10 problems

Type D BMW dimension conjecture. The dimension of $B({\rm D}_n)$ is

Goncharov's polynomiality conjecture. If $p$ is a prime, then $d_w(p)$ and $d_{w,m}(p)$ are polynomials in $p$.

Let $w_1,\ldots,w_7$ be the fundamental weights, let $\lambda=\sum_{i=1}^7 n_iw_i$, let $\rho$ be the Weyl vector, and let $\Delta_+$ be the set of positive roots. Let…

Let $E$ be a real quadratic field and let $\Gamma\subset\operatorname{SL}_3(\mathbb{Z})$ be a finite-index subgroup. Let $H(\Gamma,E)$ be the Steinberg-generated subspace of…

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…

Fix integers $r\ge1$, $c>2-r$, $t\ge2$, and $d\ge1$. Let $\mathcal{A}$ be a homogeneous $t\times(t+c-1)$ matrix with entries of degree $e$, and set $A=R/I_{t-r+1}(\mathcal{A})$. Su…

The dimension conjecture. Under these hypotheses,

For $N=2,3$, let $\mathcal S_k^{\mathrm{new}}(\Gamma_0(N))^{\pm}$ denote the subspaces of newforms of weight $k$ and level $\Gamma_0(N)$ on which the Atkin–Lehner involution acts b…

Let $F$ be the Laurent series field with valuation $v$, let $(G,G_0)$ be the symmetric pair considered in the paper, and let … be regular semisimple. Regard $XY$ as an element of…

Let $D$ be a positive integer, let $\Gamma(D,1)$ denote the non-split symplectic group in the paper, and let $S_{k,j}(\Gamma(D,1))$ be the space of Siegel cusp forms of weight…