamos os resultados algébricos de [7] e os de [12] para o cos q = 0 como também provamos uma descrição mais detalhada da estrutura geométrica da transformada ¿ o que, aliada às técnicas de [10] sugere que ela possa ser invertida. Palavras-chave: Superfícies de Riemann, Fibrados Estáveis, Teoria do Índice, Transformada de Nahm, Transformada de Fourier-Mukai, Variedades Hiper-Kähler, Variedades Abelianas, Conexões Hiperholomorfas
Abstract: We construct the Nahm transform of a stable, degree-zero Higgs bundle on a Riemann surface of genus at least 2. Atiyah-Singer¿s index theorem is the basic tool employed, along with a vanishing theorem which is due to the stability hypothesis. Our main result is that the transformed bundle is hyperholomorphic and without flat factors. This not only recovers the algebraic results of [7] and that of [12] for the cos q = 0, but also provides a more detailed description of the geometric structure of the transformed bundle. Such results suggest that this Nahm transform can be inverted, cf. [10]. Key-words:Riemann surfaces, Stable bundles, Index Theory, Nahm Transform, Fourier-Mukai Transform, Hyperk¨ahler manifolds, Abelian varieties, Hyperholomorphic connections
Mestrado
Geometria Diferencial/Geometria algebrica
Mestre em Matemática
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GROSSELLI, GIAN PAOLO. « Shimura varieties in the Prym loci of Galois covers ». Doctoral thesis, Università degli Studi di Milano-Bicocca, 2022. http://hdl.handle.net/10281/356638.
Texte intégralRésumé :
In questa tesi si studiano le sottovarietà di Shimura negli spazi di moduli delle varietà abeliane complesse. Queste sottovarietà derivano da famiglie di rivestimenti di Galois compatibili con un'azione di gruppo fissata sulla curva base tale che il quoziente della curva base per il gruppo è isomorfo alla retta proiettiva. Si da un criterio affinché l'immagine di queste famiglie tramite la mappa di Prym sia una sottovarietà speciale e, sfruttando il computer, si costruiscono numerose sottovarietà di Shimura contenute nei luoghi di Prym.In this thesis we study Shimura subvarieties in the moduli space of complex abelian varieties. These subvarieties arise from families of Galois covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to the projective line. We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, we build several Shimura subvarieties contained in the Prym loci.
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Mendes, da Costa David John. « Topics in the arithmetic of abelian varieties / David John Mendes da Costa ». Thesis, University of Bristol, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.617932.
Texte intégralRésumé :
In this thesis we study at two rather different problems within the arithmetic of abelian varieties. In the first case, we consider the problem of getting uniform bounds for the number of integer points which an elliptic curve defined over Q can obtain within a square box with sides of length N. In particular, our aim is to break the bounds of Bombieri and Pila which in this case give Oe.(N1/3+e). We accomplish this for a large family of elliptic curves using a variety of techniques including repulsion of integer points via Gap Principles, the theory of heights and the Large Sieve. As an application, we prove a result concerning the number of rational points of bounded height on a del Pezzo surface of degree L The second part of the project considers the behaviour of ranks of abelian varieties which arise as Jacobians of curves defined over a number field K. In particular, we ask for which values d are there infinitely many degree d extensions L/K such that the rank of the Jacobian increases as we base change from K to L. In the case of elliptic curves we show that this occurs for every d> 1 and for general Jacobians we show that this holds for all sufficiently large d.
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Tsui, Ho-yu. « Families of polarized abelian varieties and a construction of Kähler metrics of negative holomorphic bisectional curvature on Kodaira surfaces ». Click to view the E-thesis via HKUTO, 2006. http://sunzi.lib.hku.hk/hkuto/record/B37053760.
Texte intégral
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Tsui, Ho-yu, et 徐浩宇. « Families of polarized abelian varieties and a construction of Kähler metrics of negative holomorphic bisectional curvature on Kodairasurfaces ». Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2006. http://hub.hku.hk/bib/B37053760.
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Wilson, Ley Catherine. « Q-Curves with Complex Multiplication ». Thesis, The University of Sydney, 2010. http://hdl.handle.net/2123/6259.
