Advertisement

Nontriviality of Arithmetic Invariants

  • Haruzo Hida
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

This chapter is an introductory and historical discussion of problems concerning arithmetic invariants and L-values. Often a number-theoretic object has an associated L-function. A number field has its Dedekind zeta function, an algebraic variety has its Hasse–Weil zeta function, an automorphic form/representation has its Langlands L-functions, and we have L-functions associated with Galois representations, compatible systems of Galois representations and motives. Number theorists all agree that L-functions and L-values are useful invariants for studying the object with number-theoretic goals in mind. From L-functions, number theorists have created more invariants, for example, λ- and μ-invariant from p-adic L-functions and the -invariant from exceptional zeros of p-adic L-functions.

Keywords

Modular Form Elliptic Curve Elliptic Curf Eisenstein Series Galois Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [AAF]
    G. Shimura, Arithmeticity in the Theory of Automorphic Forms. Mathematical Surveys and Monographs, vol. 82 (American Mathematical Society, Providence, RI, 2000)Google Scholar
  2. [ALR]
    J.-P. Serre, Abelian l-Adic Representations and Elliptic Curves (A K Peters, Wellesley, MA, 1998)zbMATHGoogle Scholar
  3. [BNT]
    A. Weil, Basic Number Theory (Springer-Verlag, New York, 1974)zbMATHCrossRefGoogle Scholar
  4. [CRT]
    H. Matsumura, Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, New York, 1986)Google Scholar
  5. [EDM]
    G. Shimura, Elementary Dirichlet Series and Modular Forms. Springer Monographs in Mathematics (Springer, New York, 2007)Google Scholar
  6. [EEK]
    A. Weil, Elliptic Functions According to Eisenstein and Kronecker (Springer-Verlag, Heidelberg, 1976)zbMATHCrossRefGoogle Scholar
  7. [GME]
    H. Hida, Geometric Modular Forms and Elliptic Curves, 2nd edn. (World Scientific, Singapore, 2011)CrossRefGoogle Scholar
  8. [HMI]
    H. Hida, Hilbert Modular Forms and Iwasawa Theory (Oxford University Press, New York, 2006)zbMATHCrossRefGoogle Scholar
  9. [IAT]
    G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press, Princeton, NJ, 1971)zbMATHGoogle Scholar
  10. [ICF]
    L.C. Washington, Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, vol. 83 (Springer-Verlag, New York, 1982)Google Scholar
  11. [LFE]
    H. Hida, Elementary Theory of L-Functions and Eisenstein Series. London Mathematical Society Student Texts, vol. 26 (Cambridge University Press, Cambridge, 1993)Google Scholar
  12. [LRF]
    J.-P. Serre, Linear Representations of Finite Groups. Graduate Texts in Mathematics, vol. 42 (Springer-Verlag, New York, 1977)Google Scholar
  13. [NAA]
    S. Bosch, U. Güntzer, R. Remmert, Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry. Grundlehren der Mathematischen Wissenschaften, vol. 261 (Springer-Verlag, Berlin, 1984)Google Scholar
  14. [SGL]
    H. Hida, On the search of genuine p-adic modular L-functions for GL(n). Mém. Soc. Math. Fr. (N.S.) No. 67 (1996)Google Scholar
  15. [D6]
    P. Deligne, Valeurs des fonctions L et périodes d’intégrales. Proc. Symp. Pure Math. 33.2, 313–346 (1979)Google Scholar
  16. [FG]
    B. Ferrero, R. Greenberg, On the behavior of p-adic L-functions at s = 0. Invent. Math. 50, 91–102 (1978)Google Scholar
  17. [FW]
    B. Ferrero, L. Washington, The Iwasawa invariant \(\mu _{p}\) vanishes for abelian number fields. Ann. Math. 109, 377–395 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  18. [Gr]
    R. Greenberg, Trivial zeros of p-adic L-functions. Contemp. Math. 165, 149–174 (1994)CrossRefGoogle Scholar
  19. [GS]
    R. Greenberg, G. Stevens, p-Adic L-functions and p-adic periods of modular forms. Invent. Math. 111, 407–447 (1993)Google Scholar
  20. [GsK]
    B.H. Gross, N. Koblitz, Gauss sums and the p-adic Γ-function. Ann. Math. 109, 569–581 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  21. [GZ]
    B. Gross, D. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84, 225–320 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  22. [H81a]
    H. Hida, Congruences of cusp forms and special values of their zeta functions. Invent. Math. 63, 225–261 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  23. [H88b]
    H. Hida, On p-adic Hecke algebras for GL 2 over totally real fields. Ann. Math. 128, 295–384 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  24. [H94]
    H. Hida, On the critical values of L-functions of GL(2) and GL(2) ×GL(2). Duke Math. J. 74, 431–529 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  25. [H04b]
