Generalized Stark formulae over function fields
HTML articles powered by AMS MathViewer
- by Ki-Seng Tan PDF
- Trans. Amer. Math. Soc. 361 (2009), 2277-2304 Request permission
Abstract:
We establish formulae of Stark type for the Stickelberger elements in the function field setting. Our result generalizes work of Hayes and a conjecture of Gross. It is used to deduce a $p$-adic version of the Rubin-Stark Conjecture and the Burns Conjecture.References
- Noboru Aoki, Gross’ conjecture on the special values of abelian $L$-functions at $s=0$, Comment. Math. Univ. St. Paul. 40 (1991), no. 1, 101–124. MR 1104783
- Noboru Aoki, On Tate’s refinement for a conjecture of Gross and its generalization, J. Théor. Nombres Bordeaux 16 (2004), no. 3, 457–486 (English, with English and French summaries). MR 2144953, DOI 10.5802/jtnb.456
- D. Burns, On relations between derivatives of abelian $L$-functions at $s=0$, preprint, 2002.
- David Burns, On the values of equivariant zeta functions of curves over finite fields, Doc. Math. 9 (2004), 357–399. MR 2117419
- David Burns, Congruences between derivatives of abelian $L$-functions at $s=0$, Invent. Math. 169 (2007), no. 3, 451–499. MR 2336038, DOI 10.1007/s00222-007-0052-3
- David Burns and Joongul Lee, On the refined class number formula of Gross, J. Number Theory 107 (2004), no. 2, 282–286. MR 2072389, DOI 10.1016/j.jnt.2004.02.002
- Henri Darmon, Thaine’s method for circular units and a conjecture of Gross, Canad. J. Math. 47 (1995), no. 2, 302–317. MR 1335080, DOI 10.4153/CJM-1995-016-6
- Benedict H. Gross, On the values of abelian $L$-functions at $s=0$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 (1988), no. 1, 177–197. MR 931448
- B. Gross, A note on the refined Stark Conjecture, unpublished.
- David R. Hayes, The refined ${\mathfrak {p}}$-adic abelian Stark conjecture in function fields, Invent. Math. 94 (1988), no. 3, 505–527. MR 969242, DOI 10.1007/BF01394274
- A. Hayward, A class number formula for higher derivatives of abelian $L$-functions, Compos. Math. 140 (2004), no. 1, 99–129. MR 1984423, DOI 10.1112/S0010437X03000265
- P.-Y. Huang, Gross’ conjecture for extensions ramified over four points of $\textbf {P}^1$, manuscript, 2004, submitted.
- H. Kisilevsky, Multiplicative independence in function fields, J. Number Theory 44 (1993), no. 3, 352–355. MR 1233295, DOI 10.1006/jnth.1993.1059
- Joongul Lee, On Gross’s refined class number formula for elementary abelian extensions, J. Math. Sci. Univ. Tokyo 4 (1997), no. 2, 373–383. MR 1466351
- Joongul Lee, Stickelberger elements for cyclic extensions and the order of vanishing of abelian $L$-functions at $s=0$, Compositio Math. 138 (2003), no. 2, 157–163. MR 2018824, DOI 10.1023/A:1026128923148
- Joongul Lee, On the refined class number formula for global function fields, Math. Res. Lett. 11 (2004), no. 5-6, 583–587. MR 2106227, DOI 10.4310/MRL.2004.v11.n5.a3
- Cristian D. Popescu, Gras-type conjectures for function fields, Compositio Math. 118 (1999), no. 3, 263–290. MR 1711315, DOI 10.1023/A:1001586625441
- Cristian D. Popescu, On a refined Stark conjecture for function fields, Compositio Math. 116 (1999), no. 3, 321–367. MR 1691163, DOI 10.1023/A:1000833610462
