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1. Group problems

These are the stable problems on which workshop groups worked.

    1. Central derivative of Rankin–Selberg $L$-functions for weight-$2$ modular forms and congruent forms

          The hope is to show the non-vanishing of the central-value derivative of the Rankin–Selberg $L$-function of a weight $2$ modular form tensored with a theta series coming from a class group character via the Rankin–Selberg $L$-function with a congruent modular form on a definite quaternion algebra.

      Add remark

      Problem 1.1.

      [Philippe Michel] Let $f$ be a modular form of weight $2$ and let $\chi$ be a character of the class group of an imaginary quadratic number field $K$. Formulate a direct connection \[ L'(f \otimes \theta_\chi, \frac{1}{2}) \longleftrightarrow L(g \otimes \theta_\chi, \frac{1}{2}), \] for a modular form $g$ on a definite quaternion algebra with some congruence to $f$.
          Cf. Nicolas Templier’s thesis.

        • Non-vanishing of the central $L$-derivative for $\mathrm{U}(n) \times \mathrm{U}(n-1)$


          Add remark

          Problem 1.2.

          [Wei Zhang] Given an automorphic representation $\pi$ of $\mathrm{U}(n)$, show that there exists an automorphic representation $\sigma$ of $\mathrm{U}(n-1)$ such that \[ L'(\pi \otimes \sigma, \frac{1}{2}) \neq 0. \]
              Easier version: show that there is some pair $(\pi, \sigma)$ such that \[ L'(\pi \otimes \sigma, \frac{1}{2}) \neq 0. \]

            • $p$-indivisibility of $L(\chi, 0)$


              Add remark

              Problem 1.3.

              [Ashay Burungale] Fix a prime $p$. Let $q$ be a prime and let $\chi$ be an odd character of $(\Z/q\Z)^\times$.

              How often is $L(\chi, 0)$ indivisible by $p$?
                  Follow-up: This is the simplest scenario and we know $q^{\frac{1}{2} - \epsilon}$, but can this be improved? Expectation: positive proportion.

                • Relative conductor of $p$-adic groups


                  Add remark

                  Problem 1.4.

                  [Mladen Dimitrov] Let $H \leq G$ be groups over $\mathbb{Q}_p$, $\pi$ be an admissible representation of $G$, and $K_G$ be a maximal compact of $G$. Define \[ K_H(p^n) := \{g \in K_G \mid (g \mod p^n) \in H\}. \]

                  Does $\pi$ have a non-zero vector fixed by $K_H(p^n)$ for some $n$?

                  If so, what’s the smallest such $n$?
                      Special case: consider the case $G=\mathrm{GL}(a + b)$ and $H = \mathrm{GL}(a) \times \mathrm{GL}(b)$ (this is Jacquet–Piatetski-Shapiro–Shalika when $a = n-1, b = 1$). When can this happen? Relate this to the global non-vanishing?

                  Note: Relevant to GGP if $G$ and $H$ are dual pairs.

                  Read for motivation: Yueke Hu, Paul D. Nelson, Abhishek Saha, Some analytic aspects of automorphic forms on GL(2) of minimal type. Comment. Math. Helv. 94 (2019), no. 4, pp. 767–801

                    • Beilinson conjecture for Hecke characters of quartic CM fields


                      Add remark

                      Problem 1.5.

                      [Jennifer Johnson-Leung] Prove the Beilinson conjecture for Hecke characters of quartic CM fields.
                          Subproblem: how to generalize the Eisenstein symbol?

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