Academic literature on the topic 'Twisted elliptic genus'

We consider the holomorphic twist of the worldvolume theory of flat D[Formula: see text]-branes transversely probing a Calabi–Yau manifold. A chain complex, constructed using the BV formalism, computes the local observables in the holomorphically twisted theory. Generalizing earlier work in the case [Formula: see text], we find that this complex can be identified with the Ginzburg dg algebra associated to the Calabi–Yau. However, the identification is subtle; the complex is the space of fields contributing to the holomorphic twist of the free theory, and its differential arises from interactions. For [Formula: see text], this holomorphically twisted theory is related to the elliptic genus. We give a general description for D1-branes probing a Calabi–Yau fourfold singularity, and for [Formula: see text] quiver gauge theories. In addition, we propose a relation between the equivariant Hirzebruch [Formula: see text] genus of large-[Formula: see text] symmetric products and cyclic homology.

The genus, Duthiea Hackel (1895: 200) consists of three species, viz. D. brachypodium (P.Candargy 1901: 65) Keng & Keng (1965: 182), D. bromoides Hackel (1895: 200) and D. oligostachya (Munro ex Aitchison 1880: 108) Stapf (1896: sub Pl. 2474), distributed mainly in mountainous zones of Afghanistan to western China (Kellogg 2015). Duthiea is characterized by having pedicelled spikelets arranged to congested one-sided racemes; rachilla disarticulating above the glumes and between the florets; glumes equal to sub-equal, elliptic or lanceolate, with 5–7 nerves, persistent; lemma hirsute or villous with bifid apex and geniculate single awn with twisted column arising from the sinus of lemma; lodicules absent; ovary obovoid; style single, tomentose, longer or shorter than the stigmas; stigmas 2, terminally exerted from the floret; caryopsis cylindrical, covered with forwardly directed bristles (Bor 1953, 1960).

AbstractGross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function $$G_s(z_1,z_2)$$ G s ( z 1 , z 2 ) for the elliptic modular group at positive integral spectral parameter s are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable $$z_1$$ z 1 over all CM points of a fixed discriminant $$d_1$$ d 1 (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant $$d_2$$ d 2 . This result is deduced from more general statements for automorphic Green functions on Shimura varieties associated with the group $${\text {GSpin}}(n,2)$$ GSpin ( n , 2 ) . We also use our approach to prove a Gross–Kohnen–Zagier theorem for higher Heegner divisors on Kuga–Sato varieties over modular curves.

Fix an elliptic curve E E over a number field F F and an integer n n which is a power of 3 3 . We study the growth of the Mordell–Weil rank of E E after base change to the fields K d = F ( d 2 n ) K_d = F(\!\sqrt [2n]{d}) . If E E admits a 3 3 -isogeny, then we show that the average “new rank” of E E over K d K_d , appropriately defined, is bounded as the height of d d goes to infinity. When n = 3 n = 3 , we moreover show that for many elliptic curves E / Q E/\mathbb {Q} , there are no new points on E E over Q ( d 6 ) \mathbb {Q}(\sqrt [6]d) , for a positive proportion of integers d d . This is a horizontal analogue of a well-known result of Cornut and Vatsal [Nontriviality of Rankin-Selberg L-functions and CM points, L-functions and Galois representations, vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121–186]. As a corollary, we show that Hilbert’s tenth problem has a negative solution over a positive proportion of pure sextic fields Q ( d 6 ) \mathbb {Q}(\sqrt [6]{d}) . The proofs combine our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the “correlation trick”, which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves.

Observation of sand samples collected from an estuarine sandy beach in Southern China revealed the presence of a new biraphid diatom. It is characterized by the presence of a twisted raphe system, apically elongate areolae, and striae interrupted by longitudinal H-shaped lateral areas in a form of valve face depressions, as well as girdle composed of open, plain copulae. The structure of valve outline, the raphe sternum and striae bears some resemblance to some established genera including e.g., Lyrella, Fallacia, Scolioneis, Scoliopleura and Scoliotropis, however, these characters are uniquely combined in this novel taxon. The new species, Scoliolyra elliptica, belongs to a new biraphid genus, Scoliolyra, and it is tentatively placed within the family Scolioneidaceae.

We study elliptic curves over quadratic fields with isogenies of certain degrees. Let $n$ be a positive integer such that the modular curve $X_{0}(n)$ is hyperelliptic of genus ${\geqslant}2$ and such that its Jacobian has rank $0$ over $\mathbb{Q}$. We determine all points of $X_{0}(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, every elliptic curve over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by a quadratic extension $L$ of $K$. We determine $d$ and $L$ explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, all elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb{Q}$-curves.

