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An archimedean approach to singular moduli on Shimura curves

Mateo Crabit Nicolau

Abstract

We give a new proof of a recent generalization to Shimura curve of genus 0 of the work of Gross and Zagier in `On singular moduli'. This generalization was conjectured by Giampietro and Darmon and proved by Daas by using $p$-adic $Θ$-functions as an analogue of the $j$-invariant. Instead of working $p$-adically, we prove this result by evaluating Green's function at CM points on the Shimura curve. Our strategy is inspired by the analytic proof of Gross--Zagier. We put a special emphasis on both the similarities and the differences with the $p$-adic proof.

An archimedean approach to singular moduli on Shimura curves

Abstract

We give a new proof of a recent generalization to Shimura curve of genus 0 of the work of Gross and Zagier in `On singular moduli'. This generalization was conjectured by Giampietro and Darmon and proved by Daas by using -adic -functions as an analogue of the -invariant. Instead of working -adically, we prove this result by evaluating Green's function at CM points on the Shimura curve. Our strategy is inspired by the analytic proof of Gross--Zagier. We put a special emphasis on both the similarities and the differences with the -adic proof.
Paper Structure (12 sections, 17 theorems, 83 equations)

This paper contains 12 sections, 17 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Rewriting the equality
  3. Eisenstein Series
  4. Holomorphic projection
  5. Quaternion algebra
  6. Green's function
  7. Embeddings
  8. An F-quadratic form
  9. End of the proof
  10. Interpretation of the higher coefficients aₙ
  11. When N=6,10,22
  12. For general N

Key Result

Theorem 1

Let $w_i$ be the order of the group of units $\mathcal{O}_{i}^\times$ and where the product is taken over the $\mathrm{SL}_2(\mathbb{Z})$-orbits of points $\tau_1$,$\tau_2$ of discriminants $D_1$ and $D_2$ respectively. For a prime $l$ with $\bigl(\frac{D}{l}\bigl)\neq -1$, define If $n=\prod l_i^{\alpha_i}$ with $\bigl(\frac{D}{l_i}\bigl)\neq -1$ for all $i$, we define $\epsilon(n)=\prod \epsil

Theorems & Definitions (36)

  • Theorem 1: Gross--Zagier
  • Theorem 2: Daas
  • Definition 3
  • Proposition 4: Holomorphic projection
  • proof
  • Remark 5
  • Corollary 6
  • proof
  • Remark 7
  • Proposition 8
  • ...and 26 more