Texte intégralRésumé :
The Hecke character of an abelian variety A/F is an isogeny invariant and the Galois action is such that A is isogenous to its Galois conjugate A^σ if and only if the corresponding Hecke character is fixed by σ. The quadratic twist of A by an extension L/F corresponds to multiplication of the associated Hecke characters. This leads us to investigate the Galois groups of families of quadratic extensions L/F with restricted ramification which are normal over a given subfield k of F. Our most detailed results are given for the case where k is the field of rational numbers and F is a field of definition for an elliptic curve with complex multiplication by K. In this case the groups which occur as Gal(L/K) are closely related to the 4-torsion of the class group of K. We analyze the structure of the local unit groups of quadratic fields to find conditions for the existence of curves with good reduction everywhere. After discussing the question of finding models for curves of a given Hecke character, we use twists by 3-torsion points to give an algorithm for constructing models of curves with known Hecke character and good reduction outside 3. The endomorphism algebra of the Weil restriction of an abelian variety A may be determined from the Grössencharacter of A. We describe the computation of these algebras and give examples in which A has dimension 1 or 2 and its Weil restriction has simple abelian subvarieties of dimension ranging between 2 and 24.
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Wilson, Ley Catherine. « Q-Curves with Complex Multiplication ». University of Sydney, 2010. http://hdl.handle.net/2123/6259.
Texte intégralRésumé :
Doctor of PhilosophyThe Hecke character of an abelian variety A/F is an isogeny invariant and the Galois action is such that A is isogenous to its Galois conjugate A^σ if and only if the corresponding Hecke character is fixed by σ. The quadratic twist of A by an extension L/F corresponds to multiplication of the associated Hecke characters. This leads us to investigate the Galois groups of families of quadratic extensions L/F with restricted ramification which are normal over a given subfield k of F. Our most detailed results are given for the case where k is the field of rational numbers and F is a field of definition for an elliptic curve with complex multiplication by K. In this case the groups which occur as Gal(L/K) are closely related to the 4-torsion of the class group of K. We analyze the structure of the local unit groups of quadratic fields to find conditions for the existence of curves with good reduction everywhere. After discussing the question of finding models for curves of a given Hecke character, we use twists by 3-torsion points to give an algorithm for constructing models of curves with known Hecke character and good reduction outside 3. The endomorphism algebra of the Weil restriction of an abelian variety A may be determined from the Grössencharacter of A. We describe the computation of these algebras and give examples in which A has dimension 1 or 2 and its Weil restriction has simple abelian subvarieties of dimension ranging between 2 and 24.
33
Krämer, Thomas [Verfasser], et Rainer [Akademischer Betreuer] Weissauer. « Tannakian Categories of Perverse Sheaves on Abelian Varieties / Thomas Krämer ; Betreuer : Rainer Weissauer ». Heidelberg : Universitätsbibliothek Heidelberg, 2013. http://d-nb.info/1177249111/34.
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34
Cesarano, Luca [Verfasser], et Fabrizio [Akademischer Betreuer] Catanese. « Canonical Surfaces and Hypersurfaces in in Abelian Varieties / Luca Cesarano ; Betreuer : Fabrizio Catanese ». Bayreuth : Universität Bayreuth, 2018. http://d-nb.info/1160301913/34.
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Kriel, Marelize. « Endomorphism rings of hyperelliptic Jacobians ». Thesis, Link to the online version, 2005. http://hdl.handle.net/10019/1077.
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Castilho, Tiago Nunes 1983. « Variedades de Prym e semigrupos de Weierstrass ». [s.n.], 2013. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306009.
Texte intégralRésumé :
Orientador: Marcos Benevenuto JardimTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Esta tese trata de variedades de Prym e de semigrupos de Weierstrass, ambos no contexto de recobrimentos duplos de curvas ramificados. A partir da descrição da variedade de Prym em termos de um conjunto de fibrações lineares do recobrimento, estuda-se a dualidade entre o lugar onde a aplicação de Gauss sobre o divisor Prym-Theta se degenera e o divisor de ramos do recobrimento duplo, em que provarse uma relação entre as fibras da aplicação de Gauss e os semigrupos de Weierstrass das ramificações do recobrimento
Abstract: ln this thesis we present results about Prym varieties and Weierstrass semigroups, both in the context of ramified double covers of curves. From the description of the Prym variety by a set of linear fibrations, we study the duality between the place where the Gauss map on the Prym-Theta divisor degenerates and the branch divisor of the double covering, in which we prove a relation between the fibers of the Gauss map and the Weierstrass semigroups of branched points of the double covering
Doutorado
Matematica
Doutor em Matemática
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Ludsteck, Thomas. « P-adic vector bundles on curves and abelian varieties and representations of the fundamental group ». [S.l. : s.n.], 2008. http://nbn-resolving.de/urn:nbn:de:bsz:93-opus-35588.
