Let $E$ be an elliptic curve defined over $\bQ$. Its quadratic twists are denoted by $E_D$. In 1960, Honda made a surprising conjecture that $\rank_\bZ E_D(\bQ)$ is bounded as $D$ varies over all integers. At present, there is no evidence for or against this

conjecture.

Recently, Rubin and Silverberg derived an equivalent formulation of Honda's conjecture. Given an elliptic curve $E$, they construct certain infinite series related to $E$. They show that the ranks of elliptic curves in a family of quadratic twists are bounded if and only if these series converge.

Their theorem suggests the study of the cognate series for hyperelliptic curves. In this case, the convergence is an immediate consequence of a conjecture of Caporaso, Harris and Mazur. We prove this unconditionally applying Mumford's gap principle. This is a joint work with Ram Murty.