Siegel Modular Forms and the Sato-Tate Conjecture

Kevin McGoldrick

Advisor: Professor Nathan Ryan

Bucknell University

Abstract

Distribution

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 )

mod p

1 2 0 mod p

n

f

(

z

)

a

(

n

)

q

and has a Fourier expansion of the form

n 1

then f is a modular form of weight k.

We consider the modular form delta, defined as:

q(1 q )

24

a ( n ) q

Obtaining these will allow us to derive the corresponding Hecke

eigenvalues for the Siegel Modular form and proceed to examine

their distributions.

Satake Parameters

f ( z ) f ( z 1)

n

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:

p

k1

p 0 (1 1 )

2

0 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

Thus,

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:

0 p

The Sato-Tate conjecture claims that these angles are distributed

as 2

such: 2

0

sin ( )

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.PosterPresentations.com

1

1

2

20 2 3 5 7 11 V ( 12 E4 10 E4 E6 )

2

2

9

2

2

10

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

8

1 p

g

k 1

2

0

2 p

g

k 1

2

01

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:

4k forms

l

E 2 k 1

2k 1 (l )q

B2 k

l 1

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, )

Sato-Tate Conjecture

( 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

a

(

n

,

r

,

m

)

q

q

F

r , n , m

r 2 4 mn0

n , m 0

Classical Modular Forms

1

k

f

z

f ( z)

z

A Siegel Modular Form has the Fourier expansion

F

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

2k 1

d |l

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.

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 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

k 2

p

3

(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.

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