In analytic number theory, the Petersson trace formula is a kind of orthogonality relation between coefficients of a holomorphic

modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...

. It is a specialization of the more general

Kuznetsov trace formula In analytic number theory, the Kuznetsov trace formula is an extension of the Petersson trace formula. The Kuznetsov or ''relative trace'' formula connects Kloosterman sums at a deep level with the spectral theory of automorphic forms. Originally ...

. In its simplest form the Petersson trace formula is as follows. Let

\mathcal

be an

orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For ex ...

S_k(\Gamma(1))

, the space of cusp forms of weight

k>2

SL_2(\mathbb)

. Then for any positive integers

m,n

we have :

\frac \sum_ \bar(m) \hat(n) = \delta_ + 2\pi i^ \sum_\frac J_\left(\frac\right),

where

\delta

is the

Kronecker delta function In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...

S

is the

Kloosterman sum In mathematics, a Kloosterman sum is a particular kind of exponential sum. They are named for the Dutch mathematician Hendrik Kloosterman, who introduced them in 1926 when he adapted the Hardy–Littlewood circle method to tackle a problem involvi ...

and

J

is the

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

of the first kind.

References

Henryk Iwaniec Henryk Iwaniec (born October 9, 1947) is a Polish-American mathematician, and since 1987 a professor at Rutgers University. Background and education Iwaniec studied at the University of Warsaw, where he got his PhD in 1972 under Andrzej Schinz ...

: ''Topics in Classical Automorphic Forms''. Graduate Studies in Mathematics 17, American Mathematics Society, Providence, RI, 1991. * Theorems in analytic number theory {{Numtheory-stub Automorphic forms