Siegel’s Main Theorem for Quadratic Forms

Abstract

A classical question in the Theory of Numbers is one of expressing a positive integer as a sum of squares of integers. The qualitative aspects of this problem require at times no more than rudimentary congruence considerations e.g. a prime number leaving remainder 3 on division by 4 cannot be a sum of two squares of integers; however, in general, subtle arguments are called for. Fermat’s Principle of Descent needs to come into play for a proof of the (Euler-Fermat-) Lagrange theorem that every positive integer is a sum of four squares of integers. Skillful use of elliptic theta functions was made by Jacobi to obtain a quantitative refinement of that assertion, viz. according as n is an odd or even natural number, the number of ways of expressing n as a sum of four squares of integers is 8σ* (n) or 24σ*(n), where σ* (t) for any natural number t is the sum of all the odd natural numbers dividing t; Jacobi’s famous identity linking θ 43 with other theta constants θ 2, θ 4 and their derivatives is an analytic encapsulation of all these formulae for varying n.KeywordsQuadratic FormModular FormEisenstein SeriesCusp FormSpringer Lecture NoteThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.