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Effective Brauer-Siegel on some curves in $Y(1)^n$

Georgios Papas

TL;DR

The paper addresses the problem of making Siegel's lower bounds for class numbers effective in a CM-setting within families of curves on $Y(1)^n$. It uses André's G-functions method to relate CM points to relative periods of 1-parameter families of elliptic curves, yielding height bounds and effective Galois-orbit estimates. The main contribution is an effective Brauer-Siegel-type bound for the fields of definition of CM points on certain CM-generic curves, together with a framework that ties CM-point growth to endomorphism discriminants. This advances the understanding of CM points in arithmetic geometry and provides quantitative control consistent with conjectures in unlikely intersections and André-Oort, with explicit dependence on the geometric data of the curve and chosen boundary points.

Abstract

We establish an effective version of Siegel's lower bounds for class numbers of imaginary quadratic fields in certain cures in $Y(1)^n$. Our proof goes through the G-functions method of Yves André.

Effective Brauer-Siegel on some curves in $Y(1)^n$

TL;DR

The paper addresses the problem of making Siegel's lower bounds for class numbers effective in a CM-setting within families of curves on . It uses André's G-functions method to relate CM points to relative periods of 1-parameter families of elliptic curves, yielding height bounds and effective Galois-orbit estimates. The main contribution is an effective Brauer-Siegel-type bound for the fields of definition of CM points on certain CM-generic curves, together with a framework that ties CM-point growth to endomorphism discriminants. This advances the understanding of CM points in arithmetic geometry and provides quantitative control consistent with conjectures in unlikely intersections and André-Oort, with explicit dependence on the geometric data of the curve and chosen boundary points.

Abstract

We establish an effective version of Siegel's lower bounds for class numbers of imaginary quadratic fields in certain cures in . Our proof goes through the G-functions method of Yves André.
Paper Structure (24 sections, 17 theorems, 52 equations)

This paper contains 24 sections, 17 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Siegel's lower bounds for class numbers
  3. Connection to CM elliptic curves
  4. Our main result
  5. Connection to Unlikely Intersections
  6. Summary
  7. Notation
  8. Recollections on the G-functions method
  9. Our setting
  10. Relative periods
  11. G-functions and relative periods
  12. Family of G-functions associated to $s_0$
  13. Independence from archimedean embedding
  14. Height bounds
  15. Working with a good model
  16. ...and 9 more sections

Key Result

Theorem 1.1

Given $\epsilon>0$ there exists a constant $c(\epsilon)>0$ such that for all imaginary quadratic fields $K$, where $h(K)$ denotes the class number of the field $K$.

Theorems & Definitions (45)

  • Theorem 1.1: Siegel, siegelog
  • Theorem 1.2: Tatuzawa, tatuzawa
  • Theorem 1.3: Siegel's Theorem V. $2$
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Remark 1.8
  • Definition 2.1
  • Remark 2.3
  • ...and 35 more