Constants in Titchmarsh divisor problems for elliptic curves

Abstract

Inspired by the analogy between the group of units F × p of the finite field with p elements and the group of points E(Fp) of an elliptic curve E/Fp, E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum p≤x τ (p + a) ∼ Cx. In this paper, we present a comprehensive study of the constants C(E) emerging in the asymptotic study of these elliptic curve divisor sums. Specifically, by analyzing the division fields of an elliptic curve E/Q, we prove upper bounds for the constants C(E) and, in the generic case of a Serre curve, we prove explicit closed formulae for C(E) amenable to concrete computations. Moreover, we compute the moments of the constants C(E) over two-parameter families of elliptic curves E/Q. Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics. ,p (E) are uniquely determined positive integers satisfying d 1,p | d 2,p . Determining the asymptotic behaviour of sums over p ≤ x of arithmetic functions evaluated at the elementary divisors d 1,p and d 2,p may be viewed as Titchmarsh divisor problems for elliptic curves. Such problems unravel striking similarities, but also intriguing contrasts, to the original Titchmarsh divisor problem, as illustrated in [AkGh], [AkFe], [Co], [CoMu], [Fe], [FeMu], [FrKu], [FrPo], [Ki], [Ko], [Po], and [Wu].The focus of our paper is on the constants emerging in the following three Titchmarsh divisor problems for elliptic curves. In all the expressions below, the letters p and ℓ denote primes, with p being a prime of good reduction for the given elliptic curve. Conjecture 1. (Kowalski [Ko, Section 3.2])Let E/Q be an elliptic curve. Then, as x → ∞, 3

Elliptic curves over a finite field and the trace formula

“…The method of proof of Theorem also allows one to compute averages of several types of arithmetic functions of over , such as the divisor function, , or a Dirichlet character, . Such applications were suggested to us by Cojocaru at the Arizona Winter School 2016, see the related work of her project group . On the other hand, computing averages of arithmetic functions of over seems more challenging.…”

A bound for the conductor of an open subgroup of GL2 associated to an elliptic curve

“…Following Serre's open image result, there has been much interest in the nature of ρ E,K (G K ), for instance regarding its mod p reductions (see [10], [11], [9], [1], [2], and [21]) and also more recently its reductions at composite levels (see [5], [18] and [12]). In addition to this connection, Theorem 1.2 also has analytic relevance; for instance in [3] it is applied to the study averages of constants appearing in various elliptic curve conjectures.…”

“…Now assume that 4 | m G . In this case, when (3) does not hold, we insert the result of Lemma 4.5 into (22), concluding that…”