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
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