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Register a new userAbelian $n$-division fields of elliptic curves and Brauer groups of product Kummer & abelian surfaces
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Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $text{Br}, Y/ text{Br}_1, Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Neron-Severi lattices. Over a field of characteristic $0$, we prove that the existence of a strong uniform bound on the size of the odd-torsion of $text{Br}, Y / text{Br}_1, Y$ is equivalent to the existence of a strong uniform bound on integers $n$ for which there exist non-CM elliptic curves with abelian $n$-division fields. Using the same methods we show that, for a fixed prime $p$, a number field $k$ of fixed degree $r$, and a fixed discriminant of the geometric Neron-Severi lattice, $(text{Br}, Y / text{Br}_1, Y)[p^infty]$ is bounded by a constant that depends only on $p$, $r$, and the discriminant.
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Let $E$ be an elliptic curve without CM that is defined over a number field $K$. For all but finitely many nonarchimedean places $v$ of $K$ there is the reduction $E(v)$ of $E$ at $v$ that is an elliptic curve over the residue field $k(v)$ at $v$. The set of $v$s with ordinary $E(v)$ has density 1 (Serre). For such $v$ the endomorphism ring $End(E(v))$ of $E(v)$ is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers $N$ and $M$ there are infinitely many nonarchimedean places $v$ of $K$ such that the discriminant $Delta(v)$ of $End(E(v))$ is divisible by $N$ and the ratio $Delta(v)/N$ is relatively prime to $NM$. We also discuss similar questions for reductions of abelian varieties. The subject of this paper was inspired by an exercise in Serres Abelian $ell$-adic representations and elliptic curves and questions of Mihran Papikian and Alina Cojocaru.
Let $k$ be a field of characteristic $q$, $cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$ or $>2d+1$. Let $p e q$ be a prime number and $cactocac$ a finite geometrically textsc{Galois} and etale cover defined over $k$ with function field $K:=k(cac)$. Let $(tau,B)$ be the $K/k$-trace of $A/K$. We give an upper bound for the $bbz_p$-corank of the textsc{Selmer} group $text{Sel}_p(Atimes_KK)$, defined in terms of the $p$-descent map. As a consequence, we get an upper bound for the $bbz$-rank of the textsc{Lang-Neron} group $A(K)/tauB(k)$. In the case of a geometric tower of curves whose textsc{Galois} group is isomorphic to $bbz_p$, we give sufficient conditions for the textsc{Lang-Neron} group of $A$ to be uniformly bounded along the tower.
We discuss a non-computational elementary approach to a well-known criterion of divisibility by 2 in the group of rational points on an elliptic curve.
We prove new results on splitting Brauer classes by genus 1 curves, settling in particular the case of degree 7 classes over global fields. Though our method is cohomological in nature, and proceeds by considering the more difficult problem of splitting $mu_N$-gerbes, we use crucial input from the arithmetic of modular curves and explicit $N$-descent on elliptic curves.
Given an abelian variety over a number field, its Sato-Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the L-function of the abelian variety. It was previously shown by Fite, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato-Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato-Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato-Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimuras theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations in Gap, SageMath, and Magma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated L-functions.
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