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.

GAUSSIAN LAWS ON DRINFELD MODULES

2009 ◽  
Vol 05 (07) ◽  
pp. 1179-1203 ◽  
Author(s):  
WENTANG KUO ◽  
YU-RU LIU
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Valuation Ring ◽  
Finite Extension ◽  
Drinfeld Modules ◽  
Good Reduction ◽  
Prime Divisors ◽  
Frobenius Automorphism ◽  
Tate Module ◽  
Rank 2

Let A = 𝔽q[T] be the polynomial ring over the finite field 𝔽q, k = 𝔽q(T) the rational function field, and K a finite extension of k. Let ϕ be a Drinfeld A-module over K of rank r. For a place 𝔓 of K of good reduction, write [Formula: see text], where [Formula: see text] is the valuation ring of 𝔓 and [Formula: see text] its maximal ideal. Let P𝔓, ϕ(X) be the characteristic polynomial of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ. Let χϕ(𝔓) = P𝔓, ϕ(1), and let ν(χϕ(𝔓)) be the number of distinct primes dividing χϕ(𝔓). If ϕ is of rank 2 with [Formula: see text], we prove that there exists a normal distribution for the quantity [Formula: see text] For r ≥ 3, we show that the same result holds under the open image conjecture for Drinfeld modules. We also study the number of distinct prime divisors of the trace of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ and obtain similar results.

scholarly journals Sato–Tate distributions and Galois endomorphism modules in genus 2

2012 ◽  
Vol 148 (5) ◽  
pp. 1390-1442 ◽  
Author(s):  
Francesc Fité ◽  
Kiran S. Kedlaya ◽  
Víctor Rotger ◽  
Andrew V. Sutherland
Keyword(s):  
Hyperelliptic Curves ◽  
Abelian Surface ◽  
Module Structure ◽  
Galois Module ◽  
Tate Group ◽  
Usp 4 ◽  
Galois Action ◽  
Tate Module ◽  
Genus 2

AbstractFor an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato–Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato–Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the ℝ-algebra generated by endomorphisms of $A_{{\overline {\mathbb Q}}}$ (the Galois type), and establish a matching with the classification of Sato–Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato–Tate groups for suitable A and k, of which 34 can occur for k=ℚ. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over ℚ whenever possible), and observe numerical agreement with the expected Sato–Tate distribution by comparing moment statistics.

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
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