25 problems

Generalized Riemann Hypothesis. The non-trivial zeros of $L(s,f)$ all lie on the critical line

Let $n$ and $k$ satisfy $k>n+1$, and let $f\in S_{2k-n}(\Gamma_1)$ be a normalized elliptic Hecke eigenform. Let $L(s,F;st)$ denote the standard $L$-function of a Hecke eigenform…

Ikeda's period conjecture. One has

Let $k$ be such that the indicated spaces are defined, and let $f \in S_{2k-4}(\operatorname{SL}_2(\mathbb{Z}))$ and $g \in S_k(\operatorname{SL}_2(\mathbb{Z}))$ be normalized Heck…

Consider a reasonable family of $L$-functions for which all functional equations have even sign and for which there is no corresponding family with odd functional equations. Folklo…

Let $F$ be the underlying local field, let $G_m=GL_m(F)$, and let $N$ and $\psi$ be the subgroup and character defining the twisted Jacquet module. Let $\pi$ be an irreducible smoo…

Let $G$ be a reductive $p$-adic group and let $\phi$ be a Langlands parameter, with associated L-packet $\Pi_\phi(G)$. A representation is generic if it admits a Whittaker model. G…

Let $bbG$ be a connected reductive group over $bbF$, let $bbphi$ be an $bbSL_2$-parameter, and let $L(s,bboperatorname{Ad}\circbbphi)$ denote its adjoint $L$-factor. Gross–Prasad–R…

Gross–Prasad–Rallis conjecture. A Langlands parameter is generic if and only if its adjoint $L$-function is holomorphic at $s=1$.

Let $(G,H)$ be a strongly tempered spherical pair that is the Whittaker induction of $(G_0,H_0,\xi)$, with $G$ admitting a quasi-split pure inner form $G_{qs}$. Assume that…

Gross-Prasad--Rallis conjecture. The parameter $phi$ is generic if and only if

Langlands' Hecke theory conjecture. The Hecke theory should hold for the general automorphic $L$-functions $L(s,\pi,\rho)$.

Let $(M_A,N_A)$ be a relevant pair of global A-parameters satisfying the automorphy conditions for the representation $\pi_\chi$, and let $F(\nu)$ be the regularized period integra…

Let $M_A\times N_A$ be a discrete A-parameter of the quasi-split group $G(\mathbb{A})=G(V)(\mathbb{A})\times G(W)(\mathbb{A})$, with decompositions … … Let $(I\times J)^\heartsuit$…

Let $F$ be a number field, let $\mathbb A_F$ denote its ring of adeles, and let $\pi$ be a cuspidal automorphic representation of ${\rm GL}_n(\mathbb A_F)$ with $n>2$. The adjoint…

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…

Let $\mathcal S$ be the Selberg class, and let $d_F$ denote the degree of $F\in\mathcal S$. An $L$-function is called automorphic when it arises from an automorphic representation…

Let $F$ be a totally real number field of degree $n$, let $E/F$ be a CM extension, and for each CM type $\Phi\in\Phi(E)$ let $X_{\Phi}$ be a CM abelian variety of type…

Special case of global Gan–Gross–Prasad. If there is an automorphic form $\phi$ in the space of $\pi$ such that $B(\phi,\Lambda)\neq0$, then

Refined Böcherer conjecture. For every such $K$ and $\Lambda$,

Orthogonal-family moment conjecture. For $k\equiv0\pmod 4$,

Let $f$ be a cusp form with Fourier coefficients $c(n)$, and let … be its associated $L$-function. A Riemann hypothesis for $L(s,f)$ asserts that its non-trivial zeros lie on the c…

Tempered $L$-function conjecture. All the functions $L(s,\sigma,r_i)$ are holomorphic for $\operatorname{Re}(s)>0$ whenever $\sigma$ is tempered.

Let $f$ be a $p$-new eigenform, let $\mathfrak{p}$ be a prime above $p$, let $V_{\mathfrak{p}}(f)$ be the corresponding $p$-adic representation, let ${\Greekmath 011F}$ be a charac…

Let $k$ be an even positive integer. Consider Hecke eigenforms in $S_{2k-2}(\Gamma_1)$ and Hecke eigenforms in $M_k(\Gamma_2)$, with their spinor zeta functions. Kurokawa–Saito con…