72 problems

Let $k \geqslant 4$ be an even integer. For the Eisenstein series $E_{k,5}^{*}(z)$ associated with the Fricke group $\Gamma_0^{*}(5)$, let $\mathbb{F}^{*}(5)$ be the fundamental do…

Let $G$ be a split simple adjoint group over a number field. At certain points of reducibility, consider the spherical constituents of principal series representations of $G$. Arth…

Let $M$ be the parameter governing the supports of the test functions $\phi_j^{\scriptscriptstyle{(M)}}$, and let $\delta$ denote the exponent obtained in the preceding estimate fo…

Let $G$ be a reductive group, let $P\subset G$ be a standard parabolic with Levi subgroup $M$, and let $\bP\subset \bG$ be the corresponding parabolic in the Langlands dual group w…

Laumon's compatibility conjecture. The restriction of $\Eis^n_{n_1,\ldots,n_k}({\mathcal S}')$ to $\Bun_{GL(n)}\subset\operatorname{Coh}_n$ is canonically isomorphic to…

Irreducibility conjecture. Moreover, if ${\mathcal S}$ is irreducible, then so is $\Eis({\mathcal S})$.

Coincidence conjecture. The isomorphisms supplied by Proposition 'IC calculation' and Theorem 'FFKM thm' (1) coincide. This is a compatibility assertion internal to the proof of th…

Let $G$ be a reductive group with Langlands dual group $\cG$, let $E_\cT$ be a $\cT$-local system on $X$, and let $\DefbE$ be the DG formal scheme parametrizing its universal Borel…

Let $G$ be a reductive group with Langlands dual group $\cG$, let $E_\cT$ be a $\cT$-local system on $X$, and let $\DefbE$ be the DG formal scheme parametrizing deformations of…

Let $I_d^g$ be the integral of the $(d,d)$ lattice partition function over the fundamental domain of the moduli space of genus-$g$ Riemann surfaces, and let…

Let $d>2$, let $E_{d+1(d+1)}(\mathbb{Z})$ be the U-duality group, and let $\mathcal{E}_{\mathrm{string};3/2}$, $\mathcal{E}_{\mathrm{particle};d/2-1}$, and…

For $d\geq 2$, let $\mathcal{E}^{SL(d,\mathbb{Z})}_{[2];s=1/2}$ denote the Eisenstein series in the antisymmetric 2-tensor representation and let…

Let $\{f_i\}_{i=1}^I$ be an orthogonal family of Hecke–Maass cusp forms and let $\{\widetilde{E_{T_j}}\}_{j=1}^J$ be an orthogonal family of normalized real Eisenstein series with…

Let $s_1,s_2\in\mathbb{N}$ with $s_1,s_2\geq 2$, and let $s$ be an eigenvalue having the same parity as $s_1+s_2$. Let $\mathcal{E}(s;s_1,s_2;\tau)$ denote the corresponding genera…

Let $r$ be an integer, let $\tau$ and $\psi$ be Dirichlet characters, let ${\mathbf{f}}$ denote the relevant Eisenstein family, and let ${\mathbf j}$ denote the cyclotomic-variable…

Let $r$ be an integer and let $\tau$ and $\psi$ be Dirichlet characters satisfying $(-1)^s=-\psi(-1)$. Let $L_p'(f_{\beta})(s)$ denote the weight derivative of the beta $p$-adic…

Let $n$ be a positive integer, let $f_B(x)=(1+x)(1+2x)$ and $f_G(x)=(1+x)(1+2x)(1+3x)(2+3x)$, and let $E_{2k}(\tau)$ denote the Eisenstein series for $\tau$ in the upper half-plane…

Eisenstein conjecture. The Whittaker coefficients of a metaplectic Eisenstein series are Weyl group multiple Dirichlet series; in the setting above, the first Whittaker coefficient…

Let $\Gamma=SL_2(\mathbb{Z})$, let $\Omega\subset\Gamma\backslash\mathbb{H}$ be a fixed compact regular set, and let $\mu$ denote hyperbolic measure. Write $\xi(s)$ for the complet…

Pole classification conjecture. For $\operatorname{Re}(s)>0$, if $\chi$ is trivial, then $\mathcal{E}_{P_2}^{E_8}(s,\chi)$ can have a double pole at $s=\frac32,\frac52,\frac72$ and…

Let $E^*_{\mathcal P,\Phi}(g,s)$ be a normalized Langlands Eisenstein series for $\operatorname{SL}(n,\mathbb Z)$, associated with a parabolic…

Let $S\subset\mathbb{P}^1(\mathbb{F}_q)$ be a nonempty finite set of tame ramification points, let $C_{Eis}$ be the corresponding Eisenstein Hecke module, and let…

Let $W$ be the Weyl group, let $\Lambda$ be the coweight lattice, and let $\widetilde{C}$ be the Hecke module generated by the relevant tamely ramified Eisenstein series. Freeness…

Hecke-trick conjecture. The modular forms $F_{k,S}$ arise from $E_{k,S}$ via the Hecke trick.

Overconvergence-rate conjecture. For all $i,j\geq 0$,