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Torsion cycles on Fermat varieties

Ramesh Sreekantan

Abstract

A theorem of Manin and Drinfeld states that any divisor of degree $0$ on the cusps of a modular curve is torsion in the Jacobian. An elegant proof of this result was provided by Elkik using mixed Hodge theory. Rohrlich proved a generalization of this to Fermat curves. In this note we reprove his results along the lines of the work of Elkik. We then use the same methods to generalize it to higher codimensional null-homologous cycles as well as higher Chow cycles on Fermat varieties.

Torsion cycles on Fermat varieties

Abstract

A theorem of Manin and Drinfeld states that any divisor of degree on the cusps of a modular curve is torsion in the Jacobian. An elegant proof of this result was provided by Elkik using mixed Hodge theory. Rohrlich proved a generalization of this to Fermat curves. In this note we reprove his results along the lines of the work of Elkik. We then use the same methods to generalize it to higher codimensional null-homologous cycles as well as higher Chow cycles on Fermat varieties.
Paper Structure (15 sections, 12 theorems, 59 equations)

This paper contains 15 sections, 12 theorems, 59 equations.

Table of Contents

  1. Introduction.
  2. Hodge structures.
  3. Mixed Hodge structures.
  4. Extensions of Mixed Hodge structures.
  5. Extensions and the Abel-Jacobi map.
  6. Fermat varieties.
  7. Rohrlich's theorem and its generalizations.
  8. $G_d^n$-module structure of $H^n(F_d^n,{\mathds C})$.
  9. $G_d^n$-module structure of $H^{2p}_{S_d^p}(F_d^n,{\mathds Q})$.
  10. Splitting of the Hodge Structure.
  11. Motivic cycles.
  12. Fermat varieties.
  13. Extensions and motivic cycles.
  14. Motivic cycles on Fermat Varieties.
  15. Non-torsion cycles on Fermat varieties.

Key Result

Theorem 1.1

Let $F_d=F_d^1$ be the Fermat curve of degree $d$. Let $S_d$ be the set of $3d$ 'Fermat points' - which are points determined by the hyperplanes $X_i=0$. These are the 'trivial' solutions to Fermats Last Theorem. Then any divisor of degree $0$ supported on $S_d$ is torsion is the Jacobian. That is, if $D=\sum_{P \in S_d} a_P P$ such that $\sum_{P \i

Theorems & Definitions (20)

  • Theorem 1.1: Rohrlich
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Carlson
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3: Shioda
  • ...and 10 more