4 problems

Oura's conjecture. The following analogous properties hold for $Th(\widetilde{\varphi_\ell^{G_{\rm{II}}}})$: (1) all zeros lie on a segment of the circle…

Let $E$ be a totally real number field, and let $\chi:\mathbb A_E^\times/E^\times\rightarrow\bar{\mathbb Z}_p^\times$ be a finite-order character cutting out a non-real CM extensio…

Let $F$ be a totally real number field, and let $\bar\rho: \operatorname{Gal}(\bar{\mathbb Q}/F)\rightarrow\operatorname{GL}_n(\bar{\mathbb F}_p)$ be a semisimple modular represent…

Let $F$ be a totally real number field, and let $\rho: \operatorname{Gal}(\bar{\mathbb Q}/F) \rightarrow \operatorname{GL}_n(\bar{\mathbb Q}_p)$ be an irreducible representation cu…