- Cristian D. Popescu, Base change for Stark-type conjectures “over $\Bbb Z$”, J. Reine Angew. Math. 542 (2002), 85–111. MR 1880826, DOI 10.1515/crll.2002.010
- Cristian D. Popescu, The Rubin-Stark conjecture for a special class of function field extensions, J. Number Theory 113 (2005), no. 2, 276–307. MR 2153279, DOI 10.1016/j.jnt.2004.10.002
- Michael Reid, Gross’ conjecture for extensions ramified over three points of $\Bbb P^1$, J. Math. Sci. Univ. Tokyo 10 (2003), no. 1, 119–138. MR 1963800
- Karl Rubin, A Stark conjecture “over $\mathbf Z$” for abelian $L$-functions with multiple zeros, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 33–62 (English, with English and French summaries). MR 1385509, DOI 10.5802/aif.1505
- Jonathan W. Sands, Stark’s conjecture and abelian $L$-functions with higher order zeros at $s=0$, Adv. in Math. 66 (1987), no. 1, 62–87. MR 905927, DOI 10.1016/0001-8708(87)90030-2
- H. M. Stark, Values of $L$-functions at $s=1$. I. $L$-functions for quadratic forms, Advances in Math. 7 (1971), 301–343 (1971). MR 289429, DOI 10.1016/S0001-8708(71)80009-9
- H. M. Stark, $L$-functions at $s=1$. II. Artin $L$-functions with rational characters, Advances in Math. 17 (1975), no. 1, 60–92. MR 382194, DOI 10.1016/0001-8708(75)90087-0
- H. M. Stark, $L$-functions at $s=1$. III. Totally real fields and Hilbert’s twelfth problem, Advances in Math. 22 (1976), no. 1, 64–84. MR 437501, DOI 10.1016/0001-8708(76)90138-9
- Harold M. Stark, $L$-functions at $s=1$. IV. First derivatives at $s=0$, Adv. in Math. 35 (1980), no. 3, 197–235. MR 563924, DOI 10.1016/0001-8708(80)90049-3
- John Tate, Les conjectures de Stark sur les fonctions $L$ d’Artin en $s=0$, Progress in Mathematics, vol. 47, Birkhäuser Boston, Inc., Boston, MA, 1984 (French). Lecture notes edited by Dominique Bernardi and Norbert Schappacher. MR 782485
- J. Tate, A letter to Joongul Lee, 22 July 1997.
- John Tate, Refining Gross’s conjecture on the values of abelian $L$-functions, Stark’s conjectures: recent work and new directions, Contemp. Math., vol. 358, Amer. Math. Soc., Providence, RI, 2004, pp. 189–192. MR 2088717, DOI 10.1090/conm/358/06541
- Ki-Seng Tan, On the special values of abelian $L$-functions, J. Math. Sci. Univ. Tokyo 1 (1994), no. 2, 305–319. MR 1317462
- Ki-Seng Tan, A note on Stickelberger elements for cyclic $p$-extensions over global function fields of characteristic $p$, Math. Res. Lett. 11 (2004), no. 2-3, 273–278. MR 2067472, DOI 10.4310/MRL.2004.v11.n2.a10
- B.L. Van Der Waerden, Algebra (Frederick Unger, New York, 1970).
- Masakazu Yamagishi, On a conjecture of Gross on special values of $L$-functions, Math. Z. 201 (1989), no. 3, 391–400. MR 999736, DOI 10.1007/BF01214904
Additional Information
- Ki-Seng Tan
- Affiliation: Department of Mathematics, National Taiwan University, Taipei 10764, Taiwan
- Email: tan@math.ntu.edu.tw
- Received by editor(s): June 26, 2006
- Published electronically: December 23, 2008
- Additional Notes: The author was supported in part by the National Science Council of Taiwan, NSC91-2115-M-002-001, NSC93-2115-M-002-007.
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 2277-2304
- MSC (2000): Primary 11S40; Secondary 11R42, 11R58
- DOI: https://doi.org/10.1090/S0002-9947-08-04830-7
- MathSciNet review: 2471918