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Euler characteristics and elliptic curves.

J Coates1, S Howson

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

Proceedings of the National Academy of Sciences of the United States of America
|October 19, 2001
PubMed
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This study conjectures the Ginfinity-Euler characteristic of Selmer groups for elliptic curves over specific fields. The conjecture is proven under standard assumptions about Selmer group behavior and Iwasawa algebra modules.

Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Arithmetic Geometry

Background:

  • Focuses on modular elliptic curves E over the rational numbers Q, excluding those with complex multiplication.
  • Considers primes p where E exhibits good ordinary reduction.
  • Introduces Finfinity as the field extension of Q by all p-power division points of E, with Ginfinity as its Galois group over Q.

Purpose of the Study:

  • To formulate a precise conjecture regarding the Ginfinity-Euler characteristic of the Selmer group of E over Finfinity.
  • To investigate the relationship between the L-series of E and the structure of its Selmer group.
  • To prove the conjecture under standard assumptions in Iwasawa theory.

Main Methods:

  • The study formulates a conjecture based on the non-vanishing of the complex L-series of E at s=1.

Related Experiment Videos

  • It leverages standard conjectures concerning the behavior of the Selmer group as a module over the Iwasawa algebra.
  • Crucial local calculations are performed, building upon recent joint work with R. Greenberg.

    • Main Results:

    • A precise conjecture is proposed for the Ginfinity-Euler characteristic of the Selmer group of E over Finfinity for p >/= 5.
    • The conjecture is proven, contingent upon standard assumptions about the Selmer group's structure within Iwasawa theory.
    • The proof relies on significant local computations in arithmetic geometry.

    Conclusions:

    • The study establishes a significant result in the arithmetic of elliptic curves, connecting L-series values to Selmer group invariants.
    • The findings provide evidence for the validity of standard conjectures in Iwasawa theory concerning Selmer groups.
    • This work deepens the understanding of the interplay between analytic and algebraic properties of elliptic curves.