Critical L-values of Level p Newforms (modp)

Suppose that p≥ 5 is prime, that ℱ(z) ∈ S2k(Γ0(p)) is a newform, that v is a prime above p in the field generated by the coefficients of ℱ, and that D is a fundamental discriminant. We prove non-vanishing theorems modulo v for the twisted central critical values L(ℱ⊗ χD,k). For example, we show that if k is odd and not too large compared to p, then infinitely many of these twisted L-values are non-zero (mod v). We give applications for elliptic curves. For example, we prove that if E/ℚ is an elliptic curve of conductor p, where p is a sufficiently large prime, there are infinitely many twists D with Ш(ED/ℚ)[p] = 0, assuming the Birch and Swinnerton-Dyer conjecture for curves of rank zero as well as a weak form of Hall’s conjecture. The results depend on a careful study of the coefficients of half-integral weight newforms of level 4p, which is of independent interest.