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Sumi, Ken. « Tropical Theta Functions and Riemann-Roch Inequality for Tropical Abelian Surfaces ». Doctoral thesis, Kyoto University, 2021. http://hdl.handle.net/2433/263432.
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Pal, Aprameyo [Verfasser], et Otmar [Akademischer Betreuer] Venjakob. « Functional Equation of Characteristic Elements of Abelian Varieties over Function Fields / Aprameyo Pal ; Betreuer : Otmar Venjakob ». Heidelberg : Universitätsbibliothek Heidelberg, 2013. http://d-nb.info/1177248506/34.
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Keil, Stefan. « On non-square order Tate-Shafarevich groups of non-simple abelian surfaces over the rationals ». Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2014. http://dx.doi.org/10.18452/16901.
Texte intégralRésumé :
Bei elliptischen Kurven E/K über einem Zahlkörper K zwingt die Cassels-Tate Paarung die Ordnung der Tate-Shafarevich Gruppe Sha(E/K) zu einem Quadrat. Ist A/K eine prinzipal polarisierte abelschen Varietät, so ist bewiesen, daß die Ordnung von Sha(A/K) ein Quadrat oder zweimal ein Quadrat ist. William Stein vermutet, daß es für jede quadratfreie positive ganze Zahl k eine abelsche Varietät A/Q gibt, mit #Sha(A/Q)=kn². Jedoch ist es ein offenes Problem was zu erwarten ist, wenn die Dimension von A/Q beschränkt wird. Betrachtet man ausschließlich abelsche Flächen B/Q, so liefern Resultate von Poonen, Stoll und Stein Beispiele mit #Sha(B/Q)=kn², für k aus {1,2,3}. Diese Arbeit studiert tiefgehend nicht-einfache abelsche Flächen B/Q, d.h. es gibt elliptische Kurven E_1/Q und E_2/Q und eine Isogenie phi: E_1 x E_2 -> B. Relativ zur quadratischen Ordnung der Tate-Shafarevich Gruppe von E_1 x E_2 soll die Ordnung von Sha(B/Q) bestimmt werden. Um dieses Ziel zu erreichen wird die Isogenie-Invarianz der Vermutung von Birch und Swinnerton-Dyer ausgenutzt. Für jedes k aus {1,2,3,5,6,7,10,13,14} wird eine nicht-einfache, nicht-prinzipal polarisierte abelsche Fläche B/Q konstruiert, mit #Sha(B/Q)=kn². Desweiteren wird computergestützt berechnet wie oft #Sha(B/Q)=5n², sofern die Isogenie phi: E_1 x E_2 -> B zyklisch vom Grad 5 ist. Es stellt sich heraus, daß dies bei circa 50% der ersten 20 Millionen Beispielen der Fall ist. Abschließend wird gezeigt, daß wenn phi: E_1 x E_2 -> B zyklisch ist und #Sha(B/Q)=kn², so liegt k in {1,2,3,5,6,7,10,13}. Bei allgemeinen Isogenien phi: E_1 x E_2 -> B bleibt es unklar, ob k nur endlich viele verschiedene Werte annehmen kann. Im Anhang wird auf abelsche Flächen eingegangen, welche isogen zu der Jacobischen J einer hyperelliptischen Kurve über Q sind. Mit den in dieser Arbeit entwickelten Techniken können, anhand gewisser zyklischer Isogenien phi: J -> B, für jedes k in {11,17,23,29} Beispiele mit #Sha(B/Q)=kn² gegeben werden.For elliptic curves E/K over a number field K the Cassels-Tate pairing forces the order of the Tate-Shafarevich group Sha(E/K) to be a perfect square. It is known, that if A/K is a principally polarised abelian variety, then the order of Sha(A/K) is a square or twice a square. William Stein conjectures that for any given square-free positive integer k there is an abelian variety A/Q, such that #Sha(A/Q)=kn². However, it is an open question what to expect if the dimension of A/Q is bounded. Restricting to abelian surfaces B/Q, then results of Poonen, Stoll and Stein imply that there are examples such that #Sha(B/Q)=kn², for k in {1,2,3}. In this thesis we focus in depth on non-simple abelian surfaces B/Q, i.e. there are elliptic curves E_1/Q and E_2/Q and an isogeny phi: E_1 x E_2 -> B. We want to compute the order of Sha(B/Q) with respect to the order of the Tate-Shafarevich group of E_1 x E_2, which has square order. To achieve this goal, we explore the invariance under isogeny of the Birch and Swinnerton-Dyer conjecture. For each k in {1,2,3,5,6,7,10,13,14} we construct a non-simple non-principally polarised abelian surface B/Q, such that #Sha(B/Q)=kn². Furthermore, we compute numerically how often the order of Sha(B/Q) equals five times a square, for cyclic isogenies phi: E_1 x E_2 -> B of degree 5. It turns out that this happens to be the case in approx. 