Analytic and Geometric Number Theory Seminar

The Ramanujan conjecture states that for a holomorphic cusp form $f(z) =\sum_{n \in N} \lambda_f(n)e(nz)$ of weight $k$, the coefficients $\lambda_f(n)$ satisfy the bound $|\lambda_f(n)| \ll_\epsilon n^{(k−1)/2+\epsilon}$. In the case where $k$ is an integer this is a celebrated theorem of Deligne which he proved by reducing to a case of the Weil conjectures. In the case of half-integral weight the conjecture remains wide open, though non-trivial bounds towards it have been established by Iwaniec and Duke. We will focus on the function field case $F_p(T)$, where we formulate and prove the analogue of the half-integral Ramanujan conjecture. Our proof makes use of Drinfeld’s results relating cusp forms on $GL(2)/F_p(T)$ to galois representations, as well as developing the Shimura correspondence and a Waldspurger formula in this function setting. As our main application, we give a solution (which is moreover effective) for representing elements in $F_p[T]$ by a given ternary quadratic form with coefficients in this ring. This is joint work with Ali Altug.