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

被引：2

作者：

Chen E. ^{[1
]}

Park P.S. ^{[2
]}

Swaminathan A.A. ^{[3
]}

机构：

[1] Department of Mathematics, Massachusetts Institute of Technology, Cambridge, 02139, MA

[2] Department of Mathematics, Princeton University, Princeton, 08544, NJ

[3] Department of Mathematics, Harvard College, Cambridge, 02138, MA

来源：

Research in Number Theory | / 1卷 / 1期

基金：

美国国家科学基金会;

关键词：

Prime Ideal; Elliptic Curve; Complex Multiplication; Elliptic Curf; Class Number;

D O I：

10.1007/s40993-015-0028-0

中图分类号：

学科分类号：

摘要：

Let E/ ℚ be an elliptic curve with complex multiplication (CM), and for each prime p of good reduction, let aE(p) = p+ 1 − #E(Fp) denote the trace of Frobenius. By the Hasse bound, aE(p)=2pcosθp for a unique θp∈ [0,π]. In this paper, we prove that the least prime p such that θp∈ [α,β]⊂ [0,π] satisfies p≪(NEβ−α)A, where NE 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≪qL for an absolute constant L>0. © 2015, The Author(s).

引用

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