Siegel Modular Forms and the Sato-Tate Conjecture Kevin McGoldrick

1. Siegel Modular Forms and the Sato-Tate Conjecture Kevin McGoldrick Advisor: Professor Nathan RyanBucknell University Abstract Siegel Modular Forms Goodness of Fit 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. Then, we will compute a large number of coefficients for the Siegel Modular Form Υ20. 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 conjecture for Siegel Modular Forms as well. 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. 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. 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,p2). Also necessary will be finding Obtaining these will allow us to derive the corresponding Hecke eigenvalues for the Siegel Modular form and proceed to examine their distributions. Frobenius Angles for Prime Coefficients of Delta Classical Modular Forms Satake Parameters If k is a positive integer and f(z) is a holomorphic function on the complex upper half place which satisfies and has a Fourier expansion of the form then f is a modular form of weight k. We consider the modular form delta, defined as: Delta is classified as an eigen-cuspform given that it has the following properties: 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. 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 other will be uniformly distributed. Indeed, if we find Satake parameters for the modular form delta, the following distributions are observed: Computing Siegel Modular Forms Computing a Siegel Modular Form requires a number of preliminary computations. Indeed, we have that: where χ10 is itself a SMF, Φ10 and Φ12 are Jacobi forms, and finally E4 and E6 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 Υ20. Now, by various theorems we have the following: 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 Υ20. Again with SAGE we find each of the aforementioned components. Finally, we can compute Υ20 and attempt to find the Hecke eigenvalues. Eigenvalues forUpsilon 20 α0 angles for Delta α1 angles for Delta 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 Upsilon20, and to analyze their coefficients. Sato-Tate Conjecture In terms of classical modular forms, the Sato-Tate conjecture concerns eigen-cuspforms. In 1970, Deligne proved that Thus, for some angle φp. 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. Lifted Modular Forms By the Saito-Kurokawa lift, when given a modular form with weight 2k-g, where k and g are positive even integers, we can determine the Satake parameters β0, β1, β2 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 observe the distributions of the corresponding angles: Hecke Eigenvalues of SMF With the Fourier coefficients of Υ20 in hand we proceed to find the corresponding Hecke eigenvalues. Each eigenvalue λp can be found using the fact that As previously mentioned, it we also require the eigenvalues for each p2 as well, which are obtained in a similar fashion. 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 [3] Stein, William. SAGE mathematics software system Made possible by Bucknell Program for Undergraduate Research β1 angles for lifted form β2 angles for lifted form