Murmurations in Arithmetic
Institute for Computational and Experimental Research in Mathematics (ICERM)
July 6, 2023  July 8, 2023
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Thursday, July 6, 2023

8:50  9:00 am EDTWelcome11th Floor Lecture Hall
 Session Chair
 Brendan Hassett, ICERM/Brown University

9:00  9:45 am EDTMurmurations of Lfunctions11th Floor Lecture Hall
 Speaker
 Andrew Sutherland, MIT
 Session Chair
 KyuHwan Lee, University of Connecticut
Abstract
I will present joint work with YangHui He, KyuHwan Lee, Thomas Oilver, and Alexey Pozdnyakov that extends their disovery of osillations in Frobenius traces of elliptic curves when organized by parity and conductor to other arithmetic objects of interest, including modular forms and higher genus curves.

10:00  10:30 am EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTTBA11th Floor Lecture Hall
 Speaker
 Jonathan Bober, University of Bristol
 Session Chair
 KyuHwan Lee, University of Connecticut

11:30 am  12:15 pm EDTMurmurations of Dirichlet Characters11th Floor Lecture Hall
 Speaker
 Alexey Pozdnyakov, University of Connecticut
 Session Chair
 KyuHwan Lee, University of Connecticut
Abstract
Although murmurations were originally observed in the context of elliptic curves, it was soon discovered that this phenomenon occurred in other arithmetic contexts. Recent work with He, Lee, Oliver, and Sutherland showed that one can observe murmurations in many different families of Lfunctions. Simultaneously, Zubrilina provided a much needed theoretical explanation for murmurations by computing a local average over weight 2 modular newforms coming from all Galois orbit sizes. In my talk, I will provide a similar theoretical explanation in the context of degree 1 Lfunctions. In particular, I will compute a local average over Dirichlet characters of a fixed parity coming from all Galois orbit sizes, allowing us to predict the corresponding murmuration. This is based on recent work with Lee and Oliver.

12:30  2:00 pm EDTLunch/Free Time

2:00  3:30 pm EDTProblem Session/ Panel DiscussionPanel Discussion  11th Floor Lecture Hall
 Session Chair
 KyuHwan Lee, University of Connecticut
 Panelists
 Kimball Martin, University of Oklahoma
 Steven Miller, Williams College
 Andrew Sutherland, MIT

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm EDTFrom data science to murmurations11th Floor Lecture Hall
 Speaker
 Thomas Oliver, Teesside University
 Session Chair
 KyuHwan Lee, University of Connecticut
Abstract
Murmurations were first observed in the context of data scientific experimentation. I will outline some of the machine learning techniques that were applied to the LMFDB prior to their discovery. I hope to give some explicit examples and discuss some relevant statistics for elliptic curves.

5:00  6:30 pm EDTReception11th Floor Collaborative Space
Friday, July 7, 2023

9:00  9:45 am EDTBiases of modular forms from the trace formula11th Floor Lecture Hall
 Speaker
 Kimball Martin, University of Oklahoma
 Session Chair
 Thomas Oliver, Teesside University
Abstract
I will discuss some biases of certain spectral biases of modular forms that one can prove with the trace formula. These include biases with respect to root number, AtkinLehner eigenvalues and Hecke eigenvalues. These biases explain some aspects of the murmuration phenomena.

10:00  10:30 am EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTMurmurations and explicit formulas11th Floor Lecture Hall
 Speaker
 Alexander Cowan, Columbia university
 Session Chair
 Thomas Oliver, Teesside University
Abstract
We propose a heuristic explanation for murmurations based on the "explicit formula" from analytic number theory. A crucial ingredient in this heuristic is that the distribution of the zeros of the associated Lfunctions has a quasiperiodic structure. We present empirical results for both a family of elliptic curves as well as a family of quadratic Dirichlet characters that also exhibit murmurations.

11:30 am  12:15 pm EDTRoot numbers and murmurations11th Floor Lecture Hall
 Speaker
 Peter Sarnak, Institute for Advanced Study and Princeton University
 Session Chair
 YangHui He, London Institute for Mathematical Sciences & Merton College, Oxford University

12:25  12:30 pm EDTGroup Photo (Immediately After Talk)11th Floor Lecture Hall

12:30  2:00 pm EDTLunch/Free Time

2:00  2:45 pm EDTApplications of Moments of Dirichlet Coefficients in Elliptic Curve Families11th Floor Lecture Hall
 Speaker
 Steven Miller, Williams College
 Session Chair
 Thomas Oliver, Teesside University
Abstract
We consider consequences of the first and second moments of the Dirichlet coefficients in oneparameter families of elliptic curves. Assuming standard conjectures (known for rational surfaces), Rosen and Silverman proved a conjecture of Nagao that the first moment is related to the rank; we use this to construct families of moderate rank by having large first moment sums. For nonCM oneparameter families, Michel proved the second moment of the Fourier coefficients is p^2 + O(p^{3/2}). Cohomological arguments show that the lower order terms are of sizes p^3/2, p, p^1/2 and 1. In every case we are able to analyze, the largest lower order term in the second moment expansion that does not average to zero is on average negative. The negative bias in these lower order terms has implications toward the excess rank conjecture and the behavior of zeros near the central point of elliptic curve Lfunctions.

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space

3:30  5:00 pm EDTProblem Session/ Panel DiscussionPanel Discussion  11th Floor Lecture Hall
 Session Chair
 Thomas Oliver, Teesside University
 Panelists
 Kimball Martin, University of Oklahoma
 Steven Miller, Williams College
 Andrew Sutherland, MIT
Saturday, July 8, 2023

9:00  9:45 am EDTStarlike configurations in data related to the computation of Lvalues11th Floor Lecture Hall
 Virtual Speaker
 Barry Mazur, Harvard University
 Session Chair
 YangHui He, London Institute for Mathematical Sciences & Merton College, Oxford University
Abstract
In computation of special values of Lfunctions Karl Rubin and I have come up with some data that give pictures that are somewhat surprising to us. So far our computations aren’t substantial enough to make firm conjectures—let alone statements that we can prove—so this is work in the very early stages of . . . ‘progress.’ We would be grateful for any advice, and help, in accumulating more data.

10:00  10:30 am EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTAverage multiplicities11th Floor Lecture Hall
 Speaker
 David Rohrlich, Boston University
 Session Chair
 YangHui He, London Institute for Mathematical Sciences & Merton College, Oxford University
Abstract
A wellknown folk conjecture predicts that the average rank of an elliptic curve over Q is 1/2. Given an irreducible Artin representation r of Q, one can ask for the average multiplicity of r in the MordellWeil group over Qbar of an elliptic curve over Q. If r is the trivial onedimensional representation then one recovers the average rank. To illustrate some issues that arise if one wants to formulate an ""average multiplicity"" conjecture, we shall look at an example where r has dimension 4 and Schur index 1 and factors though a Galois group over Q of order 32.

11:30 am  12:15 pm EDTTBD11th Floor Lecture Hall
 Virtual Speaker
 Nina Zubrilina, Princeton University
 Session Chair
 Thomas Oliver, Teesside University

12:30  2:00 pm EDTLunch/Free Time

2:00  2:45 pm EDTFree DiscussionFree Discussion  11th Floor Collaborative Space

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space

3:30  4:15 pm EDTFree DiscussionFree Discussion  11th Floor Collaborative Space
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