## Abstract

The main results of this article concern the definition of a compactly

supported cohomology class for the congruence group $\Gamma_0(p^n)$ with values in the second Milnor K-group (modulo 2-torsion) of the ring of p-integers of the cyclotomic extension $\mathbb{Q}(\mu){p^n}). We endow this cohomology group with a natural action of the standard Hecke operators and discuss the existence of special Hecke eigenclasses in its parabolic cohomology. Moreover, for n = 1, assuming the non-degeneracy of a certain pairing on p-units induced by the Steinberg symbol when (p, k) is an irregular pair, i.e. $p|\frac{B_k}{k}$, we show that the values of the above pairing are congruent mod p to the L-values of a weight k, level 1 cusp form which satisfies Eisenstein-type congruences mod p, a result that was predicted by a conjecture of R. Sharifi.

supported cohomology class for the congruence group $\Gamma_0(p^n)$ with values in the second Milnor K-group (modulo 2-torsion) of the ring of p-integers of the cyclotomic extension $\mathbb{Q}(\mu){p^n}). We endow this cohomology group with a natural action of the standard Hecke operators and discuss the existence of special Hecke eigenclasses in its parabolic cohomology. Moreover, for n = 1, assuming the non-degeneracy of a certain pairing on p-units induced by the Steinberg symbol when (p, k) is an irregular pair, i.e. $p|\frac{B_k}{k}$, we show that the values of the above pairing are congruent mod p to the L-values of a weight k, level 1 cusp form which satisfies Eisenstein-type congruences mod p, a result that was predicted by a conjecture of R. Sharifi.

Original language | English |
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Pages (from-to) | 5999–6015 |

Number of pages | 17 |

Journal | Transactions of the American Mathematical Society |

Volume | 360 |

Issue number | 11 |

Early online date | 21 Sept 2006 |

DOIs | |

Publication status | Published - Nov 2008 |