CHAPTER 1

Introduction

The aim of this article is to study the family of the so-called Brandt matrices. In

the number field case, the entries of these matrices are essentially the representation

number of positive integers by reduced norm forms on definite quaternion algebras

over Q. Let N0

−

be a square-free positive integer with an odd number of prime

factors. Let D = DN0

−

be the definite quaternion algebra over Q which is ramified

precisely at the prime factors of N0

−.

Let N0

+

be another square-free positive integer

prime to N0

−.

Take an Eichler order R of type (N0

+,N0 −),

i.e. R is an order in D

satisfying that Rp := R ⊗Z Zp is a maximal Zp-order in Dp := D ⊗Q Qp for every

p N0

+

, and for p |

N0+

Rp

∼

=

a b

c d

∈ Mat2(Zp) c ∈ N0

+Zp

.

Choose {I1, ..., In} to be a complete set of representatives of locally-principal right

ideal classes of R. For each positive integer m and 1 ≤ i, j ≤ n, set

Bij(m) :=

#{b ∈ IiIj

−1

: Nr(b)Nij

−1

= m}

#(Rj

×)

∈ Z≥0,

where Rj is the left order of Ij, Nr(b) is the reduced norm of b, and Nij is the

positive generator of the fraction ideal

Nr(Ii) Nr(Ij)−1 in Q. We call B(m) :=

(Bij(m))1≤i,j≤n the m-th Brandt matrix associated to R. For convenience, set

Bij(0) := 1/#(Rj

×

).

It is known that for each pair (i, j), 1 ≤ i, j ≤ n, the following theta series

m≥0

Bij(m) exp(2π

√

−1mz)

is a weight-2 modular form of level N0

+N0 −.

Recall that the basis problem (cf. [3])

is about finding a ”natural” basis for the space of modular forms. Here ”natural”

means that these linearly independent forms are arithmetically distinguished and

whose Fourier coeﬃcients are known or easy to obtain. The celebrated Eichler’s

trace formula (cf. [2] and [3]) connects the Brandt matrices with the Hecke operators

on the space of weight-2 modular forms. This implies, among other things, that

the theta series from definite quaternion algebras over Q generate the whole space

of weight-2 modular forms of the corresponding level. In other words, these theta

series provide us a natural basis and give a solution of the basis problem for weight-2

modular forms.

By a global function field k, we mean k is a finitely generated field extension of

transcendence degree one over a finite field. Fix a place ∞ of k, we are interested in

Drinfeld type automorphic forms, which are automorphic forms on GL2 satisfying

the so-called harmonic property with respect to ∞ (cf. Chapter 3 Section 3). In

1