7 problems

Let $f\in H_k(R(1))$, let $\psi\in C_0^\infty(0,\infty)$, and let $q\in\operatorname{\mathbb{Q}}[x]$ be an irreducible quadratic integer-valued polynomial. Nelson's quadratic Hecke…

Fix an integer $N\geq 2$. Let $\ell_1,\ell_2,\ldots,\ell_N$ be integers greater than one, and let $\eta_1,\eta_2,\ldots,\eta_N$ be distinct Maass forms, with $\eta_j$ a Maass form…

First negative eigenvalue conjecture. For every $\epsilon>0$, one has

Let $(\varphi_k)_k$, $(Q_k)_k$, and $(f_k)_k$ satisfy the hypotheses of the quadratic Hecke-sum conjecture: the forms $\varphi_k$ are non-dihedral, the $Q_k$ are irreducible intege…

Let $(\varphi_k)_k$ be the sequence of holomorphic Hecke eigenforms under consideration, with Hecke eigenvalues $\lambda_{\varphi_k}(n)$. Let $\mathcal{S}_N$ denote the Sobolev nor…

Let $n_0(q,T)$ be the least integer such that two nonmonomial Maaß newforms of weight $0$, level $q$, and principal character, with Laplacian eigenvalues in $(1/4,T]$ and identical…

Let $q$ be a positive squarefree integer, let $k$ be a positive even integer, and let $n_0(q,k)$ be the least integer such that equality of Hecke eigenvalues at all primes…