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On primes represented by $aX^2+bY^3$

Jori Merikoski

Abstract

Let $a,b>0$ be coprime integers. Assuming a conjecture on Hecke eigenvalues along binary cubic forms, we prove an asymptotic formula for the number of primes of the form $ax^2+by^3$ with $x \leq X^{1/2}$ and $y \leq X^{1/3}$. The proof combines sieve methods with the theory of real quadratic fields/indefinite binary quadratic forms, the Weil bound for exponential sums, and spectral methods of GL(2) automorphic forms. We also discuss applications to elliptic curves.

On primes represented by $aX^2+bY^3$

Abstract

Let be coprime integers. Assuming a conjecture on Hecke eigenvalues along binary cubic forms, we prove an asymptotic formula for the number of primes of the form with and . The proof combines sieve methods with the theory of real quadratic fields/indefinite binary quadratic forms, the Weil bound for exponential sums, and spectral methods of GL(2) automorphic forms. We also discuss applications to elliptic curves.

Paper Structure

This paper contains 35 sections, 20 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Applications to elliptic curves
  3. Overview
  4. Type I sums
  5. Type II sums
  6. Type I$_2$ sums
  7. Generalizations to other sequences
  8. Acknowledgements
  9. Restricted representations by indefinite binary quadratic forms
  10. Real quadratic fields
  11. Indefinite binary quadratic forms
  12. Lemmas
  13. Set-up and arithmetic information
  14. Proof of Proposition \ref{['prop:typeI']}
  15. Proof of Proposition \ref{['prop:typeII']}: initial reductions
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $a,b > 0$ be coprime integers. Assume that Conjecture $\mathrm{C}_a(\varepsilon)$ holds for all $\varepsilon >0$. Then

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 3.1: Truncated Poisson summation formula
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 19 more