The error term in the Sato-Tate conjecture

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Abstract

Let f(z) = ∑n=1 a(n)e2πinz ∈ Snewk (Γ(N)) be a newform of even weight k ≥ 2 that does not have complex multiplication. Then a(n) ∈ ℝ for all n; so for any prime p, there exists θp ∈ [0, π] such that a(p) = 2p(k-1)/2 cos(θp).Let π(x) = ≠ {p ≤ x}. For a given subinterval [α, β] ⊂ [0, π], the now-proven Sato-Tate conjecture tells us that as x → ∞. # {p ≤ x: θp ∈ I} ~ μST([α, β]) π (x), μST([α, β]) = ∫βα 2\πsin2(θ) dθ.Let ∈ > 0. Assuming that the symmetric power L-functions of f are automorphic, we prove that as x → ∞, #{p ≤ x: θp ∈ I} = μST([α, β])π(x) + O (x/(log x)9/8- ∈) where the implied constant is effectively computable and depends only on k,N, and ∈.

Original languageEnglish (US)
Pages (from-to)147-156
Number of pages10
JournalArchiv der Mathematik
Volume103
Issue number2
DOIs
StatePublished - Aug 2014
Externally publishedYes

Keywords

  • Automorphic forms
  • Sato-Tate conjecture

ASJC Scopus subject areas

  • General Mathematics

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