Relevant talks and papers by Steven J. Miller (sjm1 AT williams.edu) for the Simons Symposium
"Families of Automorphic Forms and the Trace Formula" (Jan 26 - Feb 1, Rio Grande, Puerto Rico)

• Notes from all the talks (pdf)        Notes from all the talks (tex)        Needed image (eps)

• My proposed problems -- day 1

• My talk -- day 2 (results on low-lying zeros in families)

• My talk -- day 5 (results on lower order terms in moments of Satake parameters)

Quick Summary: The following are some talks and papers that I've given that are related to my work on L-functions that has a direct bearing on the symposium. Most of my research has focused on low-lying zeros of GL(1) and GL(2) families (click here for a great paper on the subject by Iwaniec, Luo and Sarnak), or Rankin-Selberg convolutions of such. Two particularly relevant ones are work with Duenez on convolving families of L-functions (where we can determine the symmetry of the convolution based on the symmetries of the families), and work with Duenez, Huynh, Keating and Snaith on the behavior of zeros in elliptic curve families. This builds on earlier numerical observations of mine which showed the first few zeros above the central point exhibited far more repulsion than you would see in the limit. In other words, while we know in the limit as the conductors tend to infinity the low-lying zeros behave as in the scaling limits of classical compact groups, for finite conductors the behavior is not the same as that for finite classical compact groups. We have introduced new ensembles, the excised orthogonal ensembles, which do a beautiful job fitting the data. The key idea is that the value of L-functions at the central point are discretized, and if they are non-zero then they must be at least a certain amount. We apply a similar cut to the orthogonal groups, and consider the subset where the characteristic polynomial is at least a certain size (which depends, among other items), on the size of the conductor. I am currently working with Nathan Ryan and his students to extend these computations to other families. I am very interested in seeing if a similar behavior occurs in natural families of L-functions on GL(3) and higher. Finally, Levent Alpoge and I are writing a paper on Maass forms for a Springer volume in honor of Maier's 60th birthday; the introductory section has a nice review of the history of studies of zeros of L-functions leading to the Katz-Sarnak Density Conjectures (note the paper is still very much in draft form)..

• RELATED TALKS THAT I'VE GIVEN
  • From the Manhattan Project to Elliptic Curves: This is a semi-general colloquium talk I gave at Dartmouth in 2013. There are nine sections.
    1. Introduction: States the goal is to understand spacings between events, and sketches the three ingredients in all the proofs in the diverse systems.
    2. Classical RMT: Reviews some of the classical theory of random matrix ensembles (these are NOT with Haar measure).
    3. Block circulant: A fun fat-thin family of matrices: fat enough so that we can still average, but thin enough that new behavior emerges.
    4. L-functions: Quick review of L-functions, elliptic curves, what zeros of L-functions tell us, and the explicit formula (the analogue of the eigenvalue trace lemma).
    5. Katz-Sarnak Conjecture: Describes their conjectures for low-lying zeros in families. Summarizes my work with Duenez on identifying the symmetry of the convolution of two families from the symmetries of the two families. Talks about similarities b/w low-lying zeros and the CLT: all that matters in the limit are the first two moments of the Satake parameters; the higher moments, similar to Berry-Essen, just control the rate of convergence. Ends with how I look at the subject (analogies with nuclear physics).
    6. Cusp forms: Quick sketch of n-level calculations for cuspidal newforms. Joint with Chris Hughes and David Montague.
    7. Elliptic curves: General results about low-lying zeros in limit of large conductors. Talks about issues with small conductors (excess rank, repulsion of first zero). Briefly mentions work of mine with Duenez, Huynh, Keating and Snaith to model with excised orthogonal ensembles.
    8. Questions and references: Some open questions and some references (many hyperlinked).
    9. Dirichlet characters: Calculates the 1-level density for Dirichlet characters, so you can see how these computations are done.
  • Low-lying zeros of GL(2) L-functions: A talk I gave at the Quebec-Vermont Number Theory Seminar in 2013. There is a lot of overlap with the above talk; below I'll mention some of the key differences. There is also an expanded version, which I'll describe afterwards.
    1. Elliptic Curves: Repeats much of the stuff above, but also talks about the Ratios Conjecture's prediction.
    2. Maass forms: Computes the 1-level density for Maass forms. Joint with Nadine Amersi, Geoff Iyer, Oleg Lazarev, and especially Levent Alpoge.
    3. Cuspforms: More detail on the n-level density for cuspforms, where we expand the support. Joint with Geoff Iyer and Nicholas Triantafillou.
    4. Excised: More details on the excised ensemble of matrices, which does a much better job modeling elliptic curves with finite conductor.
  • Low-lying zeros of GL(2) L-functions: Part I: Expanded version of  the QVNTS talk given at Michigan. Part I.
    1. Introduction: States the goal is to understand spacings between events, describes applications of knowing zeros of L-functions, and sketches the three ingredients in all the proofs in the diverse systems.
    2. Classical RMT: Reviews classical RMT, both density of states and spacings between adjacent eigenvalues, talks about fat-thin families (fat enough to average over, thin enough to see something new).
    3. Intro to L-functions: Reviews basic properties of L-functions, n-level correlations, spacings b/w zeros, n-level densities, Katz-Sarnak Philosophy, interpretation via physics.
    4. Results: Briefly summarizes results, gives an application of knowing n-level densities with high support (estimating percent in a family vanishing at the central point).
    5. Dirichlet characters: Calculates the 1-level density for Dirichlet characters, so you can see how these computations are done.
  • Low-lying zeros of GL(2) L-functions: Part II: Expanded version of  the QVNTS talk given at Michigan. Part II.
    1. Compound families: Describes my work with Duenez on attaching a symmetry constant to each family, and proving (in some cases) that the symmetry constant of the Rankin-Selberg convolution is the product of the symmetry constants. Shows how only the first two moments of the Satake parameters enter.
    2. Cuspidal newforms: My work with Hughes to expand Iwaniec-Luo-Sarnak to the n-level density for cuspidal newforms; more details than the earlier talks, get support of the Fourier transform of the test function up to \(\left(-\frac{1}{n-1}, \frac{1}{n-1}\right)\).
    3. New cuspidal results: Joint with Iyer and Triantafillou, increases to \(\left(-\frac{1}{n-2}, \frac{1}{n-2}\right)\). Also of course deals with the number theory / exponential sum results needed to compute the new main terms.
    4. New RMT: Joint with Iyer and Triantafillou, the combinatorics needed to compare the expanded number theory and RMT. In the spirit of Hughes-Miller we do not directly compare with the Katz-Sarnak determinant expansions. Though those are valid for all support, from a computational perspective they are not pleasant to work with (see also the issues with Gao's thesis, where he had number theory and RMT calculated but couldn't show the two were the same). We instead use an alternative expansion. Though valid only for smaller support, it's significantly easier for comparisons. This is due to how the number theory is calculated: we change variables and replace the n-dimensional integrals in the n-level density with a 1-dimensional integral of a new test function (which is an n-fold convolution of our original test functions). Though this increases the complexity of the combinatorics, it simplifies the Bessel-Kloosterman expansion as we can just invoke the 1-level result from Iwaniec-Luo-Sarnak.

