Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Lang-Trotter conjecture on average for genus-$2$ curves with $S_3$ reduced automorphism group

Chihiro Ando, Shushi Harashita

Abstract

For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjectured that the number of primes $p <X$ at which $E$ has a supersingular reduction is asymptotically equal to $c\sqrt{X}/\log X$, where $c>0$ is a constant depending only on $E$. While it remains an open question, an average estimation related to the Lang-Trotter conjecture was established by Fouvry and Murty. This result is called the Lang-Trotter conjecture on average. We extend the Lang-Trotter conjecture to curves of genus $2$ and obtain a similar result to the Lang-Trotter conjecture on average for the family of curves $C_λ:y^2=x(x-1)(x-λ)(x-(λ-1)/λ)(x-1/ (1-λ))$. These curves are characterized as curves of genus $2$ with reduced automorphism group containing symmetric group $S_3$.

The Lang-Trotter conjecture on average for genus-$2$ curves with $S_3$ reduced automorphism group

Abstract

For an elliptic curve over without complex multiplication, Lang and Trotter conjectured that the number of primes at which has a supersingular reduction is asymptotically equal to , where is a constant depending only on . While it remains an open question, an average estimation related to the Lang-Trotter conjecture was established by Fouvry and Murty. This result is called the Lang-Trotter conjecture on average. We extend the Lang-Trotter conjecture to curves of genus and obtain a similar result to the Lang-Trotter conjecture on average for the family of curves . These curves are characterized as curves of genus with reduced automorphism group containing symmetric group .

Paper Structure

This paper contains 17 sections, 42 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Hilbert class polynomials and modular polynomials
  4. Singular moduli by Gross-Zagier
  5. The degree-$3$ isogeny
  6. Superspecial genus-$2$ curves.
  7. Analytic results
  8. Isogenies between $E_{\Lambda^{-}(\lambda)}$ and $E_{\Lambda^{+}(\lambda)}$
  9. Endomorphism rings of supersingular $E_{\Lambda^{\pm}(\lambda)}$ with $\lambda \in \mathbb{F}_p$
  10. Elliptic curves over $\overline{\mathbb{F}_p}$ with the endomorphism ring containing $\mathbb{Z} \left[ \frac{1+\sqrt{-3p}}{2}\right]$
  11. The $j$-invariants of supersingular $E_{\Lambda^{-}(\lambda)}$ and $E_{\Lambda^{+}(\lambda)}$
  12. Factorization of $P_{3p}(X)$ mod $p$ for $p\equiv 1 \bmod 4$.
  13. Proof of Theorem \ref{['thm:A']}
  14. The case of $p \equiv 1 \bmod 4$
  15. The case of $p \equiv 4 \bmod 3$
  16. ...and 2 more sections

Key Result

Theorem A

Let $p\ge 5$ be a prime. Then we have

Theorems & Definitions (83)

  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 2.1: Deuring:Deu
  • Theorem 2.2: The Deuring lifting theorem
  • proof
  • Proposition 2.3: Elkies:Elkies
  • Proposition 2.4: Lang:SL
  • Proposition 2.5
  • proof
  • ...and 73 more