An alternate proof of Cohn's four squares theorem

✍ Scribed by Jesse Ira Deutsch

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 264 KB
• Volume: 104
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2003.08.005

✦ Synopsis

While various techniques have been used to demonstrate the classical four squares theorem for the rational integers, the method of modular forms of two variables has been the standard way of dealing with sums of squares problems for integers in quadratic fields. The case of representations by sums of four squares in Qð ffiffi ffi 5 p Þ was resolved by Go¨tzky, while those of Qð ffiffi ffi 2 p Þ and Qð ffiffi ffi 3 p Þ were resolved by Cohn. These efforts utilized modular forms. In previous work, the author was able to demonstrate Go¨tzky's theorem by means of the geometry of numbers. Here Cohn's theorem on representation by the sum of four squares for Qð ffiffi ffi 2 p Þ is proven by a combination of geometry of numbers and quaternionic techniques.