Siegel Modular Forms and the SatoTate Conjecture Kevin

Siegel Modular Forms and the Sato-Tate Conjecture Kevin Mc. Goldrick Advisor: Professor Nathan Ryan Bucknell University Abstract Distribution The Sato-Tate conjecture makes a statement about the distribution of certain numbers. In this project, we will first explore the Sato-Tate conjecture about Satake parameters for classical and lifted modular forms in order to become familiar with both modular forms and the conjecture itself. Frobenius Angles for Prime Coefficients of Delta Then, we will compute a large number of coefficients for the Siegel Modular Form Υ 20. With these coefficients, we can again find the corresponding Satake parameters and study their distributions. The goal is to formulate a version of the Sato-Tate Classical Modular Forms conjecture for Siegel Modular Forms as well. If k is a positive integer and f(z) is a holomorphic function on the complex upper half place which satisfies We consider the modular form delta, defined as: Delta is classified as an eigen-cuspform given that it has the following properties: A Siegel Modular Form has the Fourier expansion as before, we are interested in specific coefficients of the expansion. In this case we wish to have: a(1, 1, 1), a(p, p, p), a(1, p, p 2). Also necessary will be finding Satake Parameters For each prime p, it is also possible to derive Satake parameters for the modular form by solving the following equations: where k is the weight and λp is the p-th Fourier coefficient. and has a Fourier expansion of the form then f is a modular form of weight k. Siegel Modular Forms Since the Satake parameters are complex numbers, we may associate an angle with each of them. Sato. Tate asserts that one parameter will follow the distribution observed in the Frobenius angles, while the αother will be uniformly distributed. Indeed, if we α 1 angles for Delta 0 angles for Delta find Satake parameters for the modular form delta, the following distributions are observed: Obtaining these will allow us to derive the corresponding Hecke eigenvalues for the Siegel Modular form and proceed to examine their distributions. Computing a Siegel Modular Form requires a number of preliminary computations. Indeed, we have that: where χ10 is itself a SMF, Φ 10 and Φ 12 are Jacobi forms, and finally E 4 and E 6 are elliptic modular forms. Important to the process will be two mappings: the I map which maps the direct sum of a cusp form of weight k and a cusp form of weight k+2 to a Jacobi form of weight k, and the V map which sends Jacobi forms of weight k to Siegel Modular Forms also of weight k. We see the V map in the explicit formula for Υ 20. Sato-Tate Conjecture Thus, angle φp. for some The Sato-Tate conjecture claims that these angles are distributed as such: where We consider the previously defined form delta of weight 12. Using SAGE we compute its coefficients and find the corresponding angles. Then we can verify the Sato-Tate conjecture with a histogram of the angles. The results are shown to the right. TEMPLATE DESIGN © 2008 www. Poster. Presentations. com Lifted Modular Forms By the Saito-Kurokawa lift, when given a modular form with weight 2 k-g, where k and g are positive even integers, we can determine the Satake parameters β 0, β 1, β 2 of a lifted modular form of weight k from the parameters of the original modular form through the following formulas: Again, Sato-Tate makes a claim about the distribution of these lifted Satake parameters. We explore the lifted modular form of weight 10 by setting k=10 and g=2. We find the parameters for the classical modular form of weight 18 by the previous procedure and then derive the lifted parameters. Finally, we β 1 angles for lifted form β 2 angles for lifted form observe the distributions of the corresponding angles: Given the complexity of Siegel Modular Forms, we were only able to compute Hecke eigenvalues for the first 168 primes. Such a small sample will not allow us to determine the distribution, however we perform a Kolmogorov-Smirnov goodness of fit hypothesis test to find the probability that the eigenvalues are distributed in the fashion suggested by recent research. Note that we do not have an explicit formula for this distribution, but a list of points which lie upon the distribution. The eigenvalues are distributed as hypothesized. The eigenvalues have some other distribution. Computing Siegel Modular Forms Now, by various theorems we have the following: In terms of classical modular forms, the Sato-Tate conjecture concerns eigen-cuspforms. In 1970, Deligne proved that Goodness of Fit The necessary elliptic modular forms are given by the following: where Bk is the kth Bernoulli number, and Given this, we have the tools to compute Υ 20. Again Hecke Eigenvalues of SMF with SAGE we find each of the aforementioned components. Finally, we canofcompute Υ 20 and With the Fourier coefficients Υ 20 in hand we attempt eigenvalues. proceedto tofindthe the. Hecke corresponding Hecke Each eigenvalue λp can be found using the fact that The critical value for D at α = 0. 10 for a sample size of 168 is 0. 0934. Thus, we fail to reject the null hypothesis and conclude that there is not significant evidence of the Hecke eigenvalues having some other distribution. For visual demonstration, we have a histogram of eigenvalues imposed on a plot of the distribution suggested by recent research. Eigenvalues for Upsilon 20 Continuing Work Future work will seek to optimize the code which computes Siegel Modular Forms. As of now, the computing algorithm is too memory expensive for most computers to calculate a sufficient amount of coefficients that would allow us to make a satisfactory formulation of the Sato-Tate Conjecture for Siegel Modular Forms. Since most coefficients are of no consequence to the conjecture, saving only those which are necessary may be more efficient. Finally, another possible extension of this project is to compute other Siegel Modular Forms, rather than only Upsilon 20, and to analyze their coefficients. Acknowledgments [1] Breulmann, Stefan and Michael Kuss. “On a Conjecture of Duke-Imamoglu”. Proc. of AMS. 2000 [2] Skoruppa, Nils-Peter. “Computations of Siegel Modular Forms of Genus Two”. Math. Comp. 1992 As previously mentioned, it we also require the eigenvalues for each p 2 as well, which are obtained in a similar fashion. [3] Stein, William. SAGE mathematics software system Made possible by Bucknell Program for Undergraduate Research