We specialize the Eichler-Selberg trace formula to obtain trace formulas for the prime-to-level Hecke action on cusp forms for certain congruence groups of arbitrary level. As a consequence, we determine the asymptotic in the prime p of the number of (weighted) SL2(Z)-conjugation orbits of 2 × 2 matrices of determinant p whose reductions modulo N lie in a given conjugacy class in GL2(Z/NZ). Thi...

If X,Y are normed spaces, let B(X,Y ) be the set of all bounded linear maps X → Y . If T : X → Y is a linear map, I take it as known that T is bounded if and only if it is continuous if and only if E ⊆ X being bounded implies that T (E) ⊆ Y is bounded. I also take it as known that B(X,Y ) is a normed space with the operator norm, that if Y is a Banach space then B(X,Y ) is a Banach space, that ...

Trace class operators Cont Last time I define and operator to be of 'trace class' if it is a finite sum of products of Hilbert-Schmidt operators (1) T =