Rank statistics for a family of elliptic curves over a function field (Q2268326)

The authors consider the parametric family of elliptic curves \(E_d : y^2 + xy = x^3 - t^d\) over the function field \(\mathbb{F}_q(t)\), where \(d\) is a positive integer. Let \(\mathcal{U}_p\) be the set of positive integers dividing some member of the sequence \(p^n+1\), \(R_q(d)\) be the rank of \(E_q(d)\) over \(\mathbb{F}_q(t)\). The main results of the paper are the following Theorem 1. There exists an absolute constant \(\alpha > 1/2\) such that for all finite fields \(\mathbb{F}_q\) and all sufficiently large values of \(x\) (depending only on the characteristic \(p\) of \(\mathbb{F}_q\)) \[ \frac{1}{x}\sum_{d \leq x} R_q(d) \geq x^\alpha. \] Moreover, for \(x\) sufficiently large depending on \(q\), \[ \left( \sum_{d \leq x,d \in \mathcal{U}_p} \right)^{-1}\sum_{d \leq x,d \in \mathcal{U}_p} R_q(d) \leq x^{1-(\log\log\log x)/(2\log\log x)}. \] Theorem 2. Let \(\mathbb{F}_q\) be a finite field of characteristic \(p\) and let \(\varepsilon > 0\) be arbitrary. As \(x \to \infty\), except for \(o_{p,\varepsilon}(x)\) values of \(d\leq x\), we have \[ R_d(x) \geq (\log d)^{(1/3-\varepsilon) \log\log\log d}. \] It follows that the average and typical ranks in the family \(\{E_d\}\) tend to \(\infty\) as \(d \to \infty\). The key ingredients of the proofs are: the results of \textit{D. Ulmer} [Ann. Math. (2) 155, No. 1, 295--315 (2002; Zbl 1109.11314)], Bombieri-Vinogradov theorem, a version of Chebotarev density theorem due to Lagarias and Odlyzko, and subtle estimates based on the results of Erdös, Pomerance and Schmutz on Euler function and Carmichal numbers.