A Titchmarsh divisor problem for elliptic curves

Abstract

Let E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that $$\begin{equation} \sum_{\substack{p \leq x \\p \equiv 1\pmod{4}}} \tau(\#E({\bf{F}}_p)) \sim \left(\frac{5\pi}{16} \prod_{p > 2} \frac{p^4-\chi(p)}{p^2(p^2-1)}\right)x, \quad\text{as $x\to\infty$}. \end{equation}$$ Here χ is the nontrivial Dirichlet character modulo 4. The proof uses number field analogues of the Brun–Titchmarsh and Bombieri–Vinogradov theorems, along with a theorem of Wirsing on mean values of nonnegative multiplicative functions.Now suppose that E/Q is a non-CM elliptic curve. We conjecture that the sum of τ(#E(Fp)), taken over p ⩽ x of good reduction, is ~cEx for some cE > 0, and we give a heuristic argument suggesting the precise value of cE. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that this sum is ≍Ex. The proof uses combinatorial ideas of Erdős.

Non-CM elliptic curves with infinitely many almost prime Frobenius traces

“…In particular, it draws on the analogy between the interpretation of the divisibility m | a p (E) for some non-zero integer m as a Chebotarev condition at p in the division field Q(E[m]) of E and the interpretation of the divisibility m | (p + a) for some non-zero integer m as a Chebotarev condition at p in the cyclotomic field Q(ζ m ). For example, in light of this analogy, analytic methods, such as Turán's normal order method and Titchmarsh' conditional approach to the divisor problem for p + a, have already been used in the study of the divisors of a p (E) in works such as [GuMu14], [MuMu84], and [Po16].…”

ECM and the Elliott–Halberstam conjecture for quadratic fields

“…Conjecture 1 (parametric EH, [LWX20], [Pol16] 1 ). Let δ(x) be a decreasing function such that (log 2 x)/(η log x) ≤ δ(x) < η (x ≥ x 0 (η)), (2) for any η ∈ (0, 1/2] Let K be either Q or an imaginary quadratic field of class number one.…”

“…Huxley [Hux71] proved a number field variant of the BV theorem, so it was natural to make a number field EH conjecture which states Huxley's result for the same bound on q as in the original EH conjecture. Pollack [Pol16] wrote the statement of the conjecture in the case of imaginary quadratic fields of class number one. Finally, Liu et al [LWX20] extended EH by replacing δ with a decreasing function.…”

Two generalisations of the Titchmarsh divisor problem

“…(i) Very recently Xi [18] considered a quadratic analogue of (1.1) and established the following inequalities (ii) Pollack [11] studied analogues of (1.1) for elliptic curves and proved the following two results:…”