32 problems

Let $\xi$ and $\xi'$ be the cycles associated with Algorithm, and suppose $n\leq4$ and $\|\xi\|>1$. For each modular symbol ${\mathbf v}$ in step A of the algorithm, choose…

Equivariant bundle conjecture. There exists an $F$-equivariant vector bundle $(\mathcal E,g)$ of rank $N$, defined over $\mathbb Q$, such that for $z\in X_n$ the eigenvalue of…

Let $f$ and $g$ be Siegel modular forms of genus $2$, of weights $k>4$ and $l=k-2$, with Satake parameters $(\alpha_0,\alpha_1,\alpha_2)$ and $(\beta_0,\beta_1,\beta_2)$, respectiv…

Ulmer's conjecture. The action of $U_p$ on $S_k(\Gamma_0(Np),\mathbb Q_p)$ is semisimple. In particular, the polynomial

Spectral decomposition conjecture. There exist a set $I^E$, a function $d^E:I^E\to\mathbb N$, and distinct functions $\lambda_i^E:\mathcal D\to\mathbb C$ for $i\in I^E$, such that,…

Let $M^n_{\mathrm{cusp}}=M(\overline{\mathcal{M}}_{1,n})_{\varepsilon}[n]$ be the motive associated to the space of cusp forms for $\Gamma=\operatorname{SL}_2(\mathbb{Z})$ of weigh…

Let $F$ be the number field in the paper, let $\mathbb{X}_F$ be the associated quotient, and let $\bar{\operatorname{G}}(F)\backslash\bar{\operatorname{G}}(\mathbb{A})$ carry the v…

Base-3 digit conjecture.

Base-7 digit conjecture.

Degree and nilpotence-index conjecture. The degree $D_{19}(k,5)$ is given by the displayed residue-class formulas in the source; moreover, $c_0=0$, $c_1=2$, and

Fix integers $m\geq 1$ and $r\geq 1$. Let $T_m(N,k)(x)$ be the Hecke polynomial, with its $r$-th coefficient understood in the source's indexing. Generalized Lehmer conjecture for…

Let $P$ be a parabolic subgroup of $G$ with Levi subgroup $M$ and unipotent radical $U$, let $V\in\operatorname{Rep}({}^LG)$, and restrict $V$ to ${}^LP$. Choose a finite filtratio…

Let $m\geq 2$ be a fixed non-square integer, and let $T_m^{\operatorname{new}}(N,k)$ denote the restriction of the $m$-th Hecke operator to the new subspace…

Clayton et al.'s nonvanishing conjecture. Then

Let $A=\mathbb{F}_q[T]$, let $\mathfrak p\trianglelefteq A$ be a maximal ideal of degree $d$, and let $n\geq 1$. For integers $k,l$, write $\mathbf{T}_{\mathfrak p}$ for the correspo…

Let $\Gamma$ be a Fuchsian group, let $\Sigma$ be a double coset of $\Gamma$, and let $X$ be a $\Gamma$-conjugacy class contained in $\Sigma$. For $M\in X$, define the discriminant…

Let $C$ be a smooth proper curve over a non-archimedean local field $K$, let $G$ be a split reductive group, and let $H(C)$ be the commutative algebra of Hecke operators acting on…

Density hypothesis, Hecke eigenvalue version. For every $l\geq 0$, $T\geq 1$, and $\delta>0$, one has

Let $K$ be the field and level data used in the paper, let $M_0(K)$ be the cuspidal space, and let $HE(K)$ be its Hecke-eigenform basis. Write … For a prime $\ell\ne p$, let…

Let $\mathcal C$ be a compact complex curve, let $G^{\vee}$ be the Langlands dual group, and let $o$ be a real $G^{\vee}$-oper. For a dominant coweight $\lambda$, let $V_\lambda$ b…

Let $\mathcal C$ be a smooth projective curve over a local field $F$ of genus at least $2$, let $G$ be a split connected semi-simple group, and let $\operatorname{Bun}$ be the modu…

Let $F$ be a non-archimedean or archimedean local field, let $\mathcal C$ be a smooth complete curve over $F$, and let $\operatorname{Bun}$ be the moduli stack of principal $G$-bun…

Let $\chi$ be an $SL_n$-oper with real monodromy. Let $(\mathcal V_{\omega_1},\nabla_\chi)$ be its associated flat vector bundle, with determinant identified with the trivial flat…

Let $G$ be a reductive group, let $X$ be a curve over a local field $F$, and let $\lambda\ne0$. Write $V_G(\lambda)$ for the dense domain on which the Hecke operators are initially…

Let $j,k,l\geq0$, let $\nu$ be a Hecke parameter, and let $\mu\vdash4$ be a partition indexing an $\mathfrak S_4$-isotypic component. The modular-9 trace congruence conjecture. … T…