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Relevant bibliographies by topics / Abelian surfaces

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Academic literature on the topic 'Abelian surfaces'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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Journal articles on the topic "Abelian surfaces"

1

Hulek, Klaus, and Steven H. Weintraub. "Bielliptic abelian surfaces." Mathematische Annalen 283, no. 3 (1989): 411–29. http://dx.doi.org/10.1007/bf01442737.

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Garbagnati, Alice. "On K3 Surface Quotients of K3 or Abelian Surfaces." Canadian Journal of Mathematics 69, no. 02 (2017): 338–72. http://dx.doi.org/10.4153/cjm-2015-058-1.

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Abstract The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group G (respectively of a K3 surface by an Abelian group G) if and only if a certain lattice is primitively embedded in its Néron-Severi group. This allows one to describe the coarse moduli space of the K3 surfaces that are (rationally) G-covered by Abelian or K3 surfaces (in the latter case G is an Abelian group). When G has order 2 or G is cyclic and acts on an Abelian surface, this result is already known; we extend it to the other cases. Moreover, we prove that a K3 surface XG is the minimal model of the quotient of an Abelian surface by a group G if and only if a certain configuration of rational curves is present on XG . Again, this result was known only in some special cases, in particular, if G has order 2 or 3.

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Alfonso Bonfanti, Matteo, and Bert van Geemen. "Abelian Surfaces with an Automorphism and Quaternionic Multiplication." Canadian Journal of Mathematics 68, no. 1 (2016): 24–43. http://dx.doi.org/10.4153/cjm-2014-045-4.

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AbstractWe construct one-dimensional families of Abelian surfaces with quaternionic multiplication, which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)- polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space that parametrizes Abelian surfaces with real multiplication by .

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Calegari, Frank, Shiva Chidambaram, and Alexandru Ghitza. "Some modular abelian surfaces." Mathematics of Computation 89, no. 321 (2019): 387–94. http://dx.doi.org/10.1090/mcom/3434.

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Chepelev, Iouri. "Non-Abelian Wilson Surfaces." Journal of High Energy Physics 2002, no. 02 (2002): 013. http://dx.doi.org/10.1088/1126-6708/2002/02/013.

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Yoshihara, Hisao. "Quotients of abelian surfaces." Publications of the Research Institute for Mathematical Sciences 31, no. 1 (1995): 135–43. http://dx.doi.org/10.2977/prims/1195164795.

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González-Diez, G., G. A. Jones, and D. Torres-Teigell. "Beauville Surfaces with Abelian Beauville Group." MATHEMATICA SCANDINAVICA 114, no. 2 (2014): 191. http://dx.doi.org/10.7146/math.scand.a-17106.

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A Beauville surface is a rigid surface of general type arising as a quotient of a product of curves $C_{1}$, $C_{2}$ of genera $g_{1},g_{2}\ge 2$ by the free action of a finite group $G$. In this paper we study those Beauville surfaces for which $G$ is abelian (so that $G\cong \mathsf{Z}_{n}^{2}$ with $\gcd(n,6)=1$ by a result of Catanese). For each such $n$ we are able to describe all such surfaces, give a formula for the number of their isomorphism classes and identify their possible automorphism groups. This explicit description also allows us to observe that such surfaces are all defined over $\mathsf{Q}$.

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Lesfari, A. "Abelian surfaces and Kowalewski's top." Annales scientifiques de l'École normale supérieure 21, no. 2 (1988): 193–223. http://dx.doi.org/10.24033/asens.1556.

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Silverberg, A., and Yu G. Zarhin. "Inertia groups and abelian surfaces." Journal of Number Theory 110, no. 1 (2005): 178–98. http://dx.doi.org/10.1016/j.jnt.2004.05.015.

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Birkenhake, C., H. Lange, and D. van Straten. "Abelian surfaces of type (1,4)." Mathematische Annalen 285, no. 4 (1989): 625–46. http://dx.doi.org/10.1007/bf01452051.

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Dissertations / Theses on the topic "Abelian surfaces"

1

Marini, A. "On the degenerations of (1,7)-polarised abelian surfaces." Thesis, University of Bath, 2002. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.394153.

