A classical question asks: consider the set of elliptic curves $E/Q$ with conductor less than or equal to $n$. Do the root numbers of the corresponding $L$-functions $L(s,E)$ equidistribute between $\pm 1$ as $n \to \infty$? Assuming the BSD conjecture, this determines which fraction of elliptic curves have even or odd-rank groups of rational points.
We discuss an automorphic-side variant of this question generalized to higher rank. Specifically, consider the set of self-dual cuspidal automorphic representations on GL_n with specific weight at infinity and conductor. We show that for most conductors, the root numbers of the corresponding $L$-functions equidistribute between $\pm 1$ as the weight goes to infinity and exactly classify the conductors for which such equidistribution doesn't hold. We also prove similar results in the conjugate self-dual setting and for the corresponding families of Galois representations whenever they are known to exist. |