Sums of three squares under congruence condition modulo a prime

JOURNAL OF NUMBER THEORY

卷:159

Article

Tsuyumine, Shigeaki

关键词: Sums of three squares;    Modular forms;    Half integral weight;    Shimura lift;    Hilbert modular forms;   

DOI  :  10.1016/j.jnt.2015.07.011

来源: Elsevier

【 摘 要 】

Let p be an odd prime. We show that the integral points on the sphere with radius n are equidistributed modulo p as n -> infinity where n is not of the shape 4(l)(8m + 7) and its 2-adic valuation is bounded. In particular if n is sufficiently large and if n satisfies a congruence equation alpha(2)(1) + alpha(2)(2) + alpha(3)(2) equivalent to n (mod p) where p(2)vertical bar n if all alpha(i) equivalent to 0 (mod p), then there are integers x(i) with x(i) equivalent to alpha(i) (mod p) (i = 1, 2, 3) satisfying x(1)(2) + x(2)(2) + x(2)(3) = n. The similar result holds also in the case modulo 8. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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