13 problems

Let $\varphi\in\mathcal{B}(q)$ be a Hecke–Maaß cusp form, and let $\lambda_{\varphi}(m)$ denote its Hecke eigenvalue for $(m,q)=1$. Ramanujan–Petersson conjecture. For every…

Let $k$ be a large even integer, let $\mathcal{H}_k$ be the set of holomorphic Hecke cusp forms of weight $k$ for $\operatorname{SL}(2,\mathbb{Z})$, and for $f\in\mathcal{H}_k$ let…

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…

Let $F$ be a Hecke–Maass cusp form for $SL(n,\mathbb{Z})$, with normalized Fourier coefficients $\mathcal{A}_{F}(m,1,\ldots,1)$. The coefficients are multiplicative. Generalized Ra…

Serre's Sato–Tate conjecture. For any $f\in S_k(\Gamma_0(N))$, as $x\to\infty$, the values $\lambda_p(f)$ for primes $p\leq x$ are equidistributed in $[-2,2]$ with respect to the S…

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 $m_p(0,k,\chi,N)$ denote the number of newforms $f\in S_{k,\chi}(N)$ with $a_p(f)=0$. Let $m_{p,\operatorname{CM}}(0,k,\chi,N)$ denote the number of CM newforms in…

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…

Let two generic Maass newforms have Hecke eigenvalues that coincide on a set of primes of density $\frac{1}{2}$. Density-half twist conjecture. They should be related to each other…