The Square Sieve and a Lang-Trotter Question for Generic Abelian\n Varieties

2017/02/09 by Samuel W. Bloom, Bloom, Samuel
Arts and Humanities · Mathematics · #11G10 #11G35 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1702.03017

openalex publication_date 2017/02/09 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

Let A be a g-dimensional abelian variety over \ℚ whose adelic\nGalois representation has open image in \GSp_2g widehat\ℤ.\nWe investigate the endomorphism algebras \End(A_p) \⊗ \ℚ =\n\ℚ( \π_p ) of the reduction of A modulo primes p at which this\nreduction is ordinary and simple. We obtain conditional and unconditional\nasymptotic upper bounds on the number of primes at which this "Frobenius field"\nis a specified number field and, when A is two-dimensional, at which the\nFrobenius field contains a specified real quadratic number field. These\ninvestigations continue the investigations of variants of the Lang-Trotter\nConjectures on elliptic curves.\n

Related

• Applications of the square sieve to a conjecture of Lang and Trotter for\n a pair of elliptic curves over the rationals
• Arithmetic properties of the Frobenius traces defined by a rational abelian variety (with two appendices by J-P. Serre)
• Explicit open image theorems for abelian varieties with trivial endomorphism ring
• The Large Sieve and Galois Representations
• Ordinary reductions of abelian varieties