A Geometric Variant of Titchmarsh Divisor Problem

Abstract

Abstract. We formulate a geometric analogue of the Titchmarsh Divisor Problem in the context of abelian varieties. For any abelian variety A defined over Q, we study the asymptotic distribution of the primes of Z which split completely in the division fields of A. For all abelian varieties which contain an elliptic curve we establish an asymptotic formula for such primes under the assumption of GRH. We explain how to derive an unconditional asymptotic formula in the case that the abelian variety is a CM elliptic curve.

Artin's Primitive Root Conjecture – A Survey

“…In this setup the problem specializes to the Titchmarsh divisor problem for Abelian varieties and it is analogous to the classical case for τ (p − 1). Akbary and Ghioca [5] believe that in this case …”

Cyclicity and Titchmarsh divisor problem for Drinfeld modules

“…where the sum is over monic square-free polynomials m of A. If ℘ splits completely in F (φ[m]), then from Lemma 3.3 we obtain that m r | P φ,℘ (1). Since deg…”

“…Finally, the results regarding Serre's cyclicity question from [15], [2], and [11] were extended to arbitrary abelian varieties defined over number fields in [18] and to arbitrary generic Drinfeld A-modules in this article. We remark that Theorem 1.2 is an analogue of the Titchmarsh divisor problem for Drinfeld modules of rank r ≥ 2 (see [1], [17] for details). We remark that the methods of this article could be used to generalize [1] and [5], where the authors were able to prove their results only for the very particular case when the abelian variety A from [1] is defined over Q and contains an abelian subvariety E of dimension 1 also defined over Q (see [1, Theorem 1.2 and Remark 4.1] and also [5, the last sentence of Section 1.1], where the authors say that they can prove their results only for "abelian varieties defined over Q which have a 1-dimensional subvariety which is also defined over Q").…”

Average of the first invariant factor of the reductions of abelian varieties of CM type

“…As it was pointed out by [AG,(4.5 We turn our interest to the average behavior of d 1 (p). Now, we consider the case g ≥ 2.…”

“…Instead, we look for the density of primes p which A(F p ) have d 1 (p) = 1. Applying R. Murty's framework for abelian varieties, A. Akbary and D. Ghioca (see [AG,Theorem 1.4]) obtained the analogous theorem for abelian varieties: Let A be an abelian variety defined over Q, and assume that the GRH holds for each extension Q(A[m])/Q. Then the number of primes p ≤ x such that d 1 (p) = 1 satisfies the asymptotic formula…”