1,226 problems

Selberg's eigenvalue conjecture. For every $j$, $\theta_j=0$.

Density conjecture. The Katz–Sarnak philosophy predicts

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 $\pi$ be an unramified cuspidal representation of $\operatorname{GL}_N$ and let $\alpha_1,\ldots,\alpha_N$ be its Satake parameters at a prime $\mathfrak{p}$. Generalized Raman…

Exact dimension formula. The dimension is

Shahidi's conjecture. Every tempered $L$-packet contains a generic member.

Improved error-term conjecture. In all terms above, the error term $o(X)$ can be replaced by $O(X^{1/2+\varepsilon})$ for any $\varepsilon>0$, where the implied constant only depen…

Church–Farb–Putman vanishing conjecture. If $i\leq n-2$, then

Mazur–Tate–Teitelbaum conjecture. One has $L_p(f,1)=0$ and

Let $G=\mathrm{PSL}_2(\mathbb{R})^d$ and let $\Gamma$ be a lattice in $G$. For a bounded family $\mathfrak{B}\subseteq\widehat{G}$ of non-tempered irreducible unitary representatio…

Let $X$ be a complex Shimura variety, let $X^{\text{BB}}$ be its Baily--Borel compactification, and write $IH^k(X^{\text{BB}};\mathbb{C})$ for its intersection cohomology and…

Let $E$ be the quadratic field under consideration, let $h$ be the Hecke algebra acting on the relevant cohomology, let $\mathfrak{m}\subset h$ be a non-Eisenstein maximal ideal, a…

Harder's conjecture. If $p_{\mathfrak{p}}>2k+j-2$ and $\mathfrak{p}\mid L^{\operatorname{alg}}(k+j,f)$, then there exist a Hecke cusp eigenform

Iwaniec–Sarnak conjecture. For every $\varepsilon>0$,

Let $\mathbb{F}_q$ be a finite field of characteristic $p$, let $\ell$ and $\ell'$ be distinct primes with $\ell,\ell'\neq p$, and let $X$ be a normal variety over $\mathbb{F}_q$.…

Let $\pi$ and $\pi'$ be irreducible cuspidal automorphic representations of $\operatorname{GL}(m)$ and $\operatorname{GL}(m')$ over $\mathbb{Q}$, respectively, both with unitary ce…

Let $piinmathfrak{F}m$ be an automorphic representation, and let $alpha{r,pi}(p)$ denote its Satake parameters at a prime $p$. Generalized Ramanujan–Petersson conjecture. The Sat…

Ramanujan–Selberg conjecture. The Langlands parameters satisfy

An abelian surface $A_{/\mathbf{Q}}$ has conductor $N$, and write $\operatorname{End}_{\mathbf{Q}} A$ for its endomorphism ring over $\mathbf{Q}$. A QM abelian surface is an abelia…

Let $\mathbb{H}:=\{x+iy:x\in\mathbb{R},y>0\}$, let $\Gamma=\operatorname{SL}(2,\mathbb{Z})$, and let $\mathbb{X}=\Gamma\backslash\mathbb{H}$ with measure…

Buzzard–Gee conjecture. Galois representations attached to $C$-algebraic automorphic forms should take values in a $C$-group.

Let $E/F$ be a CM extension, let $\Pi_n$ and $\Pi_{n+1}$ be RACSDC automorphic representations as above, and set $\Pi=\Pi_n\boxtimes\Pi_{n+1}$. Assume that their weights are perfec…

Lapid–Mao's local conjecture. For every $\eta\in E_v^\times$ such that $\bar\eta=-\eta$, one has

Let $H_g$ be the number field cut out by the Artin representation attached to the adjoint of $g$, and let $H_{g,p}$ be a completion of $H_g$ at a prime above $p$. Let…

Böcherer's conjecture. For every imaginary quadratic field $E$,