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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)


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              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


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                  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


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                      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.