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Generalized Sato-Tate and quadratic residues

Sergey Vladuts

Abstract

We show that the Generalized Sato-Tate Conjecture permits to obtain rather precise information on the distribution of the consecutive quadratic residues modulo large primes.

Generalized Sato-Tate and quadratic residues

Abstract

We show that the Generalized Sato-Tate Conjecture permits to obtain rather precise information on the distribution of the consecutive quadratic residues modulo large primes.

Paper Structure

This paper contains 10 sections, 7 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. The Generalized Sato-Tate conjecture
  3. Consecutive quadratic residues and algebraic curves
  4. Asymptotic behaviour
  5. Asymptotics under GS
  6. Case $t=4$
  7. Case $t=5$
  8. Unconditional results
  9. $t$=4.
  10. $t$=5.

Key Result

Theorem 4.1

$(i)$ For p running the set $\mathcal{P}_3:=\{p=4m+3\;\; is \;\ prime\}$ the quantity $\delta_p(4)$ is equidistributed on $[-2, 2]$ with respect to the measure $(ii)$ Assume GST for $B(4)=E_0\times E_1\times E_4.$ Then the quantity $\delta_p(4)$ for $p$ running the set $\mathcal{P}_1:=\{p=4m+1\;\; is \;\ prime\}$ is equidistributed on $[-10, 10]$ with respect to the measure where $\mu*\lambda$ d

Theorems & Definitions (8)

  • Definition 1
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3
  • Corollary 4.4
  • Lemma 5.1
  • Proposition 5.2
  • Proposition 5.3