Linnik’s theorem for Sato-Tate laws on elliptic curves with complex multiplication
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Author(s)
Park, Peter S.
Swaminathan, Ashvin A.
Chen, Evan
Date Issued
December 2015
Journal
Research in Number Theory
Publisher
Springer International Publishing
Citation
Chen, Evan, Peter S. Park, and Ashvin A. Swaminathan. “Linnik's Theorem for Sato-Tate Laws on Elliptic Curves with Complex Multiplication.” Research in Number Theory 1.1 (2015): n. pag.
Version
Final published version
Abstract
Let E/ℚ be an elliptic curve with complex multiplication (CM), and for each prime p of good reduction, let a[subscript E](p) = p + 1 − #E(𝔽[subscript p])
denote the trace of Frobenius. By the Hasse bound, a[subscript E] (p) = 2
√pcosθ[subscript p] for a unique θ[subscript p] ∈ [0,π]. In this paper, we prove that the least prime p such that θ[subscript p]∈ [α,β]⊂ [0,π] satisfies p ≪ (N[subscript E]/β − α)[superscript A], where N[subscript E] is the conductor of E and the implied constant and exponent A>2 are absolute and effectively computable. Our result is an analogue for CM elliptic curves of Linnik’s Theorem for arithmetic progressions, which states that the least prime p≡a (mod q) for (a,q)=1 satisfies p≪q[superscript L] for an absolute constant L>0.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s40993-015-0028-0
