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Congruences between modular forms: raising the level and dropping Euler factors.

F Diamond1

  • 1Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, University of Cambridge, Cambridge CB2 1SB, United Kingdom.

Proceedings of the National Academy of Sciences of the United States of America
|October 19, 2001
PubMed
Summary

This study explores congruences between modular forms, connecting L-function values and Galois cohomology groups. It reveals how formulas at nonminimal levels derive from minimal levels by adjusting Euler factors.

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Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Modular Forms

Background:

  • Discusses generalizations of Hida, Ribet, and Wiles' theorems.
  • Focuses on congruences between modular forms.
  • Introduces L-functions and their properties.

Purpose of the Study:

  • To elucidate the connections between modular form congruences and arithmetic objects.
  • To analyze the relationship between L-function values and Galois cohomology groups.
  • To investigate the derivation of formulas at nonminimal levels from minimal levels.

Main Methods:

  • Leverages Wiles' theory on class number formulas.
  • Examines the behavior of periods in L-function values.
  • Applies concepts of Euler factors and modular form levels.

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Main Results:

  • Establishes a link between L-function values and Galois cohomology group sizes.
  • Demonstrates how nonminimal level formulas are obtained from minimal level ones.
  • Highlights the role of Euler factors in these derivations.

Conclusions:

  • The study deepens the understanding of congruences in number theory.
  • It provides insights into the structure of modular forms and their associated L-functions.
  • The findings contribute to the broader theory of Galois representations.