    H. Hida, Non-vanishing modulo p of Hecke L-values, in Geometric Aspects of Dwork Theory, ed. by A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz, F. Loeser (Walter de Gruyter, Berlin, 2004), pp. 731–780 (a preprint version posted at www.math.ucla.edu/~hida)
  26. [H06a]
    H. Hida, Anticyclotomic main conjectures. Doc. Math. Extra Volume Coates, 465–532 (2006)Google Scholar
  27. [H07a]
    H. Hida, Non-vanishing modulo p of Hecke L-values and application, in L-Functions and Galois Representations. London Mathematical Society Lecture Note Series, vol. 320 (Cambridge University Press, Cambridge, 2007), pp. 207–269Google Scholar
  28. [H09b]
    H. Hida, Quadratic exercises in Iwasawa theory. Int. Math. Res. Not. 2009, 912–952 (2009). doi:10.1093/imrn/rnn151MathSciNetzbMATHGoogle Scholar
  29. [H10b]
    H. Hida, Central critical values of modular Hecke L-functions. Kyoto J. Math. 50, 777–826 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  30. [Hs1]
    M.-L. Hsieh, On the non-vanishing of Hecke L-values modulo p. American Journal of Mathematics, 134, 1503–1539 (2012)Google Scholar
  31. [Hs2]
    M.-L. Hsieh, On the \(\mu\)-invariant of anticyclotomic p-adic L-functions for CM fields. J. Reine Angew. Math. (in press). doi:10.1515/crelle-2012-0056, http://www.math.ntu.edu.tw/~mlhsieh/research.htm)
  32. [Hs3]
    M.-L. Hsieh, Eisenstein congruence on unitary groups and Iwasawa main conjecture for CM fields (preprint posted at http://www.math.ntu.edu.tw/~mlhsieh/research.htm)
  33. [Hz1]
    A. Hurwitz, Ueber die Entwicklungscoefficienten der lemniscatischen Functionen. Göttingen Nachr., 273–276 (1897)Google Scholar
  34. [Hz2]
    A. Hurwitz, Ueber die Entwickelungscoefficienten der lemniscatischen Functionen. Math. Ann. 51, 196–226 (1899)MathSciNetzbMATHCrossRefGoogle Scholar
  35. [K2]
    N.M. Katz, p-Adic L-functions for CM fields. Invent. Math. 49, 199–297 (1978)Google Scholar
  36. [Kh2]
    C. Khare, Serre’s conjecture and its consequences. Jpn. J. Math. 5, 103–125 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  37. [KhW]
    C. Khare, J.-P. Wintenberger, Serre’s modularity conjecture. I, II. I: Invent. Math. 178, 485–504 (2009); II: Invent. Math. 178, 505–586 (2009)Google Scholar
  38. [Mh]
    K. Mahler, An interpolation series for continuous functions of a p-adic variable. J. Reine Angew. Math. 199, 23–34 (1958) (Correction: J. Reine Angew. Math. 208, 70–72 (1961))Google Scholar
  39. [Mz1]
    B. Mazur, Courbes elliptiques et symboles modulaires. Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 414. Lecture Notes in Mathematics, vol. 317 (Springer-Verlag, Berlin, 1973), pp. 277–294Google Scholar
  40. [MzTT]
    B. Mazur, J. Tate, J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84, 1–48 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  41. [Se2]
    J.-P. Serre, Formes modulaires et fonctions zêta p-adiques. Lect. Notes Math. 350, 191–268 (1973) (Œuvres III 95–172, No. 97)Google Scholar
  42. [Sh6]
    G. Shimura, On some arithmetic properties of modular forms of one and several variables. Ann. Math. 102, 491–515 (1975) ([75c] in [CPS] II)Google Scholar
  43. [Sh7]
    G. Shimura, The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math. 29, 783–804 (1976) ([76b] in [CPS] II)Google Scholar
  44. [Sh8]
    G. Shimura, On the periods of modular forms. Math. Ann. 229, 211–221 (1977) ([77d] in [CPS] II)Google Scholar
  45. [Sh9]
    G. Shimura, The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J 45, 637–679 (1978) (a new version with some mathematical addenda and typographical correction incorporated: [78c] in [CPS] III)Google Scholar
  46. [Sn1]
    W. Sinnott, On the \(\mu\)-invariant of the Γ-transform of a rational function. Invent. Math. 75, 273–282 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  47. [Sn2]
    W. Sinnott, On a theorem of L. Washington. Astérisque 147–148, 209–224 (1987)MathSciNetGoogle Scholar
  48. [SU]
    C. Skinner, E. Urban, The Iwasawa main conjecture for GL2, to appear in Inventiones Math. (posted at http://www.math.columbia.edu/~urban/EURP.html)
  49. [V3]
    V. Vatsal, Special values of L-functions modulo p, in International Congress of Mathematicians, vol. II (European Mathematical Society, Zürich, 2006), pp. 501–514Google Scholar
  50. [Wa]
    J.-L. Waldspurger, Sur les valeurs de certaines fonctions L-automorphes en leur centre de symétrie. Compos. Math. 54, 173–242 (1985)MathSciNetzbMATHGoogle Scholar
  51. [Ws]
    L. Washington, The non-p–part of the class number in a cyclotomic \(\mathbb{Z}_{p}\)-extension. Invent. Math. 49, 87–97 (1978)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Haruzo Hida
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

Personalised recommendations