scholarly journals On the existence of abelian surfaces with everywhere good reduction

2021 ◽  
Author(s):  
Lassina Dembélé
Keyword(s):  
Good Reduction ◽  
Abelian Surfaces

ABELIAN SURFACES WITH SUPERSINGULAR GOOD REDUCTION AND NON-SEMISIMPLE TATE MODULE

2010 ◽  
Vol 06 (04) ◽  
pp. 811-818
Author(s):  
MAJA VOLKOV
Keyword(s):  
Good Reduction ◽  
Galois Module ◽  
Abelian Surfaces ◽  
Tate Module

We show the existence of abelian surfaces [Formula: see text] over ℚp having good reduction with supersingular special fiber whose associated p-adic Galois module [Formula: see text] is not semisimple.

scholarly journals Abelian surfaces good away from 2

2017 ◽  
Vol 13 (04) ◽  
pp. 991-1001
Author(s):  
Christopher Rasmussen ◽  
Akio Tamagawa
Keyword(s):  
Elliptic Curves ◽  
Number Field ◽  
High Dimension ◽  
Abelian Varieties ◽  
Number Fields ◽  
Sufficient Condition ◽  
Good Reduction ◽  
Abelian Surfaces ◽  
Special Case

Fix a number field [Formula: see text] and a rational prime [Formula: see text]. We consider abelian varieties whose [Formula: see text]-power torsion generates a pro-[Formula: see text] extension of [Formula: see text] which is unramified away from [Formula: see text]. It is a necessary, but not generally sufficient, condition that such varieties have good reduction away from [Formula: see text]. In the special case of [Formula: see text], we demonstrate that for abelian surfaces [Formula: see text], good reduction away from [Formula: see text] does suffice. The result is extended to elliptic curves and abelian surfaces over certain number fields unramified away from [Formula: see text]. An explicit example is constructed to demonstrate that good reduction away from [Formula: see text] is not sufficient, at [Formula: see text], for abelian varieties of sufficiently high dimension.

Genus 2 Curves with Quaternionic Multiplication

2008 ◽  
Vol 60 (4) ◽  
pp. 734-757 ◽  
Author(s):  
Srinath Baba ◽  
Håkan Granath
Keyword(s):  
Elliptic Curve ◽  
Modular Forms ◽  
Shimura Curves ◽  
Good Reduction ◽  
Abelian Surfaces ◽  
Quaternion Algebras ◽  
Rational Models ◽  
Genus 2 ◽  
Genus 2 Curves ◽  
Classical Construction

AbstractWe explicitly construct the canonical rational models of Shimura curves, both analytically in terms of modular forms and algebraically in terms of coefficients of genus 2 curves, in the cases of quaternion algebras of discriminant 6 and 10. This emulates the classical construction in the elliptic curve case. We also give families of genus 2 QMcurves, whose Jacobians are the corresponding abelian surfaces on the Shimura curve, and with coefficients that are modular forms of weight 12. We apply these results to show that our j-functions are supported exactly at those primes where the genus 2 curve does not admit potentially good reduction, and construct fields where this potentially good reduction is attained. Finally, using j, we construct the fields ofmoduli and definition for somemoduli problems associated to the Atkin–Lehner group actions.

scholarly journals Examples of abelian surfaces with everywhere good reduction

2015 ◽  
Vol 364 (3-4) ◽  
pp. 1365-1392 ◽  
Author(s):  
Lassina Dembélé ◽  
Abhinav Kumar
Keyword(s):  
Good Reduction ◽  
Abelian Surfaces

On the computation of the endomorphism rings of abelian surfaces

2021 ◽  
Author(s):  
Claus Fieker ◽  
Tommy Hofmann ◽  
Sogo Pierre Sanon
Keyword(s):  
Abelian Surfaces ◽  
Endomorphism Rings

scholarly journals RAPOPORT–ZINK UNIFORMIZATION OF HODGE-TYPE SHIMURA VARIETIES

2018 ◽  
Vol 6 ◽  
Author(s):  
WANSU KIM
Keyword(s):  
Shimura Varieties ◽  
Good Reduction ◽  
Canonical Models

We show that the integral canonical models of Hodge-type Shimura varieties at odd good reduction primes admits ‘$p$-adic uniformization’ by Rapoport–Zink spaces of Hodge type constructed in Kim [Forum Math. Sigma6(2018) e8, 110 MR 3812116].

Computing Galois representations of modular abelian surfaces

2014 ◽  
Vol 17 (A) ◽  
pp. 36-48 ◽  
Author(s):  
Jinxiang Zeng
Keyword(s):  
Abelian Variety ◽  
Efficient Algorithm ◽  
Galois Representations ◽  
Abelian Surfaces ◽  
Genus Two

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f\in S_2(\Gamma _0(N))$ be a normalized newform such that the abelian variety $A_f$ attached by Shimura to $f$ is the Jacobian of a genus-two curve. We give an efficient algorithm for computing Galois representations associated to such newforms.

Sign in / Sign up

Export Citation Format

Share Document