5 problems

Let $n$ and $k$ satisfy $k>n+1$, and let $f\in S_{2k-n}(\Gamma_1)$ be a normalized elliptic Hecke eigenform. Let $L(s,F;st)$ denote the standard $L$-function of a Hecke eigenform…

Regular geometric lifting characterization conjecture. The representation $\bar{\rho}:G_K\to\operatorname{GL}_n(k)$ has a geometric lift regular at all places above $p$ if and only…

Regular-at-infinity lifting conjecture. Every continuous representation $\bar{\rho}:G_K\to\operatorname{GL}_n(k)$ that is regular at infinity lifts to an irreducible geometric repr…

Let $f$ and $g$ be Siegel modular forms of genus $2m$, with weights $k>2m$ and $l=k-2m$. Let $(\alpha_0,\alpha_1,\ldots,\alpha_{2m})$ and $(\beta_0,\beta_1,\ldots,\beta_{2m})$ be s…

Let $S_{2k-2}(\operatorname{SL}_2(\mathbf{Z}))$ be the space of cusp forms of weight $2k-2$ for $\operatorname{SL}_2(\mathbf{Z})$, and let Siegel modular forms have weight $k$. Sai…