Modular Groups and Modular Forms

Modular symbols, Eisenstein series, and congruences

“…When ε 1 is primitive, this construction is the same (up to scaling) as the usual construction of the Eisenstein series associated to a pair of Dirichlet characters, such as the one that can be found in [10].…”

“…(For full details, see [10], chapter 4, in particular section 4.7.) Now let χ be a nontrivial primitive Dirichlet character.…”

Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms

“…Let H be the upper half complex plane, G k (N, χ) the space of holomorphic modular forms on H of the integral weight k and character χ with respect to Γ 0 (N ) and G k (N, χ) the space of holomorphic modular forms on H of the half integral weight k and character χ with respect to Γ 0 (N ), for the definition, see [3] for integral k and [10] for half integral k. The following is our main theorem.…”

“…The proof of the theorem is based on the functional equations satisfied by ζ (w, s) and the converse theorems of Weil type (Weil [15] and Miyake [3] for the case of integral weight and Shimura [10] for the case of half integral weight).…”

“…We can regard the prehomogeneous vector space studied in [14] as a special case of the prehomogeneous vector spaces (P × GL 1 (C), V ), since the group SL 2 (C) is isomorphic to Spin (1,3) and is locally isomorphic to SO (1,3). That is the case of m = 2 and A (negative) definite.…”