50% of the first 20 million examples we have checked. Finally, we prove that if there is a cyclic isogeny phi: E_1 x E_2 -> B and #Sha(B/Q)=kn², then k is in {1,2,3,5,6,7,10,13}. For general isogenies phi: E_1 x E_2 -> B it remains unclear, whether there are only finitely many possibilities for k. In the appendix, we briefly consider abelian surfaces B/Q being isogenous to Jacobians J of hyperelliptic curves over Q. The techniques developed in this thesis allow to understand certain cyclic isogenies phi: J -> B. For each k in {11,17,23,29}, we provide an example with #Sha(B/Q)=kn².
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Delfs, Christina [Verfasser], Andreas [Akademischer Betreuer] Stein et Florian [Akademischer Betreuer] Hess. « Isogenies and endomorphism rings of abelian varieties of low dimension / Christina Delfs. Betreuer : Andreas Stein ; Florian Hess ». Oldenburg : BIS der Universität Oldenburg, 2015. http://d-nb.info/1093683937/34.
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Delfs, Christina Verfasser], Andreas [Akademischer Betreuer] [Stein et Florian [Akademischer Betreuer] Hess. « Isogenies and endomorphism rings of abelian varieties of low dimension / Christina Delfs. Betreuer : Andreas Stein ; Florian Hess ». Oldenburg : BIS der Universität Oldenburg, 2015. http://nbn-resolving.de/urn:nbn:de:gbv:715-oops-28173.
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Stoffel, Martino [Verfasser], et Klaus [Akademischer Betreuer] Künnemann. « Real-valued differential forms on non-archimedean abelian varieties with totally degenerate reduction / Martino Stoffel ; Betreuer : Klaus Künnemann ». Regensburg : Universitätsbibliothek Regensburg, 2021. http://d-nb.info/1225121469/34.
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Maarouf, Mohamed Anouar. « Problèmes d'hyperbolicité sur l'espace de Douady et ses variantes ». Université Joseph Fourier (Grenoble), 1996. http://www.theses.fr/1996GRE10089.
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Nous traitons la question d'hyperbolicite de l'espace de douady des sous-varietes lisses d'un espace analytique. Nous obtenons l'hyperbolicite au sens de brody dans le cas de dimension 1 moyennant des hypotheses de mesure hyperbolicite sur l'espace ambiant. Nous etudions egalement la meme question sur d'autres variantes de l'espace de douady telles que l'espace des varietes abeliennes plongees et les espaces de modules des applications holomorphes
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Agostini, Daniele. « On syzygies of algebraic varieties with applications to moduli ». Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/19415.
Texte intégralRésumé :
Diese Dissertation beschäftigt sich mit asymptotischen Syzygien und Gleichungen Abelscher Varietäten, sowie mit deren Anwendung auf zyklische Überdeckungen von Kurven von Geschlecht zwei.
Was asymptotischen Syzygien angeht, zeigen wir für beliebige Geradenbündel auf projektiven Schemata: Wenn die asymptotischen Syzygien von Grad p eines Geradenbündels verschwinden, dann ist das Geradenbündel p-sehr ampel. Darüber hinaus verwenden wir die Bridgeland-King-Reid-Haiman Korrespondenz, um zu zeigen, dass dieses Ergebnis auch umgekehrt wahr ist, wenn es um eine glatte Fläche und kleine p geht. Dies dehnt Ergebnisse von Ein-Lazarsfeld und Ein-Lazarsfeld-Yang aus. Wir verwenden unsere Ergebnisse, um zu untersuchen, wie Syzygien verwendet werden können, um den Grad der Irrationalität einer Varietät zu begrenzen.
