14 problems

Let $S^{-}_8(\Gamma_0(9))$ denote the space of weight-$8$ cusp forms with the minus Atkin–Lehner eigenvalue, and let $S_{4,4}(\Gamma[\sqrt{-3}],{\det}^2)$ be the space of Picard cu…

Let $m$ and $n$ be in the critical range for the algebraic Hecke character $\psi_m$, and define … Suppose a prime $\ell>m$ divides the denominator of $Q(m,n)$. Harder's conjecture.…

Let $k>0$ and $l\geq0$ satisfy $k\equiv_6 l$. Let $\mathcal X^{(2)}_{\Gamma[\sqrt{-3}]}$ be the genus-2 moduli space considered in the paper, let $\mathbb W_{k,l}$ be the correspon…

Let $p$ be a prime with $p\equiv2\pmod3$. The inert-prime divisibility conjecture. The trace of the Hecke operator on the space of genuine forms should be divisible by $9$. The sou…

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…

Let $E_c(\lambda)$ and $E_{\rm extr}(\lambda)$ be the representations obtained from the compactly supported Euler characteristic and the extraneous contributions, respectively. Let…

Let $\lambda\neq(0,0,0)$, let $\nu$ and $\mu$ be as in the paper, and write $\lambda=(a,b,i)$ so that $i\equiv_2 a+b$. Let $T(\nu)$ be the Hecke operator, $F_\nu$ the corresponding…

Let $S_{n(\lambda)}(\Gamma[\sqrt{-3}])$ be the space of Picard modular cusp forms of weight $n(\lambda)$, and let $S_{n(\lambda)}^{\rm gen}(\Gamma[\sqrt{-3}])$ denote the proposed…

Let $a$ and $b$ be the parameters occurring in the indicated weights, and let $S^{\chi}_{a+2}(\Gamma_0(9))$ denote the subspace defined using the quadratic character modulo $3$. Th…

Let $a$ and $b$ be the parameters occurring in the indicated weights. The level-3 newform lift conjecture. There should be a lift … with $\mathfrak{S}_4$-representation…

Let $a$ and $b$ be the parameters occurring in the indicated weights. The level-3 minus-space lift conjecture. There should be a lift … with $\mathfrak{S}_4$-representation…

Let $a$ and $b$ be the parameters occurring in the indicated weights, let $f$ be an eigenform in $S^{-}_{a+b+3}(\Gamma_0(9))$, and write $a_p(f)$ for its Hecke eigenvalue. The two-…

Let $a$ and $b$ be the parameters occurring in the indicated weights, and let $S^{-}_{b+2}(\Gamma_0(9))$ be the relevant Fricke-eigenspace of cusp forms. The one-dimensional lift c…

Let $j\geq 0$, and let $S_{j,k+3}(\Gamma_1[\sqrt{-3}])$ denote the indicated space of Picard modular cusp forms. The $k=0$ dimension-formula conjecture. The dimension formulas abov…