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Rose, Simon Charles Florian. "Counting hyperelliptic curves in Abelian surfaces with quasi-modular forms." Thesis, University of British Columbia, 2012. http://hdl.handle.net/2429/42091.

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In this thesis we produce a generating function for the number of hyperelliptic curves (up to translation) on a polarized Abelian surface using the crepant resolution conjecture and the Yau-Zaslow formula. We present a formula to compute these in terms of P. A. MacMahon's generalized sum-of-divisors functions, and prove that they are quasi-modular forms.

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Manoharmayum, Jayanta. "Mod n representations arising from elliptic curves and abelian surfaces." Thesis, University of Cambridge, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.624411.

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Alagal, Wafa Abdullah. "Application of Bridgeland stability to the geometry of abelian surfaces." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/20440.

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A key property of projective varieties is the very ampleness of line bundles as this provides embeddings into projective space and allows us to express the variety in equational terms. In this thesis we study the general version of this property which is k- very ampleness of line bundles. We introduce the notation of critical k-very ampleness and compute it for abelian surfaces. The property of k-very ampleness is usually discussed using tools from divisor theory but we take a different approach and use methods from derived algebraic geometry as part of program to use properties of the derived category of a variety to access the geometry of the variety. In particular, we use the Fourier-Mukai transform, moduli spaces of sheaves and properties of Bridgeland stability. We compute walls for certain Bridgeland stable spaces and certain Chern characters and to complete the picture we study the moduli spaces of torsion sheaves with minimal first Chern class and we go on to compute the walls for these as well building on tools developed earlier in the thesis.

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Kazaz, Mustafa. "Finite groups and coverings of surfaces." Thesis, University of Southampton, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.264739.

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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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Biroth, Laura [Verfasser]. "Integrable systems and a moduli space for (1,6)-polarised abelian surfaces / Laura Biroth." Mainz : Universitätsbibliothek Mainz, 2019. http://d-nb.info/1200661478/34.

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Cesarano, Luca [Verfasser], and 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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Gillibert, Florence. "Surfaces abéliennes à multiplication quaternionique et points rationnels de quotients d'Atkin-Lehner de courbes de Shimura." Thesis, Bordeaux 1, 2011. http://www.theses.fr/2011BOR14374/document.

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Dans cette thèse nous étudions deux problèmes. Le premier est la non-existence de pointsrationnels non spéciaux sur des quotients d’Atkin-Lehner de courbes de Shimura. Le se-cond est l’absence de surfaces abéliennes rationnelles à multiplication potentiellementquaternioniques munies d’une structure de niveau. Ces deux problèmes sont liés car unesurface abélienne rationnelle simple à multiplication potentiellement quaternionique cor-respond à un point rationnel non spécial sur un certain quotient d’Atkin-Lehner de courbede Shimura.Dans une première partie nous expliquons comment vérifier un critère de Parent etYafaev en grande généralité pour prouver que dans les conditions du cas non ramifié deOgg, et si p est assez grand par rapport à q, alors le quotient X^pq/w_q n’a pas de pointrationnel non spécial.Dans une seconde partie nous déterminons une borne effective pour les structures deniveaux possibles pour une surface abélienne rationnelle acquérant sur un corps quadra-tique imaginaire fixé multiplication par un ordre fixé dans une algèbre de quaternions<br>In this thesis we study two problems. The first one is the non-existence of rational non-special points on Atkin-Lehner quotients of Shimura curves. The second one is the absence of rational abelian surfaces with potential quaternionique multiplication endowed with a level structure. These two problems are linked because a simple rational abelian surface with potential quaternionique multiplication is associated to a rational non-special point on an Atkin-Lehner quotients of Shimura curve. In a first part of our work we explain how to verify in wide generality a criterium of Parent and Yafaev in order to prove that in the conditions of Ogg's non ramified case, and if $p$ is big enough compared two $q$, then the quotient $X^{pq}/w_q$ has no non-special rational point. In a second part we determine an effective born for possible level structures on rational abelian surfaces having, over a fixed quadratic field, multiplication by a fixed order in a quaternion algebra

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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.