Ferner, beweisen wir eine Vermutung von Gross and Popescu über Abelsche Flächen, deren Ideal durch Quadriken und Kubiken erzeugt wird.
Außerdem verwenden wir die projektive Normalität einer Abelschen Fläche, um die Prym Abbildung, die mit zyklischen Überdeckungen von Geschlecht zwei Kurven assoziert ist, zu untersuchen. Wir zeigen, dass das Differential der Abbildung generisch injektiv ist, wenn der Grad der Überdeckung mindestens sieben ist. Wir dehnen damit Ergebnisse von Lange und Ortega aus. Abschließend zeigen wir, dass das Differential genau für bielliptische Überdeckungen nicht injectiv ist.In this thesis we study asymptotic syzygies of algebraic varieties and equations of abelian surfaces, with applications to cyclic covers of genus two curves. First, we show that vanishing of asymptotic p-th syzygies implies p-very ampleness for line bundles on arbitrary projective schemes. For smooth surfaces we prove that the converse holds, when p is small, by studying the Bridgeland-King-Reid-Haiman correspondence for the Hilbert scheme of points. This extends previous results of Ein-Lazarsfeld and Ein-Lazarsfeld-Yang. As an application of our results, we show how to use syzygies to bound the irrationality of a variety. Furthermore, we confirm a conjecture of Gross and Popescu about abelian surfaces whose ideal is generated by quadrics and cubics. In addition, we use projective normality of abelian surfaces to study the Prym map associated to cyclic covers of genus two curves. We show that the differential of the map is generically injective as soon as the degree of the cover is at least seven, extending a previous result of Lange and Ortega. Moreover, we show that the differentials fails to be injective precisely at bielliptic covers.
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Schuster, Christian [Verfasser], Jörg [Gutachter] Winkelmann et Peter [Gutachter] Heinzner. « The Kobayashi pseudometric on compact complex surfaces in abelian varieties / Christian Schuster ; Gutachter : Jörg Winkelmann, Peter Heinzner ; Fakultät für Mathematik ». Bochum : Ruhr-Universität Bochum, 2020. http://d-nb.info/1217860096/34.
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Milio, Enea. « Calcul de polynômes modulaires en dimension 2 ». Thesis, Bordeaux, 2015. http://www.theses.fr/2015BORD0285/document.
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Les polynômes modulaires sont utilisés dans le calcul de graphes d’isogénies, le calcul des polynômes de classes ou le comptage du nombre de points d’une courbe elliptique, et sont donc fondamentaux pour la cryptographie basée sur les courbes elliptiques. Des polynômes analogues sur les surfaces abéliennes principalement polarisées ont été introduits par Régis Dupont en 2006, qui a également proposé un algorithme pour les calculer, et des résultats théoriques sur ces polynômes ont été donnés dans un article de Bröker–Lauter, en 2009. Mais les polynômes sont très gros et ils n’ont pu être calculés que pour l’exemple minimal p = 2. Dans cette thèse, nous poursuivons les travaux de Dupont et Bröker–Lauter en permettant de calculer des polynômes modulaires pour des invariants basés sur les thêta constantes, avec lesquels nous avons pu calculer les polynômes jusqu’à p = 7, tout en démontrant des propriétés de ces polynômes. Mais des exemples plus grands ne semblent pas envisageables. Ainsi, nous proposons une nouvelle définition des polynômes modulaires dans laquelle l’on se restreint aux surfaces abéliennes principalement polarisées qui ont multiplication réelle par l’ordre maximal d’un corps quadratique réel afin d’obtenir des polynômes plus petits. Nous présentons alors de nombreux exemples de polynômes et des résultats théoriquesModular polynomials on elliptic curves are a fundamental tool used for the computation of graph of isogenies, class polynomials or for point counting. Thus, they are fundamental for the elliptic curve cryptography. A generalization of these polynomials for principally polarized abelian surfaces has been introduced by Régis Dupont in 2006, who has also described an algorithm to compute them, while theoretical results can been found in an article of Bröker– Lauter of 2009. But these polynomials being really big, they have been computed only in the minimal case p = 2. In this thesis, we continue the work of Dupont and Bröker–Lauter by defining and giving theoretical results on modular polynomials with new invariants, based on theta constants. Using these invariants, we have been able to compute the polynomials until p = 7 but bigger examples look intractable. Thus we define a new kind of modular polynomials where we restrict on the surfaces having real multiplication by the maximal order of a real quadratic field. We present many examples and theoretical results
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Bisson, Gaëtan. « Anneaux d'endomorphismes en cryptographie ». Thesis, Vandoeuvre-les-Nancy, INPL, 2011. http://www.theses.fr/2011INPL047N/document.
