Product Formula of the Cubic Gauss Sum Modulo the Product of the Primes

✍ Scribed by Tsuyoshi Takagi

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 916 KB
• Volume: 62
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1997.2064

✦ Synopsis

Let | be a prime in the quadratic field Q(e 2?iÂ3 ), and let G 3 (|) be the cubic Gauss sum. Matthews [Invent. Math. 52 (1979), 163 185; 54 (1979), 23 52] determined the product formula of G 3 (|) using Weierstrass' ^function. In this paper, we establish an analogous result for the cubic Gauss sum modulo the product of the primes.

( 2 )

Here, p 1Â3 is the real cube root of p, ^is Weierstrass' ^function, which satisfies the differential equation ^$2 =4^3&1, and % ( >0) is the smallest real period of ^. S is a third-set mod (|): that is, S is a set of ( p&1)Â3

article no. NT972064 298 0022-314XÂ97 25.00