Skip to main content

Modular Symbols in Iwasawa Theory

  • 1171 Accesses

Part of the Contributions in Mathematical and Computational Sciences book series (CMCS,volume 7)

Abstract

This survey paper is focused on a connection between the geometry of \(\mathop{\text{GL}}\nolimits _{d}\) and the arithmetic of \(\mathop{\text{GL}}\nolimits _{d-1}\) over global fields, for integers d ≥ 2. For d = 2 over \(\mathbb{Q}\), there is an explicit conjecture of the third author relating the geometry of modular curves and the arithmetic of cyclotomic fields, and it is proven in many instances by the work of the first two authors. The paper is divided into three parts: in the first, we explain the conjecture of the third author and the main result of the first two authors on it. In the second, we explain an analogous conjecture and result for d = 2 over \(\mathbb{F}_{q}(t)\). In the third, we pose questions for general d over the rationals, imaginary quadratic fields, and global function fields.

Keywords

  • Exact Sequence
  • Homology Group
  • Projective Limit
  • Nonzero Ideal
  • Drinfeld Module

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.

The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.1007/978-3-642-55245-8_17

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-642-55245-8_17

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-642-55245-8_5
  • Chapter length: 43 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   129.00
Price excludes VAT (USA)
  • ISBN: 978-3-642-55245-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   169.00
Price excludes VAT (USA)
Hardcover Book
USD   169.99
Price excludes VAT (USA)

Notes

  1. 1.

    This is still not quite the right object unless we invert 2. In Sect. 4, we take the point of view that the right object is the relative homology of the quotient of the space X 1(Np r) by the action of complex conjugation.

  2. 2.

    Actually, g α, β is a root of a unit, but the difficulties this causes are resolvable by passing to the projective limit and descending, so we ignore this for simplicity of presentation. We will be very careless about denominators in several places, omitting them where they occur for simplicity of the discussion that follows.

  3. 3.

    There is one potentially confusing aspect: the action of \(\varLambda \hookrightarrow \mathfrak{h}\) on \(\mathfrak{S} \subset \varLambda [\![q]\!]\) is not given by multiplication of the coefficients of q-expansions by the element of Λ. It is instead this multiplication after first applying the inversion map λλ on Λ that takes group elements to their inverses.

  4. 4.

    It should actually be possible to allow either or both of p = 3 and \(p\mid \varphi (N)\) in what follows.

  5. 5.

    The reader may wish to ignore the involutions in order to focus on the idea of the argument.

  6. 6.

    Another, more usual, way to approach injectivity is to use \(I_{\theta } +\xi _{\theta }\mathfrak{h}_{\theta }\) in place of I θ until one recovers the equality of these ideals through a proof of the main conjecture, as in 2.5.7 below.

  7. 7.

    In the projective limit, this gives another way of defining the isomorphism \(\mathfrak{h}_{\theta }/I_{\theta }\stackrel{\sim }{\rightarrow }\varLambda _{\theta }/(\xi _{\theta })\).

  8. 8.

    To make sense of this, note that the tensor product in the sum is taken over \(\mathbb{Z}_{p}[\theta ]\).

  9. 9.

    Actually, we know that the kernel coincides with (ξ θ ), but this weaker statement is enough.

  10. 10.

    Note that q appears in this sentence as the order of the residue field of \(\mathcal{O}_{\infty }\).

  11. 11.

    Actually, g α, β as we have described it is not well-defined until we take its q 2 − 1 power. The assumption that \(p \nmid (q^{2} - 1)\) is used to avoid this issue when we work with étale cohomology.

References

  • Ash, A.: Cohomology of congruence subgroups of \(\mathop{\text{SL}}\nolimits _{n}(\mathbb{Z})\). Math. Ann. 249, 55–73 (1980)

    MathSciNet  CrossRef  Google Scholar 

  • Busuioc, C.: The Steinberg symbol and special values of L-functions. Trans. Am. Math. Soc. 360, 5999–6015 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Carlitz, L.: A class of polynomials. Trans. Am. Math. Soc. 43, 167–182 (1938)

    CrossRef  MATH  Google Scholar 

  • Cremona, J.: Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields. Compos. Math. 51, 275–324 (1984)

    MathSciNet  MATH  Google Scholar 

  • Drinfeld, V.: Parabolic points and zeta functions of modular curves. Funkcional. Anal. i Priložen. 7, 83–84 (1973)

    MathSciNet  Google Scholar 

  • Drinfeld, V.: Elliptic modules. Mat. Sb. 94, 594–627, 656 (1974)

