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4. Generalizations of the moments of BDPW and MPPRW papers on moments of L-functions papers

    1. Uniform stability


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      Problem 4.1.

      [Will Sawin] Fix a finite group $G$ and an irreducible representation $\rho: G \to \operatorname{GL}_r(\mathbb C)$. One might be interested in understanding homological stability needed for moments of $L(1/2, \rho_K)$ as $K$ varies over $G$ extensions of $\mathbb F_q(t)$. Namely, one can try to understand homological stability on $\operatorname{Hur}^G_n$ of local systems associated to tensor powers of the relative $H^1$ of the universal curve. For which $G,\rho$ can we show that certain of the needed inputs into the uniform homological stability paper MPPR-W.

      A remark by Mark Shusterman relating this to the “Question by Mark Shusterman on epsilon factors” is about understand the top wedge power of the above local system.

      For example, if one was only interested in the first power of $H^1$, this would be tracking something like the average number of points on a $G$ cover, and many people have already worked on these types of questions.

        • Monodromy images


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          Problem 4.2.

          [Dan Petersen] What is the image of the Burau representation specialized at other roots of unity?

          If one specializes to the case of a cyclic group, one may want to know exactly what the image of the Burau monodromy representation. What is the exact image of the Burau representation specialized at roots of unity? It is known in the case that the group is cyclic of order $2$. Venkataramana has some results showing that these relevant monodromy groups are arithmetic, when the number of branch points is large enough. The context for this is that this could use this as one ingredient toward giving asymptotics for $L$-functions with higher degree Dirichlet characters and not just quadratic Dirichlet characters.

            • Moments of derivatives


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              Problem 4.3.

              [Jon Keating] What do the methods for computing moments of L function give if one tries to compute moments of derivatives. Even in the random matrix theory setting computing the distribution of derivatives is difficult. What can recent methods tell us about the distribution of values of derivatives of $L$ functions? This structure is related to the Painlevé V equation, and the moment (for finite $n$) is a polynomial in $N$, the size of the random matrix. Also, they correspond to solutions for Painlevé III as $n \to \infty$.

                • Higher degree cyclic covers


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                  Problem 4.4.

                  [Dan Petersen] If one starts from the number theoretic side, consider $\chi$ cyclic of order more than $2$. It is natural to think about summing $L(1/2,\chi_d)^r L(1/2, \overline{\chi}_d)^s$ for certain cyclic characters $\chi_d$. One can find a version of that for higher order cyclic extensions. This was found by Sean Howe. (One can compute the stable trace of Frobenius without computing the stable homology.)

                  The natural follow up question is whether one can use this to get the main terms in the higher degree case. There is a bisymmetric function which encodes these values. One may want to find the stable cohomology of a Schur functor applied to Burau and a different Schur functor applied to the dual of Burau.

                  There is a related paper by Erik Lindell. He calculates the cohomology of a free group in the some Schur functor of the standard representation tensor another Schur functor of the dual.

                  It might be very hard to prove uniform stability, but it might be quite doable to calculate the stable homology.

                      Cite this as: AimPL: Moments in families of L-functions over function fields, available at http://aimpl.org/momentsoverff.