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3. Fixed vectors in representations of p-adic groups

Problems from the afternoon session of January 29, 2024

    1. Construction of abelian surfaces for paramodular newforms


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      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.
          Cf. Brumer–Pacetti–Poor–Tornaria–Voight–Yuen.

      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}


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          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?
              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.

              Cite this as: AimPL: Analytic, arithmetic, and geometric aspects of automorphic forms, available at http://aimpl.org/aagaautomorphic.