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6. More problems

Problems from the afternoon session of January 29, 2024

    1. Normalization of Bianchi modular forms


      Add remark

      Problem 6.1.

      [Mladen Dimitrov] Consider a Bianchi modular form, is there a correct way to normalize its coefficients in its Whittaker expansion (in $H^2$)?

        • Applications of the Hecke orbit conjecture


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          Problem 6.2.

          [Ashay Burungale] Are there applications of the recent progress on the Hecke orbit conjecture to vertical mod-$p$ non-vanishing (as in Iwasawa theory)?

            • $p$-adic differential operators for higher-rank groups


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              Problem 6.3.

              [Chi-Yun Hsu] For a higher-rank group (e.g. $\mathrm{GSp}_4$), there are some standard $p$-adic differential operators. Do the images of these differential operators contain special automorphic forms coming from lower-rank groups (e.g. Yoshida lifts, Saito–Kurokawa lifts)?

                • Local-global description of Stark units


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                  Problem 6.5.

                  [org.aimpl.user:robinz@mit.edu] Let $f$ be a weight-$1$ modular form with associated Galois representation $\rho_f$. The Harris–Venkatesh conjecture gives a description of Stark units for $\mathrm{Ad}^0(\rho_f)$ modulo $p$ for almost all $p$. In explicit exotic cases, can this be used to describe the Stark units for $\mathrm{Ad}^0(\rho_f)$ themselves?

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