Jacobi sums and Grössencharacters
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2018), pp. 111-122.

In 1952, Weil published a paper describing how to interpret Jacobi sums in terms of Hecke Grössencharacters of cyclotomic fields. We describe an explicit version of this, with reference to our previous work concerning algorithmic implementation of Grössencharacters. We correct various errors involving root numbers in the latter, and also indicate how Jacobi sum methods can be used to understand tame primes of hypergeometric motives.

En 1952, Weil a publié un article dans lequel il donne une interprétation des sommes de Jacobi en terme de Hecke Grössencharacters de corps cyclotomiques. Nous décrivons une version explicite de cette interprétation en lien avec un travail précédent sur l’implantation algorithmique des Grössencharacters. Nous corrigeons à ce sujet quelques erreurs liées au root numbers. Nous expliquons également comment la méthode des sommes de Jacobi peut être utilisée pour comprendre le comportement de la ramification modérée des motifs hypergéométriques.

Published online:
DOI: 10.5802/pmb.25
Mark Watkins 1

1 School of Mathematics and Statistics, Carslaw Building (F07), University of Sydney, NSW 2006, Australia
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Mark Watkins. Jacobi sums and Grössencharacters. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2018), pp. 111-122. doi : 10.5802/pmb.25. https://pmb.centre-mersenne.org/articles/10.5802/pmb.25/

[1] Greg W. Anderson Cyclotomy and an extension of the Taniyama group, Compos. Math., Volume 57 (1986), pp. 153-217

[2] Handbook of Magma functions (Wieb Bosma; John Cannon; Claus Fieker; Allan K. Steel, eds.), 2016 Chapter 132 (Hypergeometric Motives)

[3] Wieb Bosma; John Cannon; Catherine Playoust The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 (also available at http://magma.maths.usyd.edu.au) | Zbl: 0898.68039

[4] John H. Coates; Andrew John Wiles On the conjecture of Birch and Swinnerton-Dyer, Invent. Math., Volume 39 (1977), pp. 223-251 | Article | Zbl: 0359.14009

[5] Robert Coleman; William McCallum Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math., Volume 385 (1988), pp. 41-101 https://eudml.org/doc/153009 | Zbl: 0654.12003

[6] Chantal David; Jack Fearnley; Hershy Kisilevsky On the vanishing of twisted L-functions of elliptic curves, Exp. Math., Volume 13 (2004) no. 2, pp. 185-198 http://projecteuclid.org/euclid.em/1090350933 | Zbl: 1115.11033

[7] Neil Dummigan; Phil Martin; Mark Watkins Euler factors and local root numbers for symmetric powers of elliptic curves, Pure Appl. Math. Q., Volume 5 (2009) no. 4, pp. 1311-1341 | Article | Zbl: 1191.11020

[8] Benedict H. Gross; Neal Koblitz Gauss sums and the p-adic Γ-function, Ann. Math., Volume 109 (1979), pp. 569-581 | Zbl: 0406.12010

[9] Despina T. Prapavessi On the conductor of 2-adic Hilbert norm residue symbols, J. Algebra, Volume 149 (1992) no. 1, pp. 85-101 | Zbl: 0773.11062

[10] D. P. Roberts; F. Rodriguez Villegas; Mark Watkins Exploring Motivic L-functions (2016) (draft)

[11] David E. Rohrlich Jacobi sums and explicit reciprocity laws, Compos. Math., Volume 60 (1986), pp. 97-114 | Zbl: 0612.12007

[12] David E. Rohrlich Root numbers, Arithmetic of L-functions (IAS/Park City Mathematics Series), Volume 18, American Mathematical Society, 2011, pp. 353-448 (also available at http://math.bu.edu/people/rohrlich/pcmi.pdf) | Zbl: 1285.11108

[13] Norbert Schappacher Periods of Hecke characters, Lecture Notes in Math., 1301, Springer, 1988, xv+160 pages | Article | Zbl: 0659.14001

[14] Victor P. Snaith Topological methods in Galois representation theory, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, 1989, xiii+299 pages | Zbl: 0673.12009

[15] Mark Watkins Computing with Hecke Grössencharacters, Publ. Math. Besançon, Algèbre Théorie Nombres, Volume 2011 (2011), pp. 119-135 | Zbl: 1280.11071

[16] André Weil Numbers of solutions of equations in finite fields, Bull. Am. Math. Soc., Volume 55 (1949), pp. 497-508 | Article | Zbl: 0032.39402

[17] André Weil Jacobi sums as “Grössencharaktere”, Trans. Am. Math. Soc., Volume 73 (1952), pp. 487-495 | Zbl: 0048.27001

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