Rank and Bias in Families of Hyperelliptic Curves via Nagao's Conjecture

Abstract

Let $\mathcal{X} : y^2 = f(x)$ be a hyperelliptic curve over $\mathbb{Q}(T)$ of genus $g\geq 1$. Assume that the jacobian of $\mathcal{X}$ over $\mathbb{Q}(T)$ has no subvariety defined over $\mathbb{Q}$. Denote by $\mathcal{X}_t$ the specialization of $\mathcal{X}$ to an integer $T=t$, let $a_{\mathcal{X}_t}(p)$ be its trace of Frobenius, and $A_{\mathcal{X},r}(p) = \frac{1}{p}\sum_{t=1}^p a_{\mathcal{X}_t}(p)^r$ its $r$-th moment. The first moment is related to the rank of the jacobian $J_\mathcal{X}\left(\mathbb{Q}(T)\right)$ by a generalization of a conjecture of Nagao: $$\lim_{X \to \infty} \frac{1}{X} \sum_{p \leq X} - A_{\mathcal{X},1}(p) \log p = \operatorname{rank} J_\mathcal{X}(\mathbb{Q}(T)).$$ Generalizing a result of S. Arms, \'A. Lozano-Robledo, and S.J. Miller, we compute first moments for various families resulting in infinitely many hyperelliptic curves over $\mathbb{Q}(T)$ having jacobian of moderately large rank $4g+2$, where $g$ is the genus; by Silverman's specialization theorem, this yields hyperelliptic curves over $\mathbb{Q}$ with large rank jacobian. Note that Shioda has the best record in this directon: he constructed hyperelliptic curves of genus $g$ with jacobian of rank $4g+7$. In the case when $\mathcal{X}$ is an elliptic curve, Michel proved $p\cdot A_{\mathcal{X},2} = p^2 + O\left(p^{3/2}\right)$. For the families studied, we observe the same second moment expansion. Furthermore, we observe the largest lower order term that does not average to zero is on average negative, a bias first noted by S.J. Miller in the elliptic curve case. We prove this bias for a number of families of hyperelliptic curves.