Hostname: page-component-f554764f5-68cz6 Total loading time: 0 Render date: 2025-04-22T02:08:12.030Z Has data issue: false hasContentIssue false
  • English
  • Français

Kato’s main conjecture for potentially ordinary primes

Part of: Arithmetic algebraic geometry Algebraic number theory: global fields

Published online by Cambridge University Press:  26 January 2024

Katharina Müller
Katharina Müller*
Affiliation:
Department of Mathematics and Statistics, Université Laval, Québec City, Canada
*
Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we prove Kato’s main conjecture for $CM$ modular forms for primes of potentially ordinary reduction under certain hypotheses on the modular form.

Keywords

Iwasawa theorymodular formsselmer groupsmain conjecture

MSC classification

Primary: 11G15: Complex multiplication and moduli of abelian varieties 11R23: Iwasawa theory 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 66 , Issue 2 , May 2024 , pp. 338 - 358
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Amice, Y. and Velu, J., Distributions p-adiques associées aux séries de Hecke, Journées arithmétiques de Bordeaux, Astérisque 24–25 (1975), 119–131. Société Mathématique de France.Google Scholar
Bloch, S. and Kato, K., L-functions and Tamagawa numbers for motives, in The Grothendieck Festschrift, Progress in Mathematics (Cartier, P., Illusise, L., Katz, N. M., Laumon, G., Manin, Y. I. and Ribet, K. A., Editos) (Springer, New York, 2007).Google Scholar
Delbourgo, D., Iwasawa theory for elliptic curves at unstable primes, Compos. Math. 113 (1998), 123–153.CrossRefGoogle Scholar
Delbourgo, D., On the p-Adic Birch, Swinnerton-Dyer Conjecture for non-semistable reduction, J. Number Theory 95 (2002), 38–71.CrossRefGoogle Scholar
Foquet, O. and Wan, X., The Iwasawa main conjecture for universal families of modular motives, Preprint arXiv: 2107.13726 (2021).Google Scholar
Greenberg, R., On the structure of certain Galois Groups, Invent. Math. 47 (1978), 85–99.CrossRefGoogle Scholar
Kato, K., p-adic Hodge theory and values of Zeta-functions, Cohomologies p-adiques et applications arithmétiques. III, Astérisque 295 (2004), 117–290.Google Scholar
Lei, A., Iwasawa Theory for modular forms at supersingular primes, Compos. Math. 147(3) (2011), 803–838.CrossRefGoogle Scholar
Ribet, K. A., Galois representations attached to eigenforms with nebentypus, in Modular functions in one variable V, Lecture Notes in Mathematics (Serre, J.-P. and Zagier, D. B., Editors) (Springer, Berlin, 1977).Google Scholar
Rubin, K., Local units, elliptic units, Heegner points and elliptic curves, Invent. Math. 88(2) (1987), 405–422.CrossRefGoogle Scholar
Rubin, K., The main conjectures of Iwasawa Theory for imaginary quadratic fields, Invent. Math. 103 (1991), 25–68.CrossRefGoogle Scholar
Rubin, K., Euler systems, Annals of Mathematics Studies, vol. 147 (Princeton University Press, Princeton, NJ, 2000).CrossRefGoogle Scholar
Rubin, K. and Pollack, R., The main conjecture for CM elliptic curves at supersingular primes, Ann. Math. 159 (2004), 447–464.Google Scholar
Vishik, M. M., Non-archimedian measures connected with Dirichlet series, Math. USSR Sbornik 28 (1976), 216–228.CrossRefGoogle Scholar