p-adicL-functions andp-adic periods of modular forms

Modular symbols, Eisenstein series, and congruences

“…This operator also appears in Section 4 of [5], but our definition is slightly different. This is because our matrix action (of GL 2 (Q on L k−2 (C) is a left action.…”

“…(The meaning of "algebraic part" will be explained below.) The discussion in this section will closely follow that of [5], Section 4.…”

“…Following [5], we define them using double coset operators. Let g be a matrix with positive determinant and integer entries, and letΓ be any congruence subgroup.…”

“…Let S Γ (A) and B Γ (A) denote the groups of A-valued modular symbols and boundary symbols over Γ respectively. Then according to [5], Section 4, there is an exact sequence as follows:…”

“…We obtain a new modular symbol which we will denote M ± f (for one choice of sign). It is a theorem of Shimura (proved in [6], or also see [5], [9], or [18]) that there exist transcendental numbers Ω ± f , called periods, such that the modular symbols 1…”