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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.<br>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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Books on the topic "Abelian surfaces"

1

H, Lange. Complex Abelian varieties. Springer-Verlag, 1992.

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1943-, Lange H., and Lange H. 1943-, eds. Complex Abelian varieties. 2nd ed. Springer, 2004.

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Lange, Herbert. Complex Abelian varieties. Springer, 1992.

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Faltings, Gerd. Degeneration of Abelian varieties. Springer-Verlag, 1990.

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Muñoz Porras, José M., Sorin Popescu, and Rubí E. Rodríguez, eds. The Geometry of Riemann Surfaces and Abelian Varieties. American Mathematical Society, 2006. http://dx.doi.org/10.1090/conm/397.

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Hulek, Klaus. Moduli spaces of Abelian surfaces: Compactification, degenerations, and theta functions. Walter de Gruyter, 1993.

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Muñoz Porras, Jose M. 1956-, Popescu Sorin 1963-, Rodríguez Rubí E. 1953-, and Recillas-Pishmish Sevín 1943-, eds. The geometery [sic] of Riemann surfaces and Abelian varieties: III Iberoamerican Congress on Geometry in honor of Professor Sevin Recillas-Pishmish's 60th birthday, June 8-12, 2004, Salamanca, Spain. American Mathematical Society, 2006.

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Complex analysis 2: Riemann surfaces, several complex variables, abelian functions, higher modular functions. Springer, 2011.

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Hironaka, Eriko. Abelian coverings of the complex projective plane branched along configurations of real lines. American Mathematical Society, 1993.

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Rodríguez, Rubí E., 1953- editor of compilation, ed. Riemann and Klein surfaces, automorphisms, symmetries and moduli spaces: Conference in honor of Emilio Bujalance on Riemann and Klein surfaces, symmetries and moduli spaces, June 24-28, 2013, Linköping University, Linköping, Sweden. American Mathematical Society, 2014.

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Book chapters on the topic "Abelian surfaces"

1

Lange, Herbert, and Christina Birkenhake. "Abelian Surfaces." In Complex Abelian Varieties. Springer Berlin Heidelberg, 1992. http://dx.doi.org/10.1007/978-3-662-02788-2_12.

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Birkenhake, Christina, and Herbert Lange. "Abelian Surfaces." In Complex Abelian Varieties. Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-662-06307-1_12.

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Voight, John. "QM abelian surfaces." In Graduate Texts in Mathematics. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-56694-4_43.

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Howard, Benjamin, and Tonghai Yang. "Moduli Spaces of Abelian Surfaces." In Intersections of Hirzebruch–Zagier Divisors and CM Cycles. Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-23979-3_3.

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Geer, Gerard. "Moduli of Abelian Schemes with Real Multiplication." In Hilbert Modular Surfaces. Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/978-3-642-61553-5_12.

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Van der Geer, G., and T. Katsura. "Formal Brauer Groups and Moduli of Abelian Surfaces." In Moduli of Abelian Varieties. Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8303-0_6.

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Edixhoven, Bas. "On the André-Oort Conjecture for Hilbert Modular Surfaces." In Moduli of Abelian Varieties. Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8303-0_4.

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Kresch, Andrew, and Yuri Tschinkel. "Integral Points on Punctured Abelian Surfaces." In Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2002. http://dx.doi.org/10.1007/3-540-45455-1_16.

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Harder, Günter. "Compact Riemann surfaces and Abelian Varieties." In Aspects of Mathematics. Springer Fachmedien Wiesbaden, 2011. http://dx.doi.org/10.1007/978-3-8348-8330-8_5.

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Ogus, Arthur. "Singularities of the Height Strata in the Moduli of K3 Surfaces." In Moduli of Abelian Varieties. Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8303-0_12.

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Conference papers on the topic "Abelian surfaces"

1

Korzec, Tomasz, and Ulli Wolff. "Simulating the Random Surface representation of Abelian Gauge Theories." In 31st International Symposium on Lattice Field Theory LATTICE 2013. Sissa Medialab, 2014. http://dx.doi.org/10.22323/1.187.0038.

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