scholarly journals Classification of a family of abelian varieties parametrized by reduction modulo $\mathfrak {P}$ of a Shimura curve

1980 ◽  
Vol 56 (7) ◽  
pp. 338-341
Author(s):  
Yasuo Morita
Keyword(s):  
Abelian Varieties ◽  
Shimura Curve ◽  
Reduction Modulo

scholarly journals Classification of nonrigid families of abelian varieties

1993 ◽  
Vol 45 (2) ◽  
pp. 159-189
Author(s):  
Masa-Hiko Saitō
Keyword(s):  
Abelian Varieties

Trace of the Generating Series

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Eisenstein Series ◽  
Abelian Varieties ◽  
Discrete Series ◽  
Theta Series ◽  
Shimura Curves ◽  
Trace Identity ◽  
Shimura Curve ◽  
Schwartz Functions ◽  
Generating Series ◽  
New Space

This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the chapter considers the new space of Schwartz functions and constructs theta series and Eisenstein series from such functions. It proceeds by discussing discrete series at infinite places, modularity of the generating series, degree of the generating series, and the trace identity. It also presents the pull-back formula for the compact and non-compact cases. In particular, it describes CM cycles on the Shimura curve, pull-back as cycles, degree of the pull-back, and some coset identities.

scholarly journals Deformations of minimal cohomology classes on abelian varieties

2016 ◽  
Vol 18 (04) ◽  
pp. 1550066
Author(s):  
Luigi Lombardi ◽  
Sofia Tirabassi
Keyword(s):  
Cohomology Class ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Infinitesimal Deformations

We show that the infinitesimal deformations of Brill–Noether loci [Formula: see text] attached to a smooth non-hyperelliptic curve [Formula: see text] are in one-to-ne correspondence with the deformations of [Formula: see text]. As an application, we prove that if a Jacobian [Formula: see text] deforms together with a minimal cohomology class out the Jacobian locus, then [Formula: see text] is hyperelliptic. In particular, this provides an evidence toward a conjecture of Debarre on the classification of ppavs carrying a minimal cohomology class.

Classification of supersingular abelian varieties

1989 ◽  
Vol 283 (2) ◽  
pp. 333-351 ◽  
Author(s):  
Ke-Zheng Li
Keyword(s):  
Abelian Varieties

scholarly journals On compatibility between isogenies and polarizations of abelian varieties

2017 ◽  
Vol 13 (03) ◽  
pp. 673-704 ◽  
Author(s):  
Martin Orr
Keyword(s):  
Abelian Variety ◽  
Abelian Varieties ◽  
Shimura Varieties ◽  
Division Algebras ◽  
Hermitian Forms ◽  
Endomorphism Algebras

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.

scholarly journals Variation of Tamagawa numbers of semistable abelian varieties in field extensions

2018 ◽  
Vol 166 (3) ◽  
pp. 487-521
Author(s):  
L. ALEXANDER BETTS ◽  
VLADIMIR DOKCHITSER ◽  
V. DOKCHITSER ◽  
A. MORGAN
Keyword(s):  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Complete Classification ◽  
Local Fields ◽  
Selmer Groups ◽  
Genus 2 ◽  
Parity Conjecture ◽  
Tamagawa Numbers ◽  
Principally Polarised Abelian Varieties

AbstractWe investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on thep-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the formy2=f(x), under some simplifying hypotheses.

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