# Constants in Titchmarsh divisor problems for elliptic curves

## TL;DR

This paper investigates the constants arising in the asymptotic analysis of elliptic curve divisor sums, providing bounds, explicit formulas for generic cases, and moments over families of elliptic curves, enhancing understanding of their individual and average behaviors.

## Contribution

It offers bounds and explicit formulas for the constants in elliptic divisor sums, advancing the analysis of their asymptotic and individual properties.

## Key findings

- Upper bounds for constants C(E) are established.
- Explicit formulas for C(E) are derived for Serre curves.
- Moments of C(E) are computed over families of elliptic curves.

## Abstract

Inspired by the analogy between the group of units $\mathbb{F}_p^{\times}$ of the finite field with $p$ elements and the group of points $E(\mathbb{F}_p)$ of an elliptic curve $E/\mathbb{F}_p$, 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 $\sum_{p \leq x} \tau(p + a) \sim C x$. 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 $E/\mathbb{Q}$, 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 $E/\mathbb{Q}$. 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.