38 problems

Let $G$ be a reductive group satisfying the stated restriction that, for every root $\check{\alpha}\in\Lambda^{\vee}$, the map $\Lambda\to\mathbb{Z}$ given by…

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

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…

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…

Rationality conjecture. The sheaf ${\mathcal F}_0(h)$ gives a rational point on the Jacobian if and only if, for each $k$, there exists a nonzero formal power series $R_k(q)$ such…

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…

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 $r$ 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…

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 $S\subset\mathbb{P}^1(\mathbb{F}_q)$ 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…

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

Let $\theta_{\Delta(\tau,\ell)}$ and $\theta_{\Delta(\tau,2n+\ell)}$ be theta functions, and let $\Phi(\theta_{\Delta(\tau,\ell)},\theta_{\Delta(\tau,2n+\ell)},s)$ be the proposed…

2-adic valuation conjecture.

Let $F$ be a non-archimedean local field with residue field $k$, let $\mathcal C$ be a curve over the ring of integers of $F$, and let $E_{\operatorname{Bun},1/2}$ be the Eisenstei…

Horinaga's conjecture. The following statements hold:

The conjecture. (1) $\mathcal{N}(A \backslash G)_{(M,\tau)}\subset \mathcal{E}_0(M,\tau)$. (2) The action of $\mathcal{Z}$ on $\mathcal{N}(A \backslash G)$ is semisimple. (3) If…

Multiple Eisenstein correspondence conjecture. The following properties hold: (1)…

Zero-location conjecture. All but one of the zeroes of $E_\infty(z)$ in $F_\infty$ lie on the circle

Holomorphy and shadow conjecture. The value $E_{1/2}^*(\tau,0)$ is a holomorphic modular form of weight $1/2$; moreover, it is a linear combination of the shadows of the mock Eisen…

Arc-zero conjecture. For $k\geq14$, all of the zeros of $\Delta_{k,4}$, $\Delta_{k,6}$, and $\Delta_{k,8}$ in the fundamental domain lie on the arc $|z|=1$.

Boundary-zero conjecture. All the zeros of $\Delta_{k,\ell}$ lying in the standard fundamental domain are on the boundary, $|z|=1$ or $x=\pm1/2$.