Sums of Squares and the Preservation of Modularity under Congruence Restrictions

Abstract

If s is a fixed positive integer and n is any nonnegative integer, let r _s (8n + s) be the number of solutions of the equation

$$x_1^2 + x_2^2 + ... + x_s^2 = 8n + s$$

in integers x ₁, x ₂,…, x _s , and let r ^* _s (8n + s) be the number of solutions of the same equation in odd integers. Alternatively, r ^* _s (8n + s) is the number of ways of expressing n as a sum of triangular numbers, i.e., the number of solutions of the equation

$$\frac{{{y_1}({y_1} - 1)}}{2} + \frac{{{y_2}({y_2} - 1)}}{2} + ... + \frac{{{y_s}({y_s} - 1)}}{2} = n$$

in integers y _l, y ₂,…,y _s . It is known that for 1 ≤ s ≤ 7 there exists a positive constant c _s such that

$${r_s}(8n + s) = {c_s}{r_s}^*(8n + s)$$

for all nonnegative integers n. In this paper we prove that if s > 7, then no constant c _s exists such that (*) holds, even for all sufficiently large n. The proof uses the theory of modular forms of weight s/2 and appropriate multiplier system on the group Γ₀(64) and the so-called principle of the preservation of modularity under congruence restrictions.