Rank and bias in families of hyperelliptic curves via Nagao's conjecture

Abstract

Let X:y2=f(x) be a hyperelliptic curve over Q(T) of genus g≥1. Assume that the jacobian of X over Q(T) has no subvariety defined over Q. Denote by Xt the specialization of X to an integer T=t, let aXt(p) be its trace of Frobenius, and let AX,r(p)=1p∑t=1paXt(p)r be its r-th moment. The first moment is related to the rank of the jacobian JX(Q(T)) by a generalization of a conjecture of Nagao:limX→∞⁡1X∑p≤X−AX,1(p)log⁡p=rankJX(Q(T)). Generalizing a result of S. Arms, Á. Lozano-Robledo, and S.J. Miller, we compute first moments for various families resulting in infinitely many hyperelliptic curves over 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 Q with large rank jacobian. Note that Shioda has the better record in this direction: he constructed hyperelliptic curves of genus g with jacobian of rank 4g+7. In the case when X is an elliptic curve, Michel proved p⋅AX,2=p2+O(p3/2). 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.