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Explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields

Tchamitchian Pierre

TL;DR

The paper refines Mestre's explicit formulas for analytic $L$-functions to derive sharper lower bounds on conductors for elliptic curves and abelian varieties over $\bbQ$ and general number fields, explicitly accounting for prescribed bad reduction. By separating primes of bad reduction and employing tailored Mestre test functions (notably the Odlyzko function), the authors obtain improved inequalities that yield stronger nonexistence results for everywhere good reduction on selected fields, under GRH (and BSD for elliptic curves). They extend the framework to abelian varieties of higher dimension and analyze conductor bounds across towers of number fields, including Hilbert class fields, illustrating both unconditional and GRH-conditioned outcomes. The work provides practical, computable bounds $B(K,F,\lambda)$ and their refinements, along with extensive numerical data showing fields where no abelian variety with analytic $L$-function can have everywhere good reduction. Overall, the results give a versatile, test-function–driven method to certify nonexistence of everywhere good reduction in a wide arithmetic setting, independent of modularity assumptions in many cases.

Abstract

Following the work of Mestre, we use Weil's explicit formulas to compute explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields. Moreover, we obtain bounds for the conductor of elliptic curves and abelian varieties over $\mathbb{Q}$ with specified bad reduction and over number fields. As an application, for specific fields, we prove the non-existence of abelian varieties with everywhere good reduction.

Explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields

TL;DR

The paper refines Mestre's explicit formulas for analytic -functions to derive sharper lower bounds on conductors for elliptic curves and abelian varieties over and general number fields, explicitly accounting for prescribed bad reduction. By separating primes of bad reduction and employing tailored Mestre test functions (notably the Odlyzko function), the authors obtain improved inequalities that yield stronger nonexistence results for everywhere good reduction on selected fields, under GRH (and BSD for elliptic curves). They extend the framework to abelian varieties of higher dimension and analyze conductor bounds across towers of number fields, including Hilbert class fields, illustrating both unconditional and GRH-conditioned outcomes. The work provides practical, computable bounds and their refinements, along with extensive numerical data showing fields where no abelian variety with analytic -function can have everywhere good reduction. Overall, the results give a versatile, test-function–driven method to certify nonexistence of everywhere good reduction in a wide arithmetic setting, independent of modularity assumptions in many cases.

Abstract

Following the work of Mestre, we use Weil's explicit formulas to compute explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields. Moreover, we obtain bounds for the conductor of elliptic curves and abelian varieties over with specified bad reduction and over number fields. As an application, for specific fields, we prove the non-existence of abelian varieties with everywhere good reduction.

Paper Structure

This paper contains 17 sections, 18 theorems, 99 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. analytic L-functions and Mestre's formulas
  3. Analytic L-functions
  4. Mestre's explicit formulas
  5. Application of the formulas to elliptic curves
  6. The refined explicit formulas for elliptic curves over $\mathbb{Q}$
  7. The refined formulas
  8. Numerical applications for elliptic curves
  9. The refined formulas for abelian varieties over $\mathbb{Q}$
  10. The $L$-function of an abelian variety
  11. The refined formulas
  12. Numerical applications for abelian varieties
  13. Application to abelian varieties over number fields
  14. Elliptic curves over number fields
  15. Behavior of $B(K,F,\lambda)$ in towers of number field
  16. ...and 2 more sections

Key Result

Theorem 1

Let $F(x)$ be a test function (see Definition test function) and let $\lambda >0$ be a real number. Then there is a computable bound $B_M(F,\lambda)$ such that for any abelian variety over $\mathbb{Q}$ of dimension $g$ and conductor $N$, and whose $L$-function is an analytic $L$-function, we have

Figures (1)

  • Figure :

Theorems & Definitions (42)

  • Theorem : Mestre, Proposition p. 21
  • Theorem : Theorem \ref{['Mestre inequality abelian variety over Q']}
  • Theorem : Theorem \ref{['Principal th of the article']}
  • Definition 2.1
  • Theorem 2.2: Mestre's explicit formulas Mestre
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • ...and 32 more