59 problems

Let $m\geq 1$, let $N\geq 1$ be coprime to $m$, and let $k$ be an even integer with $k=12$ or $k\geq 16$. Write $T'_m(N,k)$ for the normalized Hecke operator. Rouse's generalized L…

Let $S_{2k}(\Gamma_0(N))$ be the space of cusp forms of weight $2k$ and level $N$, and let $T_m(N,2k)$ be the $m$-th Hecke operator on this space. Write … where…

For a level-one Hecke polynomial, let $c_r$ denote its $r$-th coefficient. Clayton et al.'s nonvanishing conjecture. None of the Hecke polynomial coefficients vanish in the level-o…

Reciprocal-polynomial realization conjecture. Any reciprocal polynomial having negative roots and leading coefficient equal to $1$ should have the form $P_\Lambda(t)$ for some even…

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…

Let $\Gamma$ be a group, let $\lambda$ be a parameter, and let $m$ be an integer, with $W_{\lambda,m}$ the coefficient space used in the paper's cohomology construction. Finite-dim…

Let $N$ be a positive integer, let $M_0(N)$ be the space of weight-zero Maass wave forms of level $N$, and let $\Phi=(\nu_p\mid p\nmid N)$ be a system of Hecke eigenvalues occurrin…

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…

Schwartz–Shemanske's conjecture. For all $n\geq 3$,

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,…

Minimal-intersection conjecture. The lower bound is always attained for graphs of Hecke operators; equivalently, the intersection of $\operatorname{im}(M-\lambda)$ and…

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…

Non-repetition conjecture. As $k$ varies, the values $a_2(T_m(N,2k))$ are pairwise distinct. The conjecture is motivated by the paper's numerical computations and by strategies tha…

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

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…

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 $A=\mathbb{F}_q[T]$ and let $\mathrm{S}_{k,l}$ denote the space of Drinfeld cusp forms of weight $k$ and type $l$. The operator $\mathbf{T}_T$ is the Hecke operator associated…

Let $L_2(A_{5+1/2})$ be the intermediate vertex operator algebra appearing in the inclusion $L_2(A_5)\subset L_2(A_{5+1/2})\subset L_2(E_6)$, and let $\mathsf{T}_7$ denote the Heck…

Let $L_2(A_{5+1/2})$ be the level-two affine VOA of $A_{5+1/2}$, and let $\mathsf{T}_7$ denote the relevant Hecke operation on characters. Hecke-operation conjecture. The character…