A non-split sum of coefficients of modular forms

We shall introduce and study certain truncated sums of Hecke eigenvalues of $GL_2$-automorphic forms along quadratic polynomials. A power saving estimate is established and new applications to moments of critical $L$-values associated to quadratic fields are derived. An application to the asymptotic behavior of the height of Heegner points and singular moduli is discussed in details

The mirror conjecture for minuscule flag varieties

We prove Rietsch's mirror conjecture that the Dubrovin quantum connection for minuscule flag varieties is isomorphic to the character D-module of the Berenstein-Kazhdan geometric crystal. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. We reveal surprising relations with the works of Frenkel-Gross, Heinloth-Ng\^o-Yun and Zhu on Kloosterman sheaves. The isomorphism comes from global rigidity results where Hecke eigensheaves are determined by their local ramification. As corollaries we obtain combinatorial identities for counts of rational curves and the Peterson variety presentation of the small quantum cohomology ring

On the sup-norm of Maass cusp forms of large level. III

Let $f$ be a Hecke--Maass cuspidal newform of square-free level $N$ and Laplacian eigenvalue $\lambda$. It is shown that \pnorm{f}_\infty \ll_{\lambda,\epsilon} N^{-1/6}+\epsilon} \pnorm{f}_2 for any $\epsilon>0$

Sato-Tate equidistribution for families of Hecke-Maass forms on SL(n,R)/SO(n)

We establish the Sato-Tate equidistribution of Hecke eigenvalues on average for families of Hecke--Maass cusp forms on SL(n,R)/SO(n). For each of the principal, symmetric square and exterior square L-functions we verify that the families are essentially cuspidal and deduce the level distribution with restricted support of the low-lying zeros. We also deduce average estimates toward Ramanujan

On Fields of rationality for automorphic representations

This paper proves two results on the field of rationality \Q(\pi) for an automorphic representation $\pi$, which is the subfield of \C fixed under the subgroup of \Aut(\C) stabilizing the isomorphism class of the finite part of $\pi$. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations $\pi$ such that $\pi$ is unramified away from a fixed finite set of places, $\pi_\infty$ has a fixed infinitesimal character, and [\Q(\pi):\Q] is bounded. The second main result is that for classical groups, [\Q(\pi):\Q] grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed $L$-packet under mild conditions