Institute for Computational and Experimental Research in Mathematics (ICERM)

Abstract

I will present joint work with Yang-Hui He, Kyu-Hwan 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.

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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 L-functions. 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 L-functions. 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.

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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.

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, Atkin-Lehner eigenvalues and Hecke eigenvalues. These biases explain some aspects of the murmuration phenomena.

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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 L-functions has a quasi-periodic 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.

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Abstract

We consider consequences of the first and second moments of the Dirichlet coefficients in one-parameter 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 non-CM one-parameter 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 L-functions.

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Abstract

In computation of special values of L-functions 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.

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Abstract

A well-known 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 Mordell-Weil group over Qbar of an elliptic curve over Q. If r is the trivial one-dimensional 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.

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