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5. Fourier coefficients of Siegel modular forms

Problems from the afternoon session of January 29, 2024

    1. Computing Fourier coefficients from $L$-values


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      Problem 5.1.

      [Manami Roy] Let $d$ be a positive integer and let $F$ be a Siegel modular form with Fourier coefficients $a(F)$. In many cases, we know the (generalized) Bòˆcherer conjecture: \[ \left \lvert \sum_{S \in \mathrm{Cl}_d} a(F, S) \Lambda(S) \right \rvert^2 = L\big(\pi_F \times \theta_\Lambda, \frac{1}{2}\big) = L\big(\mathrm{BC}_{K/\mathbb{Q}} (\pi_F) \times \Lambda, \frac{1}{2}\big). \]

      Knowing this equality and values of $L(\pi_F \times \theta_\Lambda, \frac{1}{2})$ for many $\Lambda$, can we compute the Fourier coefficients $a(F, S)$?
          Interesting case: paramodular forms and symmetric cube $L$-functions, where the first relation is "semi-known".

        • Siegel modular forms


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          Problem 5.2.

          [Abhishek Saha] Let $F$ be a Siegel cusp form of full level and weight $k$ that is a Hecke eigenform. Assume any standard conjecture on $L$-values (e.g. GRH) and all conjectural period formulae.

          Prove that for some $\delta > 0$, \[ \lvert a(F, S) \rvert \ll_{F} \det(S)^{\frac{K}{2}-\frac{1}{2}-\delta}. \]
              Known: assuming GRH, proved with $\delta = 0$ using the weighted average expression of Problem 5.1.

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