41 problems

Let $n$, $k$, and $N$ be positive integers, let $r$ be an even integer, and let $p$ be a prime. Let $\chi$ be a quadratic Dirichlet character modulo $N$ satisfying…

Let $m\geq 1$, and let $f$ and $g$ be Siegel modular forms of genus $2m$, of weights $k>2m$ and $l=k-2m$, with Satake parameters $(\alpha_0,\alpha_1,\ldots,\alpha_{2m})$ and…

Let $f$ and $g$ be Siegel modular forms of genus $2$, of weights $k>4$ and $l=k-2$, with Satake parameters $(\alpha_0,\alpha_1,\alpha_2)$ and $(\beta_0,\beta_1,\beta_2)$, respectiv…

Let $(l,m)$ be regular. The endoscopic contribution $e_{\rm endo}({\mathcal A}_2,\mathbb V_{l,m})$ is defined using the cohomology of the moduli space ${\mathcal A}_2$ of principal…

Let $k$ and $n$ be natural numbers with $k-n$ even. Let $f\in S_{2k}(\Gamma_1)$ be a normalized Hecke eigenform, and let $F_{2n}\in S_{k+n}(\Gamma_{2n})$ be the Ikeda lift of $f$.…

Positive-density refinement. A positive answer to the Fourier-coefficient growth conjecture holds when the growth condition is imposed on a set $S$ with $\delta_+(S)>0$. The paper…

Let $\rho$ be a representation and let $\widetilde{M}_n(k,\rho,\widetilde{\Gamma})$ denote the corresponding space of vector-valued Siegel modular forms. Vector-valued generalizati…

Let $c\in\mathbf R$, let $k_0\in\mathbf N$ be a scalar weight, and let $F\in M_k^n(\Gamma)$, where $\Gamma\subset\operatorname{Sp}_n(\mathbf Z)$ is a congruence subgroup. For all…

Let $\Gamma_n=\operatorname{Sp}_n(\mathbf Z)$, let $T$ be a positive-definite index for the Siegel Poincaré series $P_{T,\Gamma_n}$, and assume the usual parity condition for scala…

Level-aspect sup-norm conjecture. With $F$ and $\Gamma$ as above, as $N\to\infty$,

Weight-aspect sup-norm conjecture. If $F$ is an eigenfunction of all Hecke operators, then as $k\to\infty$,

Let $n\geq 1$, let $\Gamma\subset\operatorname{Sp}_n(\mathbf Z)$ be a congruence subgroup of level $N$, and let $S^{(n)}_k(\Gamma)$ and $\mathbb B_{k,\Gamma}(Z)$ be as above. Level…

Weight-aspect Bergman-kernel size conjecture. As $k\to\infty$,

Let $k$ and $\nu$ be even integers with $k>4$. Let … be a normalized Hecke eigenform, meaning that $a(1,f)=1$. Let $L(k-1,f,\mathrm{St})$ denote the relevant standard $L$-function,…

Holomorphic QUE conjecture. Fix a bounded continuous function $\phi$ on $\operatorname{Sp}_{2n}(\mathbb Z)\backslash\mathbb H_n$. If $F\in S_k(\operatorname{Sp}_{2n}(\mathbb Z))$ t…

Let $F\in S_k(\Gamma^{(2)}_0(N))$ be a Hecke eigenform at good primes that is non-CAP, meaning that it is of general type or Yoshida type. For each prime $p$ with $\gcd(p,N)=1$, le…

Bergman-kernel growth conjecture. With this notation,

Let $S_k^n$ denote the space of Siegel cusp forms of degree $n$ and weight $k$, and let $F\in S_k^n$ be an $L^2$-normalised Hecke eigenform. Write $\|F\|_\infty$ for its sup-norm.…

Let $N\geq 1$ be square-free, let $\chi$ be a quadratic character mod $N$, let $j\geq 0$ and $k\geq 3$. Suppose $f\in S_{j+k}(\Gamma_0^{(1)}(N),\chi)$ is an eigenform with…

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

Let $N_1$ and $N_2$ be coprime squarefree positive integers, with $\omega(N_1)$ odd, and let $E_4$ be the weight-$4$ Eisenstein series on $\operatorname{SL}_2(\mathbb Z)$. Squarefr…

Let $G$ be the relevant inner form of $\operatorname{SO}(5)$, let $K$ be the maximal compact subgroup in the notation of the SO(5) mass formula, and let $S=S_1\sqcup S_2\sqcup S_3$…

Let $D$) be the quaternion algebra over $\mathbb{Q}$ that is non-split over $\mathbb{R}$ and $\mathbb{Q}_p$ for a prime $p$ and split at all other places. Let $G=GU_2(D)$ be the as…

Non-vanishing conjecture. The Rankin convolution $R(k+n+\frac{r-1}{2},H,\bar g)$ is non-vanishing. The series is known to converge absolutely when $2n>r+2$, but its non-vanishing i…

Let $k$ be positive and even, let $f \in S_{2k-2(n+r)}(\operatorname{SL}_2(\mathbb{Z}))$ be a normalized Hecke eigenform, let $F^{(2n+2r)}$ be its Ikeda lift, and let…