3. Fixed vectors in representations of $p$-adic groups
Problems from the afternoon session of January 29, 2024-
Construction of abelian surfaces for paramodular newforms
Cf. Brumer–Pacetti–Poor–Tornaria–Voight–Yuen.Problem 3.1.
[Jennifer Johnson-Leung] Given a paramodular newform $F$ of weight $2$ with rational coefficients, find a construction of the abelian surface to which it corresponds.
Follow-up: in the setting of Problem 1.4 (Relative conductor of p-adic groups): when is the space of $K_H(p^n)$-fixed vectors one-dimensional? Is there always an $H$ for which this happens? Is there a commonality between such $H$? -
Fixed vectors for $\mathrm{GSp}_{2n}$
Note: If $\pi$ is generic, this might be easier because of the paramodular newform theory. Cf. "On ratios of Petersson norms of Yoshida lifts" Saha 2015.Problem 3.2.
[Abhishek Saha] Let $\pi$ be a irreducible admissible infinite-dimensional representation of $\mathrm{GSp}_{2n}(\mathbb{Q}_p)$. Consider the subgroup \[ R(n) := \{g \equiv (\text{matrix with $1$ on the diagonal for first $n$ entries, $a$ on the diagonal for the last $n$ entries}) \pmod{p^n} \text{ for some } a \in \mathbb{Z}_p^\times \}. \]
Does $\pi$ have a $R(n)$-fixed vector for some $n$?
Cite this as: AimPL: Analytic, arithmetic, and geometric aspects of automorphic forms, available at http://aimpl.org/aagaautomorphic.