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Hirzebruch-Zagier classes and rational elliptic curves over quintic fields

Michele Fornea, Zhaorong Jin

Abstract

Conditionally on a conjecture on the étale cohomology of Hilbert modular surfaces and some minor technical assumptions, we establish new instances of the equivariant BSD-conjecture in rank $0$ with applications to the arithmetic of rational elliptic curves over quintic fields. The key ingredients are a refinement of twisted triple product $p$-adic $L$-functions, the construction of a compatible collection of Hirzebruch-Zagier cycles and an explicit reciprocity law relating the two.

Hirzebruch-Zagier classes and rational elliptic curves over quintic fields

Abstract

Conditionally on a conjecture on the étale cohomology of Hilbert modular surfaces and some minor technical assumptions, we establish new instances of the equivariant BSD-conjecture in rank with applications to the arithmetic of rational elliptic curves over quintic fields. The key ingredients are a refinement of twisted triple product -adic -functions, the construction of a compatible collection of Hirzebruch-Zagier cycles and an explicit reciprocity law relating the two.

Paper Structure

This paper contains 69 sections, 74 theorems, 477 equations.

Table of Contents

  1. Introduction
  2. The equivariant BSD-conjecture.
  3. Main results
  4. Overview of the proof
  5. Hirzebruch--Zagier classes.
  6. The motivic $p$-adic $L$-function.
  7. Explicit reciprocity law.
  8. On the conjecture
  9. Review of Hilbert cuspforms
  10. Adelic Hilbert cuspforms
  11. Adelic q-expansion.
  12. Diagonal restriction.
  13. Hecke Theory
  14. Hida families
  15. Twists of cuspforms
  16. ...and 54 more sections

Key Result

Theorem A

Suppose that $N$ is coprime to $\mathfrak{Q}$, split in $L$, and there exists an ordinary prime $p\nmid 2N\cdot\frak{Q}$ for $E_{/\mathbb{Q}}$ such that If, additionally, $\varrho$ is residually not solvable and Conjecture wishingOhta1 holds, then

Theorems & Definitions (202)

  • Theorem A
  • Remark 1.1
  • Corollary 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem B
  • Conjecture 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • ...and 192 more