5. Moments of L-functions
This is a section on questions related to the BDPW and MPPRW paper, but somewhat less related to a direct generalizations of it.-
Hyperelliptic Torelli
Problem 5.1.
[Dan Petersen] Can you give a quadratic presentation of the Maltsev completion of hyperelliptic Torelli using the computation from Bergstrom-Diaconu-Petersen-Westerland? -
Negative moments
Problem 5.2.
[Alessandro Fazzari] Can one prove a version of the pair correlation conjecture using topological methods?
Will sawin thinks this should be possible in some range.
Can one use homological stability methods to say anything about negative moments? -
Error terms
Problem 5.3.
[Jordan Ellenberg] What kind of error terms do people expect for the moments?
Diaconu suggests that there are infinitely many higher order terms. Could there be interesting secondary stable homology that can be understood? In which of the polynomial pieces associated to the moments do the relevant classes live? The simplest setting where these secondary terms is the third moment, \begin{align*} \sum_d L(1/2, \chi_d)^3. \end{align*} In this case, there is a proven secondary term in the function field case. In this and other examples, with known or expected secondary terms, can we construct secondary stable cohomology classes?
Will Sawin points out the the common point is that these are all families parameterized by algebraic varieties. (These are geometric families with secondary terms, but the vertical moments of zeta would be a harmonic family, where no secondary terms are known to exist.) -
The moments conjecture
Problem 5.4.
[Victor Wang] Assume the most optimistic bounds on unstable cohomology. can we get the full ratios conjecture without a limitation on the size of $q$. I.e. can we get it for fixed $q$ instead of taking a limit as $q \to \infty$? -
A conjecture about shifted L functions
Problem 5.5.
[Brian Conrey] Let $a = \{a_1, \ldots, a_k\}$. Define \begin{equation} \sum_n \tau_a(n) n^{-s} := \prod_{i=1}^k l(s+a_i). \end{equation} Then \begin{equation} \sum_n \chi_d(n)\tau_a(n) n^{-s} := \prod_{i=1}^k l(s+a_i,\chi_d). \end{equation} Observe \begin{equation} \sum_{d=1}^d \sum_{n=1}^x \chi_d(n) \tau_a(n)n^{-1/2}\sim \sum_{d=1}^d \sum_{\substack{n=1 \\ n \text{square }}}^x \tau_a(n) n^{-1/2}. \end{equation} In order to recover the certain moments you’re interested in one needs to take $x = d^{|a|}$. In order to get the correct estimate of this sum one can split up this sum by splitting $a$ into subsets. It is known how to split the sum to get moments corresponding to counting rational points on many subvarieties. What is the topological interpretation? can topological methods say anything about this conjecture?
Brian Conrey points out that the dominant term of the point count is coming from a subvariety, since Manin’s conjecture would predict a smaller answer for the relevant point counts on certain related varieties. -
Twisted averages of $L$ functions
Problem 5.6.
[Jordan Ellenberg] Consider $\sum_d L(1/2, \chi_d) \cdot \phi(d)$ as $d$ varies over polynomials over $\mathbb F_q$ generating quadratic fields and $\phi$ is some interesting arithmetic function. For example, you can take $\phi$ to be the mobius function or some interesting character. What would be the relevant homological stability question be and is it solvable by our current techniques in homological stability? If not, can we prove new homological stability results to answer this type of question? Potentially, the relevant local systems could be of the form $v_\lambda \otimes w$ for $w$ some representation of $s_n$ and $v_\lambda$ certain $\lambda$-schur functor applied to the burau representation, associated to a partition $\lambda$.
Diaconu believes that a similar approach should be applicable to the main terms when $\phi$ itself is a quadratic character.
Separately, perhaps one can use analytic number theory methods directly when $\phi$ is the mobius function. -
Vanishing of L functions
Problem 5.7.
[Alexandra Florea] As in the question by Jordan Ellenberg on twisted averages of $L$ functions, take $\phi$ to be a quadratic character and try to compute $\sum_d L(1/2, \chi_d) \cdot \phi(d)$. Is there some application to non-vanishing of $L(1/2, \chi_d)$? For example, fix $q$. is the proportion of $d$ so that $L(1/2,\chi_d) \neq 0$ equal to $100\%$? -
Elliptic Curves
Problem 5.8.
[Adrian Diaconu] Take $E$ over $\mathbb F_q(t)$ containing $n$th roots of unity. one can study the analogous question. what is $\sum_d L(1/2,e \otimes \chi_d)^r L(1/2, e \otimes \overline{\chi}_d)^s$. for cubic $\chi_d$ David-Lalin-Nam made a recipe. What is a homological interpretation of the main term? can one compute the stable homology and show the homology stabilizes? One can start with $\chi_d$ quadratic. A conjectural main term can be made but it is not proven. This was proven when $r = 2, s= 0$ and $\chi$ is quadratic by Florea and others.
This is related to thinking about homological stability over punctured curves as well. -
Vanishing of $L$ functions
Problem 5.9.
[Jordan Ellenberg] How often does $L(1/2, E \otimes \chi) = L'(1/2, E \otimes \chi) = 0$, for $E$ an elliptic curve and $\chi$ a quadratic character? This could perhps be doable if one could solve question by adrian diaconu on elliptic curves.
Cite this as: AimPL: Moments in families of L-functions over function fields, available at http://aimpl.org/momentsoverff.