Ranks of elliptic fibrations

In the 1950s, Erdős developed a method to estimate the average of the divisor function over the values of an integer polynomial. Nair and Tenenbaum later extended this to a substantially general class of arithmetic functions. In 1993 Heath-Brown used character sums to study the average size of the 2-Selmer group in $ty^2=x^3-x$. Combining these approaches, we prove that all exponential moments of the rank of $P(t)y^2=x^3-x$ are bounded. This is joint work with Peter Koymans and Carlo Pagano.