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Lang--Trotter Conjecture for CM Elliptic Curves

Daqing Wan, Ping Xi

Abstract

Given an elliptic curve $E$ over $\mathbb{Q}$ and non-zero integer $r$, the Lang--Trotter conjecture predicts a striking asymptotic formula for the number of good primes $p\leqslant x$, denoted by $π_{E,r}(x)$, such that the Frobenius trace of $E$ at $p$ is equal to the given integer $r$. We focus on the CM case in this memoir, and show how to realize the following two goals: (1) to give an unconditional estimate for $π_{E,r}(x)$, which confirms the upper bound part of the conjecture up to a constant multiple; (2) to give a conditional explicit asymptotic formula for $π_{E,r}(x)$ based on the Hardy--Littlewood conjecture on primes represented by quadratic polynomials. For completeness, we also summarize classical results on quadratic, cubic and quartic residues, as well as the corresponding reciprocity laws. This part should be of independent interests and could provide useful materials for more junior readers. We also highlight some possible extensions of the arguments in this memoir that may work for other statistical problems of CM elliptic curves.

Lang--Trotter Conjecture for CM Elliptic Curves

Abstract

Given an elliptic curve over and non-zero integer , the Lang--Trotter conjecture predicts a striking asymptotic formula for the number of good primes , denoted by , such that the Frobenius trace of at is equal to the given integer . We focus on the CM case in this memoir, and show how to realize the following two goals: (1) to give an unconditional estimate for , which confirms the upper bound part of the conjecture up to a constant multiple; (2) to give a conditional explicit asymptotic formula for based on the Hardy--Littlewood conjecture on primes represented by quadratic polynomials. For completeness, we also summarize classical results on quadratic, cubic and quartic residues, as well as the corresponding reciprocity laws. This part should be of independent interests and could provide useful materials for more junior readers. We also highlight some possible extensions of the arguments in this memoir that may work for other statistical problems of CM elliptic curves.

Paper Structure

This paper contains 57 sections, 59 theorems, 550 equations, 5 tables.

Table of Contents

  1. Backgrounds: history, conjectures and theorems
  2. Elliptic curves over rationals and finite fields
  3. Sato--Tate and Lang--Trotter conjectures
  4. Our main results
  5. The Hardy--Littlewood conjecture and upper bound for $\pi_{E,r}(x)$
  6. The Hardy--Littlewood conjecture and sieve methods
  7. Elliptic curves with complex multiplication
  8. Proof of Theorem \ref{['thm:upperbound']}
  9. Fixed trace in imaginary quadratic fields
  10. Power residues and laws of reciprocity
  11. Quadratic residues and reciprocity
  12. Quartic residues and reciprocity
  13. An incomplete character sum and quartic Gauß sums
  14. Cubic residues and reciprocity
  15. An incomplete character sum and cubic Gauß sums
  16. ...and 42 more sections

Key Result

Theorem 1.3

Let $r$ be a fixed non-zero integer and $E/\mathbb{Q}$ an elliptic curve with CM by the imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$ with squarefree $D\geqslant1$. Let $\varepsilon>0.$ For all sufficiently large $x,$ we have where $h_{D,r}$ is a constant depending only on $D$ and $r$, defined by with $\xi(D,r)$ taking values in $\{0, 1, 2\}$, explicitly given as in eq:xi(D,r).

Theorems & Definitions (81)

  • Conjecture 1.1: Sato--Tate Conjecture
  • Conjecture 1.2: Lang--Trotter Conjecture
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5: Comparison Conjecture
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 71 more