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School Colloquium——Stable reduction of algebraic curves and abelian varieties
北大 algebraic curves abelian varieties
2023/6/16
The stable reduction theorem was proved by Grothendieck for Abelian varieties and subsequently by Deligne and Mumford for projective curves.
Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem
Isogeny expander graph hyperelliptic curve cryptography
2017/2/20
Fix an ordinary abelian variety defined over a finite field. The ideal class group of its endomorphism ring acts freely on the set of isogenous varieties with same endomorphism ring, by complex multip...
Isogeny graphs of ordinary abelian varieties
Jacobians of hyperelliptic curves genus 2 cryptography isogeny graphs
2016/12/8
Fix a prime number ГГ. Graphs of isogenies of degree a power of ГГ are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ord...
ESSENTIAL DIMENSION OF ABELIAN VARIETIES OVER NUMBER FIELDS
ESSENTIAL DIMENSION OVER NUMBER FIELDS
2015/9/29
We affirmatively answer a conjecture in the preprint “Essential
dimension and algebraic stacks,” proving that the essential dimension of an
abelian variety over a number field is inʂ...
ABELIAN VARIETIES OVER LARGE ALGEBRAIC FIELDS WITH INFINITE TORSION
ABELIAN VARIETIES OVER LARGE ALGEBRAIC FIELDS INFINITE TORSION
2015/8/26
Let A be a non-zero abelian variety defined over a number field K and let K be a fixed algebraic closure of K. For each element σ of the absolute Galois group Gal(K/K), let K(σ) be the fixed field in ...
HIGHER-LEVEL CANONICAL SUBGROUPS IN ABELIAN VARIETIES
ABELIAN VARIETIES CANONICAL SUBGROUPS
2015/7/6
An abeloid space over a rigid space S over a non-archimedean field k is a proper
smooth S-group A → S with connected fibers. Relative ampleness (in the sense of [C3]) gives a good notion
...
A generalisation of Miller's algorithm and applications to pairing computations on abelian varieties
Pairings Abelian varieties
2013/4/18
In this paper, we use the theory of theta functions to generalize to all abelian varieties the usual Miller's algorithm to compute a function associated to a principal divisor. We also explain how to ...
On Colmez's product formula for periods of CM-abelian varieties
Fermat curve stable reduction product formula
2011/8/25
Abstract: Colmez conjectured a product formula for periods of abelian varieties with complex multiplication by a field K, analogous to the standard product formula in algebraic number theory. He prove...
Homomorphisms of abelian varieties over geometric fields of finite characteristic
Homomorphisms of abelian varieties geometric fields of finite characteristic
2011/2/25
We study analogues of Tate’s conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of ab...
Motivic zeta functions for degenerations of abelian varieties and Calabi-Yau varieties
Motivic zeta functions degenerations of abelian varieties Calabi-Yau varieties
2011/2/25
Let f ∈ Z[x1, . . . , xn] be a non-constant polynomial, and let p be a prime. Igusa’s p-adic zeta function Zp f (s) is a meromorphic function on the complex plane that encodes the number of solutions ...
Artin representations attached to pairs of isogenous abelian varieties
Artin representations pairs of isogenous abelian varieties
2011/2/21
Given a pair of abelian varieties defined over a number field k and isogenous over a finite Galois extension L/k, we define a rational Artin representation of the group Gal(L/k) that shows a global re...
On The Characteristic Polynomial of Frobenius of Supersingular Abelian Varieties Of Dimension up to 7 over Finite Fields
The Characteristic Polynomial Frobenius Supersingular Abelian
2010/11/15
In this article, we derive the list of the characteristic polynomials of the Frobenius endomorphism of simple supersingular abelian varieties of dimension $1,~2,~3,~4,~5,~6,~7$ over $\mathbb{F}_q$ whe...
A Grunwald-Wang type theorem for abelian varieties
Grunwald-Wang type theorem abelian varieties
2010/12/8
Let A be an abelian variety over a number field k. We show that weak approxima-tion holds in the Weil-Chˆatelet group, H1(k,A), but that it may fail when one restricts to the n-torsion subgroup.
We prove a strong form of the motivic monodromy conjecture for abelian varieties, by showing that the order of the unique pole of the motivic zeta function is equal to the size of the maximal Jordan b...
On the security of pairing-friendly abelian varieties over non-prime fields
pairing-friendly abelian non-prime fields pairing-based cryptosystem
2009/6/9
Let A be an abelian variety defined over a non-prime finite
field Fq that has embedding degree k with respect to a subgroup of prime
order r. In this paper we give explicit conditions on q, k, and r...