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2. Central values and central derivatives

Problems from the afternoon session of January 29, 2024

    1. Central derivative of Rankin–Selberg $L$-functions on U(n)


      Add remark

      Problem 2.2.

      [Wei Zhang] Prove an analogue of Problem 1.2 for algebraic cycles.
          This problem is implied by Problem 1.2 by some version of Gross–Zagier. This problem does not imply Problem 1.2 because we don’t know the non-degeneracy of height pairings. This is probably easier than Problem 1.2.


        • Add remark

          Problem 2.3.

          [Mladen Dimitrov] Is there a $p$-adic interpolation of $L'(\pi \otimes \sigma, \frac{1}{2})$?


            • Add remark

              Problem 2.4.

              [Chris Skinner] Given an automorphic representation $\pi$ of $\mathrm{U}(n)$, show that there exists an automorphic representation $\sigma$ (with some restriction) of $\mathrm{U}(n-1)$ such that \[ L(\pi \otimes \sigma, \frac{1}{2}) \neq 0. \]
                  Note: this is the central value version of Problem 1.2

                • Meta problem about values & derivatives

                      Generally, can the statement & proof of a known analytic number theory result for a special value be adapted to the derivative?

                  Add remark

                  Problem 2.5.

                  [Paul Nelson] Given a subconvexity result \[ L(\pi, \frac{1}{2}) \ll c(\pi)^{\frac{1}{4} - \delta}; \]

                  1) find an analogue of the result where the size of $L(\pi, \frac{1}{2})$ is replaced by some "arithmetic" analogue of size, or

                  2) find an analogue of the result for the derivative $L'(\pi, \frac{1}{2})$.
                      Special case: instead of looking for subconvex bounds, prove $p$-integrality bounds.

                  Simplest example: Gross–Zagier.

                    • Non-vanishing of products of twisted $L$-functions


                      Add remark

                      Problem 2.6.

                      [Dinakar Ramakrishnan] Let $f$ be a newform (corresponding to elliptic curve?) and let $\chi$ be a quadratic character.

                      Does there exist a quadratic character $\psi$ such that \[ L\left(f \otimes \psi, \frac{1}{2}\right) L\left(f \otimes \chi \psi, \frac{1}{2}\right) \neq 0? \]
                          Note: this is okay if $\psi$ is allowed to be of arbitrary order.

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