Linnik’s theorem for Sato-Tate laws on elliptic curves with complex multiplication
Author(s)
Park, Peter S.; Swaminathan, Ashvin A.; Chen, Evan
Download40993_2015_Article_28.pdf (467.8Kb)
PUBLISHER_CC
Publisher with Creative Commons License
Creative Commons Attribution
Terms of use
Metadata
Show full item recordAbstract
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
2015-12Department
Massachusetts Institute of Technology. Department of MathematicsJournal
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