Abstract We study twisted circle compactification of 6d (2, 0) SCFTs to 5d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ0(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

Les principaux objets étudiés dans cette thèse sont les équations décrivant le morphisme de groupe sur une variété abélienne, plongée dans un espace projectif, et leurs applications en cryptographie. Notons g sa dimension et k son corps de définition. Ce mémoire est composé de deux parties. La première porte sur l'étude des courbes d'Edwards, un modèle pour les courbes elliptiques possédant un sous-groupe de points k-rationnels cyclique d'ordre 4, connues en cryptographie pour l'efficacité de leur loi d'addition et la possibilité qu'elle soit définie pour toute paire de points k-rationnels (loi d'addition k-complète). Nous en donnons une interprétation géométrique et en déduisons des formules explicites pour le calcul du couplage de Tate réduit sur courbes d'Edwards tordues, dont l'efficacité rivalise avec les modèles elliptiques couramment utilisés. Cette partie se conclut par la génération, spécifique au calcul de couplages, de courbes d'Edwards dont les tailles correspondent aux standards cryptographiques actuellement en vigueur. Dans la seconde partie nous nous intéressons à la notion de complétude introduite ci-dessus. Cette propriété est cryptographiquement importante car elle permet d'éviter des attaques physiques, comme les attaques par canaux cachés, sur des cryptosystèmes basés sur les courbes elliptiques ou hyperelliptiques. Un précédent travail de Lange et Ruppert, basé sur la cohomologie des fibrés en droite, permet une approche théorique des lois d'addition. Nous présentons trois résultats importants : tout d'abord nous généralisons un résultat de Bosma et Lenstra en démontrant que le morphisme de groupe ne peut être décrit par strictement moins de g+1 lois d'addition sur la clôture algébrique de k. Ensuite nous démontrons que si le groupe de Galois absolu de k est infini, alors toute variété abélienne peut être plongée dans un espace projectif de manière à ce qu'il existe une loi d'addition k-complète. De plus, l'utilisation des variétés abéliennes nous limitant à celles de dimension un ou deux, nous démontrons qu'une telle loi existe pour leur plongement projectif usuel. Finalement, nous développons un algorithme, basé sur la théorie des fonctions thêta, calculant celle-ci dans P^15 sur la jacobienne d'une courbe de genre deux donnée par sa forme de Rosenhain. Il est désormais intégré au package AVIsogenies de Magma
The main objects we study in this PhD thesis are the equations describing the group morphism on an abelian variety, embedded in a projective space, and their applications in cryptograhy. We denote by g its dimension and k its field of definition. This thesis is built in two parts. The first one is concerned by the study of Edwards curves, a model for elliptic curves having a cyclic subgroup of k-rational points of order 4, known in cryptography for the efficiency of their addition law and the fact that it can be defined for any couple of k-rational points (k-complete addition law). We give the corresponding geometric interpretation and deduce explicit formulae to calculate the reduced Tate pairing on twisted Edwards curves, whose efficiency compete with currently used elliptic models. The part ends with the generation, specific to pairing computation, of Edwards curves with today's cryptographic standard sizes. In the second part, we are interested in the notion of completeness introduced above. This property is cryptographically significant, indeed it permits to avoid physical attacks as side channel attacks, on elliptic -- or hyperelliptic -- curves cryptosystems. A preceeding work of Lange and Ruppert, based on cohomology of line bundles, brings a theoretic approach of addition laws. We present three important results: first of all we generalize a result of Bosma and Lenstra by proving that the group morphism can not be described by less than g+1 addition laws on the algebraic closure of k. Next, we prove that if the absolute Galois group of k is infinite, then any abelian variety can be projectively embedded together with a k-complete addition law. Moreover, a cryptographic use of abelian varieties restricting us to the dimension one and two cases, we prove that such a law exists for their classical projective embedding. Finally, we develop an algorithm, based on the theory of theta functions, computing this addition law in P^15 on the Jacobian of a genus two curve given in Rosenhain form. It is now included in AVIsogenies, a Magma package

The most powerful known primitive in public-key cryptography is undoubtedly elliptic curve pairings. Upon their introduction just over ten years ago the computation of pairings was far too slow for them to be considered a practical option. This resulted in a vast amount of research from many mathematicians and computer scientists around the globe aiming to improve this computation speed. From the use of modern results in algebraic and arithmetic geometry to the application of foundational number theory that dates back to the days of Gauss and Euler, cryptographic pairings have since experienced a great deal of improvement. As a result, what was an extremely expensive computation that took several minutes is now a high-speed operation that takes less than a millisecond. This thesis presents a range of optimisations to the state-of-the-art in cryptographic pairing computation. Both through extending prior techniques, and introducing several novel ideas of our own, our work has contributed to recordbreaking pairing implementations.

