**The Steinberg Symbol and Special Values of L-Functions.** / Busuioc, Cecilia.

Research output: Contribution to journal › Article

Published

**The Steinberg Symbol and Special Values of L-Functions.** / Busuioc, Cecilia.

Research output: Contribution to journal › Article

Busuioc, C 2008, 'The Steinberg Symbol and Special Values of L-Functions', *Transactions of the American Mathematical Society*, vol. 360, no. 11, pp. 5999–6015. https://doi.org/10.1090/S0002-9947-08-04701-6

Busuioc, C. (2008). The Steinberg Symbol and Special Values of L-Functions. *Transactions of the American Mathematical Society*, *360*(11), 5999–6015. https://doi.org/10.1090/S0002-9947-08-04701-6

Busuioc C. The Steinberg Symbol and Special Values of L-Functions. Transactions of the American Mathematical Society. 2008 Nov;360(11):5999–6015. https://doi.org/10.1090/S0002-9947-08-04701-6

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title = "The Steinberg Symbol and Special Values of L-Functions",

abstract = "The main results of this article concern the definition of a compactlysupported 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.",

author = "Cecilia Busuioc",

year = "2008",

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doi = "10.1090/S0002-9947-08-04701-6",

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AU - Busuioc, Cecilia

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N2 - The main results of this article concern the definition of a compactlysupported 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.

AB - The main results of this article concern the definition of a compactlysupported 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.

U2 - 10.1090/S0002-9947-08-04701-6

DO - 10.1090/S0002-9947-08-04701-6

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JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

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