  • Finite conductor models for zeros near the central point of elliptic curve L-functions: From a number theory seminar at Brown in 2011. Gives data, discusses models (with an appendix by Simon Marhsall on an explanation of the observations).

    1. Introduction: General review, similar to others above, of the theory of low-lying zeros.

    2. Old Theory / Models: Describes some other models for zeros for finite conductors.

    3. Data / New Models: Reports on some numerical data and ends with the excised orthogonal model, showing how well it does.

    4. Ratios Conjecture: Ratios Conjecture prediction, ending with some excellent numerics from Stopple.

    5. Excised Ensembles: Theory of the excised ensemble and some data.

    6. Explanation: An appendix by Siman Marshall on an alternate explanation of the observed repulsion using Waldspurger's formula and some complex analysis.

   
• RELATED PAPERS THAT I'VE WRITTEN
  • History:
    • Nuclei, primes and the random matrix connection (with Frank W. K. Firk), invited paper to Symmetry 1, (2009), 64--105; doi:10.3390/sym1010064. This is a survey article on the history of the connections between nuclear physics and number theory. My co-author was one of the original experimentalists gathering data that led to the connections between RMT and nuclear physics.
  • n-level densities:
    • 1- and 2-level densities for rational families of elliptic curves: evidence for the underlying group symmetries. Compositio Mathematica 140 (2004), no. 4, 952-992. Title is pretty explanatory; the main contributions are noticing the 2-level density distinguishes the three orthogonal flavors for arbitrarily small support, and deals with the issues of having varying conductors.   
    • Lower order terms in the 1-level density for families of holomorphic cuspidal newforms, Acta Arithmetica 137 (2009), 51-98. While the main terms of families of L-functions agree with the main terms of the classical ensembles, the lower order terms (which depend on higher moments of the Satake parameters) can differ. Here some examples are shown.
    • The effect of convolving families of L-functions on the underlying group symmetries (with Eduardo Dueñez), Proceedings of the London Mathematical Society 2009; doi: 10.1112/plms/pdp018. This is one of the most important for the conference. We attach a symmetry type to each family of L-functions, and show under certain hypotheses (which can sometimes be verified) that the symmetry type of the Rankin-Selberg convolution of two families is the product of the symmetries. The dependence on the Satake parameters is explained.
  • Numerics / Computations:
    • Investigations of zeros near the central point of elliptic curve L-functions, Experimental Mathematics 15 (2006), no. 3, 257–279 (data available online). This paper reports on the behavior of the first few zeros above the central point in elliptic curve, specifically the additional observed repulsion. The numerics here helped us eventually come up with the excised ensemble.
    • The lowest eigenvalue in Jacobi ensembles and Painlevé VI (with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith), Journal of Physics A: Mathematical and Theoretical 43 (2010), 405204, 27pages. This paper develops the theory to compute the lowest eigenangle above 0 for the excised ensembles.
  • Ratios Conjecture:
    • An Orthogonal Test of the L-functions Ratios Conjecture, II (with David Montague),  Acta Arithmetica 146 (2011), 53--90.  This paper goes through how to apply the L-function Ratios Conjecture to a family of cuspidal forms.
  • New Excised Model:
    • Models for zeros near the central point of families of elliptic curves (with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith), Journal of Physics A: Mathematical and Theoretical 45 (2012),  115207 (32pp). Here we develop the theory of excised ensembles and see excellent agreement with quadratic twists of elliptic curves.

For other papers, please see http://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/papers.html