Siegel Modular Forms and the Sato-Tate Conjecture
Advisor: Professor Nathan Ryan
Siegel Modular Forms
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
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 a(1 2 , p(1 2 ), p 2 )
1 2 0 mod p
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:
q(1 q )
a ( n ) q
Obtaining these will allow us to derive the corresponding Hecke
eigenvalues for the Siegel Modular form and proceed to examine
f ( z ) f ( z 1)
Delta is classified as an eigen-cuspform given that it has the
For each prime p, it is also possible to derive Satake parameters
for the modular form by solving the following equations:
p 0 (1 1 )
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:
0 angles for Delta
1 angles for Delta
a ( 0) 0
a (mn) a (m)a (n)
a ( p ) 2 p
a ( p ) 2 p
( k 1) / 2
cos( p )
for some angle p.
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 0, 1, 2 of a lifted modular
form of weight k from the parameters of the original modular form
through the following formulas:
The Sato-Tate conjecture claims that these angles are distributed
sin ( )
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
20 2 3 5 7 11 V ( 12 E4 10 E4 E6 )
where 10 is itself a SMF, 10 and 12 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 20.
4 gk 3 g 2 2 g
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:
1 angles for lifted form
2 angles for lifted form
12 I (,0)
10 V (10 )
The necessary elliptic modular
are given by the following:
E 2 k 1
2k 1 (l )q
where Bk is the kth Bernoulli number, and
2 k 1 (l ) d
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.
H O : The eigenvalues are distributed as hypothesized.
H A : The eigenvalues have some other distribution.
D max Fn ( x) F ( x) 0.0820
The critical value for D at = 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.
Computing a Siegel Modular Form requires a number of
preliminary computations. Indeed, we have that:
10 I (0, )
( k 1) / 2
Computing Siegel Modular Forms
Now, by various theorems we have the following:
gcd(m, n) 1
In terms of classical modular forms, the Sato-Tate conjecture
concerns eigen-cuspforms. In 1970, Deligne proved that
n r 'm
r , n , m
r 2 4 mn0
n , m 0
Classical Modular Forms
f ( z)
A Siegel Modular Form has the Fourier expansion
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.
If k is a positive integer and f(z) is a holomorphic function on the
complex upper half place which satisfies
Goodness of Fit
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.
For visual demonstration, we have a histogram of eigenvalues
imposed on a plot of the distribution suggested by recent research.
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
Finally, another possible extension of this project is to compute
other Siegel Modular Forms, rather than only Upsilon20, and to
analyze their coefficients.
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
p a (1,1,1) a( p, p, p) p
(1 ( )) a(1,1,1)
As previously mentioned, it we also require the eigenvalues for
each p2 as well, which are obtained in a similar fashion.
 Breulmann, Stefan and Michael Kuss. On a Conjecture of
Duke-Imamoglu. Proc. of AMS. 2000
 Skoruppa, Nils-Peter. Computations of Siegel Modular Forms of
Genus Two. Math. Comp. 1992
 Stein, William. SAGE mathematics software system
Made possible by Bucknell Program for Undergraduate Research
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