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Barry Mazur, Glenn Stevens - 1994 - Preview
|PROOF: Let V be the associated representation space of pm and Set d = dimk V |
and consider W = V® (Hom(V",Mp) ® xfc~V) , where x is the p-cyclotomic
character and e((.) = (() e k = T/m is the Neben- typus character. By the Eichler-
|... of the complex L-function associated to 6 with Euler factors at primes in EU |
removed. The p-adic cyclotomic character of F is denoted by K i.e. it is a
homomorphism K: Gal(F/F)—> Z; defined by g f I $45'), for all g G Gal(F/F) and
allE G ,apoo.
|The cyclotomic theory. This section will be devoted to a brief account of the p-adic |
analogue of the Riemann zeta function. Recalling that X is given by (4), we write
Xalg for the subgroup of X consisting of all characters of the form E,=\|!1\x (ne Z) ...
|of N-th roots of unity for any ring A. Let Q be the totality of algebraic numbers in C. |
We consider the cyclotomic field Q(uN). ... We simply write X, for Xp. By taking the
projective limit, we get the p-adic cyclotomic character x = |imax, : Gal(Q.5/Q) ...
|We view without comment a character of Gal(Qp /F) as a character of F×. We |
denote by ε the p-adic cyclotomic character. It corresponds to the character of F×
given by x→ |x|F ∏ σ∈S σ(x). A p-adic norm on an E-vector space V is a function
|For each place p of Q, we fix embeddings ιp of Q in its completions Qp. Denote by |
χp the p-adic cyclotomic character, and ωp the Teichm ̈uller lift of the mod p
cyclotomic character χp (the latter being the reduction mod p of χp). By abuse of ...
|We fix a prime p > 2 and write | - |p for the p—adic absolute value on Q nor1 |
malized so that |p|p I 5. ... p/Qp) is upper triangular with diagonal characters Naj
on the inertia I}, for the p-adic cyclotomic character N ordered from top to bottom
as a1 ...
|Let p be a prime number, and let K be a finite extension of Qp. Let N > 1 be an |
integer relatively prime to p, f € S1(N)K a ... unramified characters of GQ, whose
value at pp are a and 3, respectively. Let X be the p-adic cyclotomic character.
|The other 0 Â is the constant term modulo Â of a -adic modular form that appears |
in a computation of the regulators of zeta elements. 2.7.2. The map HS ... The
map ÄWGQ ! Zp is the p-adic cyclotomic character, and log is the p-adic logarithm
|GROUP. CHARACTERS. Let E be an elliptic curve over Q with complex |
multiplication by the ring of integers OK. of an imaginary ... It is analogous to the
fact that the restriction to Gp — Gal(<Q^/Qp) of the p-adic cyclotomic character xP
: Go ~t ...