Anderson, Greg W. Cyclotomy and an extension of the Taniyama group. (English) Zbl 0591.14001 Compos. Math. 57, 153-217 (1986). The “Taniyama group” \(T\) classifies those “motives” that arise from the spectra of number fields or abelian varieties over \(\mathbb Q\) with potential complex multiplication. In this elegant paper, the author constructs a certain extension of \(T\) by a profinite completion of \(2\pi i\mathbb Z\). This extension classifies what the author calls “ulterior motives”. Motives factorize into ulterior motives like (in an analogy given elsewhere by the author) hadrons, e.g., protons, neutrons, factor into quarks. Through the use of ulterior motives, and results of D. Blasius and C. Siegel, the author is able to prove the \(\Gamma\)-hypothesis of Lichtenbaum which gives critical values of certain Hecke \(L\)-series in terms of special values of the classical \(\Gamma\)-function. Reviewer: David Goss (Columbus/Ohio) Cited in 1 ReviewCited in 8 Documents MSC: 11G15 Complex multiplication and moduli of abelian varieties 11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14L40 Other algebraic groups (geometric aspects) Keywords:Taniyama group; ulterior motives; \(\Gamma\)-hypothesis of Lichtenbaum; Hecke L-series × Cite Format Result Cite Review PDF Full Text: Numdam EuDML References: [1] D. Blasius : On the critical values of Hecke L-series (to appear). · Zbl 0608.10029 · doi:10.2307/1971386 [2] G. Brattström : Jacobi-sum Hecke characters of a totally real abelian field , Séminarie de Théorie des Nombres, Bordeaux, expose no. 22 (1981-82). · Zbl 0528.12011 [3] G. Brattström and S. Lichtenbaum : Jacobi-sum Hecke characters of imaginary quadratic fields , Comp. Math. (to appear). · Zbl 0584.12007 [4] R.M. Damerell : L-functions of elliptic curves with complex multiplication, I, II , Acta Arith. 17 (1970) 287-301; 19 (1971) 311-317. · Zbl 0229.12015 [5] P. Deligne : Applications de la formule des traces aux sommes trigonométriques, SGA41/ 4 , Lecture notes in math., Heidelberg: Springer. 569 (1977). · Zbl 0349.10031 [6] P. Deligne : Valeurs de fonctions L et périodes d’intégrales . Proc. Symp. Pure Math. AMS 33 (1979) 313-346. · Zbl 0449.10022 [7] P. Deligne , J.S. Milne , A. Ogus and K.Y. Shih : Hodge cycles, motives, Shimura varieties , Lecture notes in math. New York: Springer. 900 (1982). · Zbl 0465.00010 [8] D. Kubert : Jacobi sums and Hecke characters (preprint). · Zbl 0577.12004 · doi:10.2307/2374416 [9] D. Kubert and S. Lichtenbaum : Jacobi-sum Hecke characters , Comp. Math. 48 (1983) 55-87. · Zbl 0513.12010 [10] R.P. Langlands : Automorphic representations: Shimura varieties and motives. Ein märchen , Proc. Symp. Pure Math. AMS 33 (1979) 205-246. · Zbl 0447.12009 [11] N. Saavedra Rivano : Catégories tannakiennes , Lecture Notes in Math. New York: Springer. 265 (1979). · Zbl 0241.14008 [12] J.-P. Serre : Abelian l-adic representations and elliptic curves , New York: W.A. Benjamin (1968). · Zbl 0186.25701 [13] G. Shimura : On some arithmetic properties of modular forms of one and several variables , Ann. Math. 102 (1975) 491-515. · Zbl 0327.10028 · doi:10.2307/1971041 [14] T. Shioda and T. Katsura : On Fermat varieties , Tohoku math. J. 31 (1979) 97-115. · Zbl 0415.14022 · doi:10.2748/tmj/1178229881 [15] C.L. Siegel : Berechnung von Zetafunctionen an ganzzahligen Stellen , Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. II. 10 (1969) 87-102. · Zbl 0186.08804 [16] William C. Waterhouse : Introduction to affine group schemes, GTM66 , New York: Springer (1979). · Zbl 0442.14017 [17] A. Weil : Sur la théorie du corps du classes , J. Math. Soc. Japan 3 (1951) 1-35. (In collected works, New York: Springer, 1979.) · Zbl 0044.02901 · doi:10.2969/jmsj/00310001 [18] A. Weil : Jacobi sums as ”Grössencharaktere” , Trans. Am. Math. Soc. 73 (1952) 487-495. · Zbl 0048.27001 · doi:10.2307/1990804 [19] A. Weil : Basic number theory . Grundlehren Bd. 144, New York: Springer (1974). · Zbl 0326.12001 [20] A. Weil : La cyclotomie jadis et naguère , Enseignement Math. XX (1974) 247-263. · Zbl 0352.12006 [21] A. Weil : Sommes de Jacobi et caractères de Hecke , Göttingen Nachr. Nr. 1, 14 pp. (1974). · Zbl 0367.10035 [22] A. Weil : Sur les périodes des intégrales abéliennes , Comm. Pure Appl. Math. 29 (1976) 813-819. · Zbl 0342.14020 · doi:10.1002/cpa.3160290620 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.