A reciprocity map and the two-variable $p$-adic $L$-function

Abstract

For primes $p \ge 5$, we propose a conjecture that relates the values of cup products in the Galois cohomology of the maximal unramified outside $p$ extension of a cyclotomic field on cyclotomic $p$-units to the values of $p$-adic $L$-functions of cuspidal eigenforms that satisfy mod $p$ congruences with Eisenstein series. Passing up the cyclotomic and Hida towers, we construct an isomorphism of certain spaces that allows us to compare the value of a reciprocity map on a particular norm compatible system of $p$-units to what is essentially the two-variable $p$-adic $L$-function of Mazur and Kitagawa.

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      ISSN = {0037-9484},
      CODEN = {BSMFAA},
      MRCLASS = {11G18 (11F99 11R23)},
      MRNUMBER = {926532},
      MRREVIEWER = {Noburo Ishii},
      URL = {http://www.numdam.org/item?id=BSMF_1987__115__329_0},
      ZBLNUMBER = {0677.14006},
      }

Authors

Romyar Sharifi

Department of Mathematics
University of Arizona
Tuscon, AZ 85721