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La cryptographie est indispensable aux réseaux de communication modernes afin de garantir la sécurité et l'intégrité des données y transitant. Récemment, des cryptosystèmes efficaces, sûr et riches ont été construits à partir de variétés abéliennes définies sur des corps finis. Cette thèse contribue à plusieurs aspects algorithmiques de ces variétés touchant à leurs anneaux d'endomorphismes. Cette structure joue un rôle capital pour construire des variétés abéliennesmunies de bonnes propriétés, comme des couplages, et nous montrons qu'un plus grand nombre de telles variétés peut être construit qu'on ne pourrait croire. Nous considérons aussi le problème inverse qu'est celui du calcul de l'anneau d'endomorphismes d'une variété abélienne donnée. Les meilleures méthodes connues ne pouvaient précédemment résoudre ce problème qu'en temps exponentiel ; ici, nous concevons plusieurs algorithmes de complexité sous-exponentielle le résolvant dans le cas ordinaire. Pour les courbes elliptiques, nous bornons rigoureusement la complexité de nos algorithmes sous l'hypothèse de Riemann étendue et démontrons qu'ils sont extrêmement efficaces en pratique. Comme sous-routine, nous développons notamment un algorithme sans mémoire pour résoudre une généralisation du problème du sac à dos. Nous généralisons aussi notre méthode aux variétés abélienne de dimension supérieure. Concrètement, nous développons une bibliothèque qui permet d'évaluer des isogénies entre variétés abéliennes ; cet outil nous permet d'appliquer une généralisation de notre méthode à des exemples jusqu'alors incalculablesModern communications heavily rely on cryptography to ensure data integrity and privacy. Over the past two decades, very efficient, secure, and featureful cryptographic schemes have been built on top of abelian varieties defined overfinite fields. This thesis contributes to several computational aspects of ordinary abelian varieties related to their endomorphism ring structure. This structure plays a crucial role in the construction of abelian varieties with desirable properties, such as pairings, and we show that more such varieties can be constructed than expected. We also address the inverse problem, that of computing the endomorphism ring of a prescribed abelian variety. Prior state-of-the-art methods could only solve this problem in exponential time, and we design several algorithms of subexponential complexityfor solving it in the ordinary case. For elliptic curves, we rigorously bound the complexity of our algorithms assuming solely the extended Riemann hypothesis, and demonstrate that they are very effective in practice. As a subroutine, we design in particular a memory-less algorithm to solve a generalization of the subset sum problem. We also generalize our method to higher-dimensional abelian varieties. Practically speaking, we develop a library enabling the computation of isogenies between abelian varieties; this building block enables us to apply a generalization of our algorithm to cases that were previously not computable
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Terrisse, Robin. « Flux vacua and compactification on smooth compact toric varieties ». Thesis, Lyon, 2019. http://www.theses.fr/2019LYSE1144/document.
