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- W2107340925 abstract "Note that potential modularity of E [Wi, Thm. A.1] implies that the L-function L(E/F, s) has a meromorphic continuation to C and satisfies the expected functional equation ([T2, proof of Cor. 2.2]; [N2, 12.11.6]). As a result, the integer ords=1 L(E/F, s) ∈ Z is well-defined. Various special cases of Theorem A (for F 6= Q) were proved in [N2], [Ki] and [N6]. If the p-primary part of X(E/F ) is finite for some prime number p, then sp(E/F ) = rkZE(F ) and the statement of Theorem A is the conjecture of Birch and Swinnerton-Dyer for E over F modulo 2. Theorem B. Let g = ∑∞ n=1 anq n ∈ S2r(Γ0(N)) (r ≥ 1) be a normalised (a1 = 1) newform, let L = Q(a1, a2, . . .) be the (totally real) number field generated by its coefficients. For any prime p of L above a rational prime p 6= 2, denote by Vp(g) the two-dimensional representation of GQ = Gal(Q/Q) over Lp attached to g: det (1−X Frgeom(l) | Vp(g)) = 1− alX + l2r−1X2 (l pN)." @default.
- W2107340925 created "2016-06-24" @default.
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- W2107340925 date "2013-09-06" @default.
- W2107340925 modified "2023-10-14" @default.
- W2107340925 title "Some consequences of a formula of Mazur and Rubin for arithmetic local constants" @default.
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- W2107340925 doi "https://doi.org/10.2140/ant.2013.7.1101" @default.
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