### Abstract

For primes p ≥ 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.

Original language | English (US) |
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Pages (from-to) | 251-300 |

Number of pages | 50 |

Journal | Annals of Mathematics |

Volume | 173 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2011 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

**A reciprocity map and the two-variable p-adic L-function.** / Sharifi, Romyar T.

Research output: Contribution to journal › Article

*Annals of Mathematics*, vol. 173, no. 1, pp. 251-300. https://doi.org/10.4007/annals.2011.173.1.7

}

TY - JOUR

T1 - A reciprocity map and the two-variable p-adic L-function

AU - Sharifi, Romyar T

PY - 2011/1

Y1 - 2011/1

N2 - For primes p ≥ 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.

AB - For primes p ≥ 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.

UR - http://www.scopus.com/inward/record.url?scp=78751633330&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78751633330&partnerID=8YFLogxK

U2 - 10.4007/annals.2011.173.1.7

DO - 10.4007/annals.2011.173.1.7

M3 - Article

AN - SCOPUS:78751633330

VL - 173

SP - 251

EP - 300

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 1

ER -