TY - JOUR

T1 - Critical L-values of Level p Newforms (mod p)

AU - Ahlgren, Scott

AU - Rouse, Jeremy

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2010

Y1 - 2010

N2 - Suppose that p≥ 5 is prime, that ℱ(z) ∈ S2k(Γ 0(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 III(E D/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.

AB - Suppose that p≥ 5 is prime, that ℱ(z) ∈ S2k(Γ 0(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 III(E D/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.

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U2 - 10.1093/imrn/rnp180

DO - 10.1093/imrn/rnp180

M3 - Article

AN - SCOPUS:77951813293

VL - 2010

SP - 1575

EP - 1606

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 9

ER -