In this thesis we study the applications of Mathieu moonshine symmetry to compacti cations of supersymmetric string theories. These theories are compacti ed on a 6 dimensional manifold K3 T2. The main ingredient in this study is a topological index called twisted elliptic genus. For a super-conformal eld theory whose target space is a K3 there can be several automorphisms on K3 which are related to Mathieu group M24. Under these automorphisms it was observed that the twining genera of the twisted elliptic genus of K3 could be written in terms of the short and long representations of N = 4 super-conformal algebra and the characters of M24 [1, 2, 3]. We compute the twisted elliptic genus in every sector for 16 of these orbifolds using the results of [2]. Firstly we study the heterotic compacti cations of N = 2 super-symmetric strings compacti ed on orbifolds of K3 T2 and E8 E8 where g0 is an action on K3 corresponding to [M24] along with a 1=N shift on one of the circles of T2. We compute the gauge and gravitational threshold corrections in these theories. Here we need a topological index called the new supersymmetric index. The un-orbifolded result for K3 was known for gauge couplings in [4] and the gravitational ones were computed in [5]. We observe that the di erences in gauge couplings can be written in terms of the twisted elliptic genus of K3 for standard embeddings. For non-standard embeddings we studied two orbifold realizations of K3 as T4=Z2 and T4=Z4 and computed the threshold di erences. The result could be written in terms the twisted elliptic genus of K3 and the elliptic genus of K3. From the gravitational corrections we predict the Gopakumar Vafa invariants and the Euler character for the dual Calabi Yau geometries. We also observe that the conifold singularities of these manifolds are manifested in twisted sectors only and only the genus zero Gopakumar-Vafa invariants at those points are non-zero. Secondly we study the properties of 1/4 BPS dyons in type II string compacti ed on K3 T2 orbifolded with an action of g0 which corresponds to automorphisms of K3 corresponding to the conjugacy classes of Mathieu group M24 and a 1=N shift in one of the circles of T2. For these compacti cations the counting function for these dyons can be computed from Siegel modular forms given by the lift of the twisted elliptic genus. These give the correct sign as predicted from black hole physics as conjectured by Sen [6]. We also study the properties of 1/4th BPS dyons in type II string theory compacti ed on Z2 and Z3 orbifolds on T6 with 1=N shift in one of the S1 and encountered some violations to this conjecture which points to the existence of non-trivial hair modes. We associate mock modular forms corresponding to single centred black holes and extend the work of Dabholkar-Murthy-Zagier [7] to these orbifolds of K3 and also for the toroidal orbifolds. In computing the twisted elliptic genus and new super-symmetric index in various twisted sectors we encounter several identities between some 􀀀0(N) modular forms. With a bit more analysis we determine the exact location of the zeros of some weight 2 Eisenstein series of 􀀀0(N) in the fundamental domain of 􀀀0(N) where N = 2; 3; 5; 7. The location of their zeros were controlled by those of Eisenstein series of weight 4 and 6.

AbstractIn this chapter, we review basic notions about spectra, group representations, and twisted Laplace operators. We first recall how to define the spectrum and the spectral zeta function for a general symmetric second order elliptic differential operator acting on smooth sections of a Hermitian line bundle. We prove that the non-zero spectrum (i.e., the spectral zeta function) determines the entire spectrum on an odd-dimensional manifold, but also give an example showing that this is not always true for even-dimensional manifolds; the example is obstructed by the non-vanishing of some topological genus. After setting up some notation from representation theory, we discuss G-sets and weak conjugacy (“Gaßmann equivalence”) of subgroups of a group, explaining the interrelations. In the final sections, we introduce twisted Laplacians, corresponding to unitary representations of the fundamental group. After this, we focus on the case of a twisted Laplacian arising from a finite Galois cover of manifolds and we relate the spectrum on the top manifold to that of the induced representation on the bottom manifold. We relate the multiplicity of zero in the spectrum to the multiplicity of the trivial representation in the given representation, and finally we show that, contrary to the general case, the multiplicity of zero in the spectrum of a twisted Laplacian is determined from the non-zero spectrum, provided one also knows the usual Laplace spectrum of the manifold.