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The cuspidal group and special values of $L$-functions

Author: Glenn Stevens
Journal: Trans. Amer. Math. Soc. 291 (1985), 519-550
MSC: Primary 11G16; Secondary 11F11, 11G30, 11G40, 14G10
MathSciNet review: 800251
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Abstract: The structure of the cuspidal divisor class group is investigated by relating this structure to arithmetic properties of special values of $L$-functions of weight two Eisenstein series. A new proof of a theorem of Kubert (Proposition 3.1) concerning the group of modular units is derived as a consequence of the method. The key lemma is a nonvanishing result (Theorem 2.1) for values of the “$L$-function” attached to a one-dimensional cohomology class over the relevant-congruence subgroup. Proposition 4.7 provides data regarding Eisenstein series and associated subgroups of the cuspidal divisor class group which the author hopes will simplify future calculations in the cuspidal group.

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  • A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
  • V. G. Drinfel′d, Two theorems on modular curves, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 83–84 (Russian). MR 0318157
  • B. Gross and J. Lubin, The Eisenstein descent at level $N = {p^2}$ (to appear).
  • Kenkichi Iwasawa, Lectures on $p$-adic $L$-functions, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Annals of Mathematics Studies, No. 74. MR 0360526
  • S. Kamienny and G. Stevens, Special values of $L$-functions attached to $X_{1}(N)$, Compositio Math. 49 (1983), no. 1, 121–142. MR 699863
  • Daniel S. Kubert and Serge Lang, Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 244, Springer-Verlag, New York-Berlin, 1981. MR 648603
  • Daniel S. Kubert, The square root of the Siegel group, Proc. London Math. Soc. (3) 43 (1981), no. 2, 193–226. MR 628275, DOI
  • ---, The square root of the Siegel group-even case, Manuscript, 1982.
  • Serge Lang, Cyclotomic fields, Springer-Verlag, New York-Heidelberg, 1978. Graduate Texts in Mathematics, Vol. 59. MR 0485768
  • Serge Lang, Introduction to modular forms, Springer-Verlag, Berlin-New York, 1976. Grundlehren der mathematischen Wissenschaften, No. 222. MR 0429740
  • Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66 (Russian). MR 0314846
  • B. Mazur, On the arithmetic of special values of $L$ functions, Invent. Math. 55 (1979), no. 3, 207–240. MR 553997, DOI
  • B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287
  • B. Mazur and A. Wiles, Class fields of abelian extensions of ${\bf Q}$, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, DOI
  • Andrew Ogg, Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0256993
  • Hans Rademacher and Emil Grosswald, Dedekind sums, The Mathematical Association of America, Washington, D.C., 1972. The Carus Mathematical Monographs, No. 16. MR 0357299
  • Bruno Schoeneberg, Elliptic modular functions: an introduction, Springer-Verlag, New York-Heidelberg, 1974. Translated from the German by J. R. Smart and E. A. Schwandt; Die Grundlehren der mathematischen Wissenschaften, Band 203. MR 0412107
  • Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1. MR 0314766
  • Glenn Stevens, Arithmetic on modular curves, Progress in Mathematics, vol. 20, Birkhäuser Boston, Inc., Boston, MA, 1982. MR 670070
  • Lawrence C. Washington, Class numbers and ${\bf Z}_{p}$-extensions, Math. Ann. 214 (1975), 177–193. MR 364182, DOI
  • E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469

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