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Siegel Modular Forms and the Sato-Tate
Conjecture Kevin McGoldrick Advisor Professor
Nathan RyanBucknell University
Abstract
Goodness of Fit
Siegel Modular Forms
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 ?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 conjecture
for Siegel Modular Forms as well.
Frobenius Angles for Prime Coefficients of Delta
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.
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 a
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.
Classical Modular Forms
Satake Parameters
Computing Siegel Modular Forms
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 ?p 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 a Siegel Modular Form requires a number
of preliminary computations. Indeed, we have
that where ?10 is itself a SMF, F10 and F12
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 k2 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
?20. 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 ?20. Again with SAGE we find
each of the aforementioned components. Finally,
we can compute ?20 and attempt to find the Hecke
eigenvalues.
Eigenvalues forUpsilon 20
a0 angles for Delta
a1 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
fp. 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 ß0, ß1, ß2 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 k10 and g2. 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 ?20 in hand we
proceed to find the corresponding Hecke
eigenvalues. Each eigenvalue ?p 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
ß1 angles for lifted form
ß2 angles for lifted form