58 research outputs found
A non-split sum of coefficients of modular forms
We shall introduce and study certain truncated sums of Hecke eigenvalues of
-automorphic forms along quadratic polynomials. A power saving estimate
is established and new applications to moments of critical -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 be a Hecke--Maass cuspidal newform of square-free level and
Laplacian eigenvalue . It is shown that \pnorm{f}_\infty
\ll_{\lambda,\epsilon} N^{-1/6}+\epsilon} \pnorm{f}_2 for any
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 , which is the subfield of \C fixed under the
subgroup of \Aut(\C) stabilizing the isomorphism class of the finite part of
. For general linear groups and classical groups, our first main result is
the finiteness of the set of discrete automorphic representations such
that is unramified away from a fixed finite set of places,
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 -packet under mild conditions
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