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L’étude des vides avec flux est une étape primordiale afin de mieux comprendre la compactification en théorie des cordes ainsi que ses conséquences phénoménologiques. En présence de flux, l’espace interne ne peut plus être Calabi-Yau, mais admet tout de même une structure SU(3) qui devient un outil privilégié. Après une introduction aux notions géométriques nécessaires, cette thèse examine le rôle des flux dans la compactification supersymétrique sous différents angles. Nous considérons tout d’abord des troncations cohérentes de la supergravité IIA. Nous montrons alors que des condensats fermioniques peuvent aider à supporter des flux et générer une contribution positive à la constante cosmologique. Ces troncations admettent donc des vides de Sitter qu’il serait autrement très difficile d’obtenir, si ce n’est impossible. L’argument est tout d’abord employé avec des condensats de dilatini puis améliorer en suggérant un mécanisme pour générer des condensats de gravitini à partir d’instantons gravitationnels. Ensuite l’attention se tourne sur les branes et leur comportement sous T-dualité non abélienne. Nous calculons les configurations duales à certaines solutions avec D branes de la supergravité de type II, et examinons les flux ainsi que leurs charges afin d’identifier les branes après dualité. La solution supersymétrique avec brane D2 est étudiée plus en détails en vérifiant explicitement les équations sur les spineurs généralisés, puis en discutant de la possibilité d’une déformation massive. Le dernier chapitre fournit une construction systématique de structures SU(3) sur une large classe de variétés toriques compactes. Cette construction définit un fibré en sphère au-dessus d’une variété torique 2d quelconque, mais fonctionne tout aussi bien sur une base Kähler-EinsteinThe study of flux vacua is a primordial step in the understanding of string compactifications and their phenomenological properties. In presence of flux the internal manifold ceases to be Calabi-Yau, but still admits an SU(3) structure which becomes thus the preferred framework. After introducing the relevant geometrical notions this thesis explores the role that fluxes play in supersymmetric compactification through several approaches. At first consistent truncations of type IIA supergravity are considered. It is shown that fermionic condensates can help support fluxes and generate a positive contribution to the cosmological constant. These truncations thus admit de Sitter vacua which are otherwise extremely difficult to get, if not impossible. The argument is initially performed with dilatini condensates and then improved by suggesting a mechanism to generate gravitini condensates from gravitational instantons. Then the focus shifts towards branes and their behavior under non abelian T-duality. The duals of several D-brane solutions of type II supergravity are computed and the branes are tracked down by investigating the fluxes and the charges they carry. The supersymmetric D2 brane is further studied by checking explicitly the generalized spinor equations and discussing the possibility of a massive deformation. The last chapter gives a systematic construction of SU(3) structures on a wide class of compact toric varieties. The construction defines a sphere bundle on an arbitrary two-dimensional toric variety but also works when the base is Kähler-Einstein
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Scarponi, Danny. « Formes effectives de la conjecture de Manin-Mumford et réalisations du polylogarithme abélien ». Thesis, Toulouse 3, 2016. http://www.theses.fr/2016TOU30100/document.
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Dans cette thèse nous étudions deux problèmes dans le domaine de la géométrie arithmétique, concernant respectivement les points de torsion des variétés abéliennes et le polylogarithme motivique sur les schémas abéliens. La conjecture de Manin-Mumford (démontrée par Raynaud en 1983) affirme que si A est une variété abélienne et X est une sous-variété de A ne contenant aucune translatée d'une sous-variété abélienne de A, alors X ne contient qu'un nombre fini de points de torsion de A. En 1996, Buium présenta une forme effective de la conjecture dans le cas des courbes. Dans cette thèse, nous montrons que l'argument de Buium peut être utilisé aussi en dimension supérieure pour prouver une version quantitative de la conjecture pour une classe de sous-variétés avec fibré cotangent ample étudiée par Debarre. Nous généralisons aussi à toute dimension un résultat sur la dispersion des relèvements p-divisibles non ramifiés obtenu par Raynaud dans le cas des courbes. En 2014, Kings and Roessler ont montré que la réalisation en cohomologie de Deligne analytique de la part de degré zéro du polylogarithme motivique sur les schémas abéliens peut être reliée aux formes de torsion analytique de Bismut-Koehler du fibré de Poincaré. Dans cette thèse, nous utilisons la théorie de l'intersection arithmétique dans la version de Burgos pour raffiner ce résultat dans le cas où la base du schéma abélien est propreIn this thesis we approach two independent problems in the field of arithmetic geometry, one regarding the torsion points of abelian varieties and the other the motivic polylogarithm on abelian schemes. The Manin-Mumford conjecture (proved by Raynaud in 1983) states that if A is an abelian variety and X is a subvariety of A not containing any translate of an abelian subvariety of A, then X can only have a finite number of points that are of finite order in A. In 1996, Buium presented an effective form of the conjecture in the case of curves. In this thesis, we show that Buium's argument can be made applicable in higher dimensions to prove a quantitative version of the conjecture for a class of subvarieties with ample cotangent studied by Debarre. Our proof also generalizes to any dimension a result on the sparsity of p-divisible unramified liftings obtained by Raynaud in the case of curves. In 2014, Kings and Roessler showed that the realisation in analytic Deligne cohomology of the degree zero part of the motivic polylogarithm on abelian schemes can be described in terms of the Bismut-Koehler higher analytic torsion form of the Poincaré bundle. In this thesis, using the arithmetic intersection theory in the sense of Burgos, we give a refinement of Kings and Roessler's result in the case in which the base of the abelian scheme is proper