Applications of the square sieve to a conjecture of Lang and Trotter for\n a pair of elliptic curves over the rationals

2017/10/05 by Stephan Baier, Baier, Stephan, Vijay M. Patankar +1
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.1710.02125

Abstract

Let E be an elliptic curve over \ℚ. Let p be a prime of good\nreduction for E. Then, for a prime p \≠ \ℓ, the Frobenius automorphism\nassociated to p (unique up to conjugation) acts on the \ℓ-adic Tate\nmodule of E. The characteristic polynomial of the Frobenius automorphism has\nrational integer coefficients and is independent of \ℓ. Its splitting field\nis called the Frobenius field of E at p. Let E₁ and E₂ be two\nelliptic curves defined over \ℚ that are non-isogenous over\n\\ℚ and also without complex multiplication over\n\\ℚ. In analogy with the well-known Lang-Trotter conjecture\nfor a single elliptic curve, it is natural to consider the asymptotic behaviour\nof the function that counts the number of primes p \≤ x such that the\nFrobenius fields of E₁ and E₂ at p coincide. In this short note, using\nHeath-Brown's square sieve, we provide both conditional (upon the Generalized\nRiemann Hypothesis) and unconditional upper bounds.\n

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