Petersson trace formula explained

In analytic number theory, the Petersson trace formula is a kind of orthogonality relation between coefficients of a holomorphic modular form. It is a specialization of the more general Kuznetsov trace formula.

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

l{F} be an orthonormal basis of S _{k(\Gamma(1))} , the space of cusp forms of weight k> 2 on SL _{2(Z)} . Then for any positive integers m,n we have(\Gamma(k -1 ))/(( 4 \pi \sqrt{mn) ) ^{k-1} } \sum_ \bar(m) \hat(n) = \delta_ + 2\pi i^ \sum_\frac J_\left(\frac\right),where \delta is the Kronecker delta function, S is the Kloosterman sum and J is the Bessel function of the first kind.

References

• Henryk Iwaniec: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics 17, American Mathematics Society, Providence, RI, 1991.
• Book: 10.1090/amsip/037/04. Petersson and Kuznetsov trace formulas . Lie Groups and Automorphic Forms . AMS/IP Studies in Advanced Mathematics . 2006 . 37 . 147–168 . 9780821841983 .