Numerical evidence for higher order Stark-type conjectures
HTML articles powered by AMS MathViewer
- by Kevin J. McGown, Jonathan W. Sands and Daniel Vallières HTML | PDF
- Math. Comp. 88 (2019), 389-420 Request permission
Abstract:
We give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Stark’s conjecture over $\mathbb {Q}$, Rubin’s conjecture, Popescu’s conjecture, and a conjecture due to Burns that constitutes a generalization of Brumer’s classical conjecture on annihilation of class groups. Our approach is general and could be used for any abelian extension of number fields, independent of the signature and type of places (finite or infinite) that split completely in the extension.
We then employ our techniques in the situation where $K$ is a totally real, abelian, ramified cubic extension of a real quadratic field. We numerically verify the conjectures listed above for all fields $K$ of this type with absolute discriminant less than $10^{12}$, for a total of $19197$ examples. The places that split completely in these extensions are always taken to be the two real archimedean places of $k$ and we are in a situation where all the $S$-truncated $L$-functions have order of vanishing at least two.
References
- E. Artin, Über Einheiten relativ galoisscher Zahlkörper, J. Reine Angew. Math. 167 (1932), 153–156 (German). MR 1581332, DOI 10.1515/crll.1932.167.153
- Emil Artin and John Tate, Class field theory, AMS Chelsea Publishing, Providence, RI, 2009. Reprinted with corrections from the 1967 original. MR 2467155, DOI 10.1090/chel/366
- A. A. Beĭlinson, Higher regulators and values of $L$-functions, Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181–238 (Russian). MR 760999
- Spencer Bloch and Kazuya Kato, $L$-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333–400. MR 1086888
- 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, On derivatives of Artin $L$-series, Invent. Math. 186 (2011), no. 2, 291–371. MR 2845620, DOI 10.1007/s00222-011-0320-0
- D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501–570. MR 1884523
- David Burns, Masato Kurihara, and Takamichi Sano, On zeta elements for $\Bbb G_m$, Doc. Math. 21 (2016), 555–626. MR 3522250
- D. Burns and A. L. Boomla, On Selmer groups and refined Stark conjectures, Preprint, 2017.
- Pierrette Cassou-Noguès, Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta $p$-adiques, Invent. Math. 51 (1979), no. 1, 29–59 (French). MR 524276, DOI 10.1007/BF01389911
- Pierre Deligne and Kenneth A. Ribet, Values of abelian $L$-functions at negative integers over totally real fields, Invent. Math. 59 (1980), no. 3, 227–286. MR 579702, DOI 10.1007/BF01453237
- David Grant, Units from $5$-torsion on the Jacobian of $y^2=x^5+1/4$ and the conjectures of Stark and Rubin, J. Number Theory 77 (1999), no. 2, 227–251. MR 1702153, DOI 10.1006/jnth.1998.2346
- H. Johnston and A. Nickel, On the $p$-adic Stark conjecture at $s=1$ and applications, Preprint, 2017.
- Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723, DOI 10.1007/978-1-4612-0853-2
- 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
- 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
- Jonathan W. Sands, Stark’s question and Popescu’s conjecture for abelian $L$-functions, Number theory (Turku, 1999) de Gruyter, Berlin, 2001, pp. 305–315. MR 1822017
- 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
- 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
- P. Stucky, Experimental verification of the Rubin-Stark conjecture, Master’s thesis, University of Munich, Germany, 2017.
- 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
- The PARI Group, Bordeaux. PARI/GP, version 2.9.1, 2016. Available from http://pari.math.u-bordeaux.fr/.
- Daniel Vallières, The equivariant Tamagawa number conjecture and the extended abelian Stark conjecture, J. Reine Angew. Math. 734 (2018), 1–58. MR 3739312, DOI 10.1515/crelle-2015-0014
- Daniel Vallieres, On a generalization of the rank one Rubin-Stark conjecture, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–University of California, San Diego. MR 2890119
Additional Information
- Kevin J. McGown
- Affiliation: Department of Mathematics and Statistics, California State University, Chico, California 95929
- MR Author ID: 768800
- ORCID: 0000-0002-5925-801X
- Email: kmcgown@csuchico.edu
- Jonathan W. Sands
- Affiliation: Department of Mathematics, University of Vermont, Burlington, Vermont 05401
- MR Author ID: 154195
- Email: Jonathan.Sands@uvm.edu
- Daniel Vallières
- Affiliation: Department of Mathematics and Statistics, California State University, Chico, California 95929
- Email: dvallieres@csuchico.edu
- Received by editor(s): June 2, 2017
- Received by editor(s) in revised form: October 23, 2017
- Published electronically: April 12, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 389-420
- MSC (2010): Primary 11R42; Secondary 11R27
- DOI: https://doi.org/10.1090/mcom/3337
- MathSciNet review: 3854063