Holmstrom Special values of L-functions

A program for a conference on Tamagawa numbers

Stark’s conjectures: recent work and new directions, Contemp. Math., 358, Amer. Math. Soc.

See everything by Lichtenbaum, including Motives volume. Also everything by de Jeu.

See Goncharov in K-theory handbook.

http://mathoverflow.net/questions/70217/k-theory-and-rings-of-integers

http://mathoverflow.net/questions/15153/periods-and-l-values

http://londonnumbertheory.wordpress.com/2010/01/15/zeros-and-poles/

Christophe Soulé, $K$-theory and values of zeta functions, in the mysterious Kuke Bass Pedrini 1997 volume that is nowhere to be found

Deligne: Valeurs de fonctions Let periodes d’integrales (1979)

Borel: Values of zeta functions at integers, cohomology and polylogarithms. 1994 survey, see things to scan for ref.

The cyclotomic trace map and values of zeta functions, by Thomas Geisser: http://www.math.uiuc.edu/K-theory/0697

MR544704 (81i:12010) 12A60 (12B20) Schneider, Peter U¨ ber gewisse Galoiscohomologiegruppen. (German) Math. Z. 168 (1979), no. 2, 181–205.

Nobushige Kurokawa, Special values of Selberg zeta functions (pp. 133–150) (1987)

MR1086888 (92g:11063) Bloch, Spencer(1-ILCC); Kato, Kazuya(J-TOKYO) $L$-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I, 333–400, Progr. Math., 86, Birkhäuser Boston, Boston, MA, 1990. See long MR review

Interesting: MR1330935 (96k:11083) de Shalit, Ehud(IL-HEBR-isomorphism) The explicit reciprocity law of Bloch-Kato. (English summary) Columbia University Number Theory Seminar (New York, 1992). Astérisque No. 228 (1995), 4, 197–221.

MR1408541 (97g:11136) Kolster, Manfred(3-MMAS); Nguyen Quang Do, Thong(F-FRAN-M); Fleckinger, Vincent(F-FRAN-M) Twisted $S$-units, $p$-adic class number formulas, and the Lichtenbaum conjectures. Duke Math. J. 84 (1996), no. 3, 679–717. (important paper)

MR1423032 (98a:11150) Ritter, Jürgen(D-AGSB-MI); Weiss, Alfred(3-AB) Cohomology of units and $L$-values at zero. The article under review presents quite significant progress on Stark’s conjecture in the version of Tate. A considerable amount of algebraic machinery is developed to that end.

MR1609325 (99b:11051) Kings, Guido(D-MUNS) Higher regulators, Hilbert modular surfaces, and special values of $L$-functions. Duke Math. J. 92 (1998), no. 1, 61–127.

MR1685076 (2000b:11067) Huber, Annette(D-MUNS-isomorphism); Kings, Guido(D-MUNS-isomorphism) Dirichlet motives via modular curves. (English, French summary) Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 3, 313–345.

MR1737227 (2001d:11071) Otsubo, Noriyuki(J-TOKYOGM) Note on conjectures of Beilinson-Bloch-Kato for cycle classes. (English summary) Manuscripta Math. 101 (2000), no. 1, 115–124.

MR1760901 (2001i:11082) Bloch, Spencer J.(1-CHI) Higher regulators, algebraic $K$-theory, and zeta functions of elliptic curves.

MR1817645 (2002k:11101) Kings, Guido(D-MUNS-isomorphism) The Tamagawa number conjecture for CM elliptic curves. Invent. Math. 143 (2001), no. 3, 571–627.

MR2002643 (2004m:11182) Huber, Annette(D-LEIP-isomorphism); Kings, Guido(D-RGBGNS1) Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters.

MR2175640 (2006m:11168) Kolster, Manfred(3-MMAS) $K$-theory and arithmetic. (Summer school lectures on special values)

Soulein Asterisque 311: Genres de Todd et valeurs aux entiers des dérivées de fonctions $L$

arXiv:0909.0712 K_1 of products of Drinfeld modular curves and special values of L-functions from arXiv Front: math.AG by Ramesh Sreekantan Beilinson obtained a formula relating the special value of the L-function of H^2 of a product of modular curves to the regulator of an element of a motivic cohomology group - thus providing evidence for his general conjectures on special values of L-functions. In this paper we prove a similar formula for the L-function of the product of two Drinfeld modular curves providing evidence for an analogous conjecture in the case of function fields.

arXiv:0908.0171 Higher Mahler measures and zeta functions from arXiv Front: math.NT by Nobushige Kurokawa, Matilde Lalin, Hiroyuki Ochiai We consider a generalization of the Mahler measure of a multivariable polynomial $P$ as the integral of $\log^k|P|$ in the unit torus, as opposed to the classical definition with the integral of $\log|P|$. A zeta Mahler measure, involving the integral of $|P|^s$, is also considered. Specific examples are computed, yielding special values of zeta functions, Dirichlet $L$-functions, and polylogarithms.

arXiv:0908.0996 A cohomological Tamagawa number formula from arXiv Front: math.NT by Annette Huber, Guido Kings For smooth linear group schemes over $\bbZ$ we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori the cohomological and the motivic Tamagawa measures coincide, which reproves the Bloch-Kato conjecture for motives associated to tori.

http://mathoverflow.net/questions/13287/special-values-of-p-adic-l-functions

nLab page on Special values of L-functions