    Google Scholar 

  • Emerton, E.: The Eisenstein ideal in Hida’s ordinary Hecke algebra. IMRN Int. Math. Res. Not. 1999 (15), 793–802 (1999)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Fukaya, T.: Coleman power series for K 2 and p-adic zeta functions of modular forms. In: Kazuya Kato’s fiftieth birthday. Doc. Math. Extra Vol., 387–442 (2003)

    Google Scholar 

  • Fukaya, T., Kato, K.: On conjectures of Sharifi, 121pp (Preprint)

    Google Scholar 

  • Gekeler, E.: A note on the finiteness of certain cuspidal divisor class groups. Isr. J. Math. 118, 357–368 (2000)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Goncharov, A.: Euler complexes and geometry of modular varieties. Geom. Funct. Anal. 17, 1872–1914 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Harder, G., Pink, R.: Modular konstruierte unverzweigte abelsche p-Erweiterungen von \(\mathbb{Q}(\zeta _{p})\) und die Struktur ihrer Galoisgruppen. Math. Nachr. 159, 83–99 (1992)

    MathSciNet  CrossRef  Google Scholar 

  • Hayes, D.: Explicit class field theory for rational function fields. Trans. Am. Math. Soc. 189, 77–91 (1974)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Hida, H.: Iwasawa modules attached to congruences of cusp forms. Ann. Sci. École Norm. Sup. 19, 231–273 (1986)

    MathSciNet  MATH  Google Scholar 

  • Kato, K.: p-adic Hodge theory and values of zeta functions of modular forms. In: Cohomologies p-adiques et Applications Arithmétique, III. Astérisque, vol. 295, pp. 117–290. Socit Mathmatique de France, Paris (2004)

    Google Scholar 

  • Kondo, S.: Euler systems on Drinfeld modular curves and zeta values. Dissertation, University of Tokyo (2002)

    Google Scholar 

  • Kondo, S.,Yasuda, S.: Zeta elements in the K-theory of Drinfeld modular varieties. Math. Ann. 354, 529–587 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Kurihara, M.: Ideal class groups of cyclotomic fields and modular forms of level 1. J. Number Theory 45, 281–294 (1993)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Manin, J.: Parabolic points and zeta-functions of modular curves. Math. USSR Izvsetija 6, 19–64 (1972)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Mazur, B., Wiles, A.: Class fields of abelian extensions of \(\mathbb{Q}\). Invent. Math. 76, 179–330 (1984)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • McCallum, W., Sharifi, R.: A cup product in the Galois cohomology of number fields. Duke Math. J. 120, 269–310 (2003)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Ochiai, T.: On the two-variable Iwasawa main conjecture. Compos. Math. 142, 1157–1200 (2006)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Ohta, M.: On the p-adic Eichler-Shimura isomorphism for Λ-adic cusp forms. J. reine angew. Math. 463, 49–98 (1995)

    MathSciNet  MATH  Google Scholar 

  • Ohta, M.: Ordinary p-adic étale cohomology groups attached to towers of elliptic modular curves. II. Math. Ann. 318, 557–583 (2000)

    CrossRef  MATH  Google Scholar 

  • Ohta, M.: Congruence modules related to Eisenstein series. Ann. Éc. Norm. Sup. 36, 225–269 (2003)

    MathSciNet  MATH  Google Scholar 

  • Pal, A.: The rigid analytical regulator and K 2 of Drinfeld modular curves. Publ. Res. Inst. Math. Sci. 46, 289–334 (2010)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Ribet, K.: A modular construction of unramified p-extensions of \(\mathbb{Q}(\mu _{p})\). Invent. Math. 34, 151–162 (1976)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Sharifi, R.: A reciprocity map and the two variable p-adic L-function. Ann. Math. 173, 251–300 (2011)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Tate, J.: Relations between K 2 and Galois cohomology. Invent. Math. 36, 257–274 (1976)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Teitelbaum, J.: Modular symbols for F q (T). Duke Math. J. 69, 271–295 (1992)

    MathSciNet  CrossRef  MATH  Google Scholar 

  • Wiles, A.: On ordinary λ-adic representations associated to modular forms. Invent. Math. 94, 529–573 (1988)

    MathSciNet  CrossRef  MATH  Google Scholar 

Download references

Acknowledgements

The work of the first two authors (resp., third author) was supported in part by the National Science Foundation under Grant Nos. DMS-1001729 (resp., DMS-0901526).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Romyar Sharifi .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Fukaya, T., Kato, K., Sharifi, R. (2014). Modular Symbols in Iwasawa Theory. In: Bouganis, T., Venjakob, O. (eds) Iwasawa Theory 2012. Contributions in Mathematical and Computational Sciences, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55245-8_5

Download citation