26 problems

Let $\mathcal O$ be a fixed closed orbit, and let $a_{n,\lambda}$ be the coefficients in the expansion associated with $\mathcal O$ and the representation parameter $\lambda$. Clos…

Nonvanishing conjecture. The first Fourier coefficient satisfies

Unbounded denominators conjecture. The Fourier coefficients of $f$ have unbounded denominators. Equivalently, for every positive integer $n$, the Fourier coefficients of $nf$ are n…

Generalized Atkin–Serre conjecture. Under these hypotheses, the stated lower bound holds for every nonzero Fourier coefficient $a_f(m)$. This extends the Atkin–Serre conjecture fro…

Let $A$ and $B$ be positive integers such that … Write $\mathcal{J}^*_{10}(m)$ for the $m$-th Fourier coefficient of $j^*_{10}(\tau)$. Asymptotic parity conjecture. There are such…

Let $D_1,D_2$ range over fundamental discriminants with $|D1D2|to obreakter x$, and let $\eta(D1,D2)$ be the smallest prime $p$ such that the sign of the corresponding Fourier coeff…

Let $k>4$ with $k\equiv 2$ modulo $4$. For every reduced matrix $T\in\halfint^+_2$ with bottom-right entry $m$, let $S_m(n,T)$ be the set of integers $r$ such that…

Let $F$ be a number field, let ${\mathrm {U}}_{n,n}$ be the $F$-quasisplit unitary group, and let $\pi$ be an irreducible cuspidal automorphic representation of…

Let $G_n$ be a unitary group of $F$-rank ${\mathfrak r}$ with $2{\mathfrak r}< {\mathfrak n}$, and let $G_n^*$ be its $F$-quasisplit pure inner form. Define … For a representation…

Let $G_n={\mathrm {U}}_{n+1,n-1}$ be defined over $F$ with $F$-rank $n-1$, and let $G_n^*={\mathrm {U}}_{n,n}$. Let $\phi$ be a generic global Arthur parameter of $G_n^*$ that is…

Let ${\mathrm {G}}_n$ be an $F$-split ${\mathrm{SO}}_{2n+1}$ or ${\mathrm{SO}}_{2n}$, and let $\frak{p}(\pi)$ denote the partitions attached to a cuspidal automorphic representatio…

1/4-BPS index conjecture. The index is

1/2-BPS index conjecture. The index is

Let $\tau$ be the datum in the descent diagram, let $m\leq n$, and define … Let $\mathfrak p(\mathcal E_{\tau\otimes\sigma})$ denote the relevant partitions for which the residual…

Fourier-support conjecture. Every $\underline p\in\mathfrak p(\mathcal E_{\tau\otimes\sigma})$ satisfies $\underline p\leq\underline p_{\tau;\sigma,H_m^*}$; if $H_m=H_m^*$ is $F$-q…

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Let $B$ be a definite quaternion algebra ramified at exactly one finite prime $p$, and let $N=pM$ with $p mid M$ be square-free. Write … For $D\in\mathbb{N}$, let $-D$ be a fundame…

Arthur-parameter conjecture. The following hold: (1) if $\underline{p}>\eta_{{\frak{g}^\vee,\frak{g}}}(\underline{p}(\psi))$, then $\underline{p}$ does not belong to…

Uniqueness conjecture. The set $\frak{p}^m(\pi)$ contains only one partition in both the local and automorphic cases.

Conjecture 4.2. For every $\psi\in\widetilde{\Psi}_2(\operatorname{Sp}_{2n})$:

Let $n_0(N,k)$ be the smallest nonnegative integer such that, for any newforms $f,g\in S_k(\Gamma_0(N))$, equality $a_n(f)=a_n(g)$ for all $n\leq n_0(N,k)$ implies $f=g$. Let…

Fourier-coefficient conjecture. The automorphic representation of $GL_2^{(r+1)}(\mathbb{A})$ generated by all the functions $f(g)$ is $\Theta_2^{(r+1)}$. The conjecture predicts th…

Average first negative value conjecture. As $x\to\infty$, one expects

Let $G$ be an $F$-quasisplit classical group, let $\psi\in\widetilde\Psi_2(G)$ be a global Arthur parameter, and let $\widetilde\Pi_\psi(\epsilon_\psi)$ be its automorphic $L^2$-pa…

Let $G$ be an $F$-quasisplit classical group, let $\psi\in\widetilde\Psi_2(G)$ be a global Arthur parameter, and let $\widetilde\Pi_\psi(\epsilon_\psi)$ be its automorphic $L^2$-pa…