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Linnik’s theorem for Sato-Tate laws on elliptic curves with complex multiplication

Author(s)
Park, Peter S.; Swaminathan, Ashvin A.; Chen, Evan
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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.
Date issued
2015-12
URI
http://hdl.handle.net/1721.1/109820
Department
Massachusetts Institute of Technology. Department of Mathematics
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
ISSN
2363-9555

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