Abelian varieties isogenous to a power of an elliptic curve

Jordan, Bruce W.; Keeton, Allan; Poonen, Bjorn; Rains, Eric M.; Shepherd‐Barron, N. I.; Tate, John

doi:10.1112/s0010437x17007990

Cited by 11 publications

(11 citation statements)

References 24 publications

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“…The prototype of such descriptions is the Deuring-Eichler correspondence, which establishes the following bijection where p is a prime number and D p,∞ is the quaternion Q-algebra ramified exactly at {p, ∞}. We refer to Waterhouse [30], Deligne [7], Ekedahl [10], Katsura and Oort [15], C.-F. Yu [36,37,40,41], Centeleghe and Stix [3] J. Xue, T.-C. Yang and C.-F. Yu [31,33,34], Jordan-Keeton-Poonen-Shepherd-Barron-Tate [14] and others for various generalizations, and explicit formulas for numbers of certain abelian varieties.…”

Section: Analogies Between Abelian Varieties and Latticesmentioning

confidence: 99%

On Counting Certain Abelian Varieties Over Finite Fields

Xue

2020

Acta. Math. Sin.-English Ser.

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This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers [32,31,33,34], the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number √ q. This establishes a key step that one may extend our previous explicit calculations of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the results of [40] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigation.

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Section: Analogies Between Abelian Varieties and Latticesmentioning

confidence: 99%

On Counting Certain Abelian Varieties Over Finite Fields

Xue

2020

Acta. Math. Sin.-English Ser.

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“…Proof. Parts (1) -(3) follow from the corresponding properties after base-extension to k 1 [4,Thm 4.4]. For part (4), we have Hom k pC, H om RxGy pR, E k 1 qq " Hom RxGy pR, Hom k 1 pC k 1 , E k 1 qq " Hom k 1 pC k 1 , E k 1 q G " Hom k pC, Eq.…”

Section: Categorical Constructionsmentioning

confidence: 99%

“…Let E be an elliptic curve over a field k. As in [4], the theory of abelian varieties isogenous over k to a power of E is related to the theory of finitely presented torsion-free modules over the endomorphism ring R k :" End k E. To recall briefly, there is a functor H om R k p´, Eq :…”

Section: Introductionmentioning

confidence: 99%

Abelian varieties isogenous to a power of an elliptic curve over a Galois extension

Vogt¹

2019

J. Théor. Nombres Bordeaux

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Given an elliptic curve E{k and a Galois extension k 1 {k, we construct an exact functor from torsion-free modules over the endomorphism ring End E k 1 with a semilinear Galpk 1 {kq action to abelian varieties over k that are k 1 -isogenous to a power of E. As an application, we give a simple proof that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.

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“…Moreover it is always possible to identify one elliptic curve isogenous to E with minimal endomorphism ring, i.e. equal to R. We will assume from now on that this is our curve E. The functor given in [JKP+18] which associates to any A ∈ W the finitely generated torsion-free R-module (or in short R-lattice) Hom(A, E) of rank g is an equivalence of categories and provides an inverse denoted F E . Note that this functor is distinct from the one used for instance in [Mar19] (it is contravariant and exact) and there is no easy way to compare them away from projective R-modules.…”

mentioning

confidence: 99%

“…. ⊂ End(E g ) and invertible End(E i )-ideal classes I i with a given fixed product I 1 • • • I g in ICM(R) (see [Kan11, Th.1], [Mar19], [JKP+18,Th.3.2]).…”

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confidence: 99%

Spanning the isogeny class of a power of an elliptic curve

et al. 2021

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Let E E be an ordinary elliptic curve over a finite field and g g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E g E^g . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E 3 E^3 and of the Igusa modular form in dimension 4 4 . We illustrate our algorithms with examples of curves with many rational points over finite fields.

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Abelian varieties isogenous to a power of an elliptic curve

Cited by 11 publications

References 24 publications

On Counting Certain Abelian Varieties Over Finite Fields

On Counting Certain Abelian Varieties Over Finite Fields

Abelian varieties isogenous to a power of an elliptic curve over a Galois extension

Spanning the isogeny class of a power of an elliptic curve

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