Modular symbols, Eisenstein series, and congruences

Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k ≥ 2 and of the same level N, both eigenfunctions of the Hecke operators, and both normalized so that a₁ = 1. The main result we seek is that when E and f are congruent mod a prime p (which may be a prime ideal lying over a rational prime p > 2), the algebraic parts of the special values L(E,χ,j) and L(f,χ,j) satisfy congruences mod the same prime. More explicitly, the congruence result states that, under certain conditions, τ(χ ̄)L(f,χ,j)/(2πi)^(j−1)Ω_f^(sgn(E)) ≡ τ(χ ̄)L(E,χ,j)/(2πi)^(j)Ω_E (mod p) where the sign of E is ±1 depending on E, and Ω_f^(sgn(E)) is the corresponding canonical period for f. Also, χ is a primitive Dirichlet character of conductor m, τ(χ ̄) is a Gauss sum, and j is an integer with 0 < j < k such that (−1)^(j−1) · χ(−1) = sgn(E). Finally, Ω_E is a p-adic unit which is independent of χ and j. This is a generalization of earlier results of Stevens and Vatsal for weight k = 2. In this paper we construct the modular symbol attached to an Eisenstein series, and compute the special values. We give numerical examples of the congruence theorem stated above, and we sketch the proof of the congruence theorem.Science, Faculty ofMathematics, Department ofGraduat