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

In: Transactions of the American Mathematical Society, Vol. 360, No. 11, 11.2008, p. 5999–6015.

Research output: Contribution to journalArticle

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In: Transactions of the American Mathematical Society, Vol. 360, No. 11, 11.2008, p. 5999–6015.

Research output: Contribution to journalArticle

### Harvard

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

### APA

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

### Vancouver

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

### Author

Busuioc, Cecilia. / The Steinberg Symbol and Special Values of L-Functions. In: Transactions of the American Mathematical Society. 2008 ; Vol. 360, No. 11. pp. 5999–6015.

### BibTeX

@article{1512945c6e1d4de5bca660365892d9eb,
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}$, we show that the values of the above pairing are congruent mod p to the L-values of a weight k, level 1 cusp form which satisfies Eisenstein-type congruences mod p, a result that was predicted by a conjecture of R. Sharifi.", author = "Cecilia Busuioc", year = "2008", month = nov, doi = "10.1090/S0002-9947-08-04701-6", language = "English", volume = "360", pages = "5999–6015", journal = "Transactions of the American Mathematical Society", issn = "0002-9947", publisher = "American Mathematical Society", number = "11", } ### RIS TY - JOUR T1 - The Steinberg Symbol and Special Values of L-Functions AU - Busuioc, Cecilia PY - 2008/11 Y1 - 2008/11 N2 - The main results of this article concern the definition of a compactlysupported cohomology class for the congruence group$\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

M3 - Article

VL - 360

SP - 5999

EP - 6015

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 11

ER -