On compatibility between isogenies and polarizations of abelian varieties

We discuss the notion of polarized isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarizations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show that certain questions about polarized isogenies can be reduced to questions about unpolarized isogenies or vice versa. Our main theorem concerns abelian varieties [Formula: see text] which are isogenous to a fixed abelian variety [Formula: see text]. It establishes the existence of a polarized isogeny [Formula: see text] whose degree is polynomially bounded in [Formula: see text], if there exist both an unpolarized isogeny [Formula: see text] of degree [Formula: see text] and a polarized isogeny [Formula: see text] of unknown degree. As a further result, we prove that given any two principally polarized abelian varieties related by an unpolarized isogeny, there exists a polarized isogeny between their fourth powers. The proofs of both theorems involve calculations in the endomorphism algebras of the abelian varieties, using the Albert classification of these endomorphism algebras and the classification of Hermitian forms over division algebras.

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The isometry classification of hermitian forms over division algebras

Linear Algebra and its Applications ◽

10.1016/0024-3795(82)90258-0 ◽

1982 ◽

Vol 43 ◽

pp. 245-272 ◽

Cited By ~ 5

Author(s):

D.W. Lewis

Keyword(s):

Division Algebras ◽

Hermitian Forms

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Classification of nonrigid families of abelian varieties

Tohoku Mathematical Journal ◽

10.2748/tmj/1178225915 ◽

1993 ◽

Vol 45 (2) ◽

pp. 159-189

Author(s):

Masa-Hiko Saitō

Keyword(s):

Abelian Varieties

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ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

International Journal of Number Theory ◽

10.1142/s1793042110003071 ◽

2010 ◽

Vol 06 (03) ◽

pp. 579-586 ◽

Cited By ~ 3

Author(s):

ARNO FEHM ◽

SEBASTIAN PETERSEN

Keyword(s):

Finite Field ◽

Abelian Variety ◽

Galois Extension ◽

Characteristic Zero ◽

Rational Point ◽

Abelian Varieties ◽

Finitely Generated ◽

Infinite Rank ◽

Finite Set

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

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ON SELMER RANK PARITY OF TWISTS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788716000306 ◽

2016 ◽

Vol 102 (3) ◽

pp. 316-330 ◽

Cited By ~ 1

Author(s):

MAJID HADIAN ◽

MATTHEW WEIDNER

Keyword(s):

Elliptic Curves ◽

Abelian Variety ◽

Number Field ◽

Arbitrary Number ◽

Hyperelliptic Curve ◽

Abelian Varieties ◽

Hyperelliptic Curves ◽

Torsion Subgroup ◽

Tate Conjecture ◽

Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

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Holomorphic curves in Shimura varieties

Archiv der Mathematik ◽

10.1007/s00013-018-1227-4 ◽

2018 ◽

Vol 111 (4) ◽

pp. 379-388 ◽

Cited By ~ 1

Author(s):

Michele Giacomini

Keyword(s):

Abelian Varieties ◽

Holomorphic Curves ◽

Zariski Closure ◽

Shimura Varieties

Abstract We prove a hyperbolic analogue of the Bloch–Ochiai theorem about the Zariski closure of holomorphic curves in abelian varieties. We consider the case of non compact Shimura varieties completing the proof of the result for all Shimura varieties. The statement which we consider here was first formulated and proven by Ullmo and Yafaev for compact Shimura varieties.

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Endomorphism Algebras of Abelian Varieties

Algebraic Geometry and Commutative Algebra ◽

10.1016/b978-0-12-348032-3.50007-4 ◽

1988 ◽

pp. 469-502 ◽

Cited By ~ 7

Author(s):

Frans OORT

Keyword(s):

Abelian Varieties ◽

Endomorphism Algebras

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Isogenies Between K3 Surfaces Over

International Mathematics Research Notices ◽

10.1093/imrn/rnaa176 ◽

2020 ◽

Author(s):

Ziquan Yang

Keyword(s):

Abelian Varieties ◽

K3 Surfaces ◽

Main Step ◽

Shimura Varieties ◽

K3 Surface ◽

Perfect Field ◽

Finite Height ◽

Mild Assumption

Abstract We generalize Mukai and Shafarevich’s definitions of isogenies between K3 surfaces over ${\mathbb{C}}$ to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over $\bar{{\mathbb{F}}}_p$ by prescribing linear algebraic data when $p$ is large. The main step is to show that isogenies between Kuga–Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on $p$.

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