NAME:  Marvin Knopp

ADDRESS:Temple University
   Department of Mathematics
   Philadelphia, PA 19122-2585

EMAIL ADDRESS: knopp@math.temple.edu


TITLE OF TALK:  Sums of squares and the SMUCR's principle

ABSTRACT OF TALK:
Let $s\in Z^{+}$, $m$ a nonnegative integer and let $r_s(m)$ be the
number of ways to represent $m$ as a sum of squares. Let $r_s^*(m)$
be the number of ways to represent $m$ as a sum of odd squares.
Then $r_s^*(m)=0$ unless $m=8n+s$, with $n$ a nonnegative integer.
In [Ramanujan J. 2 (1998)] Paul Bateman and MK showed in an elementary
manner that for $1\ke s \le 7$,
$$
r_s(8n + s) = \{1 + \tfrac{1}{2}\binom{s}{s}\}\, r_s^*(8n + s),
$$
for all $n\ge0$. Furthermore, for $s\ge8$, the ratio
$$
r_s(8n + s) /  r_s^*(8n + s)
$$
remains bounded as $n \to \infty$.

The authors show here that when $s\ge8$ the ratio is not constant,
not even for sufficiently large $n$. The proof depends heavily
upon an instance of the principle:

\noindent
\underbar{Survival of Modularity Under Congruence Restrictions}: Let
$f(z) = \sum_{n=n_0}^\infty a(n) e^{2\pi inz}$ be a modular form
of weight $k$ on a congruence subgroup $\Gamma$ of $\mbox{SL}(2,Z)$.
For $\alpha$, $\beta \in Z$ define
$$
f(z;\alpha,\beta) 
= \sum_{n=n0 \\ n\equiv\alpha\pmod{\beta}}^\infty a(n) e^{2\pi inz}.
$$
Then, $f(z;\alpha,\beta)$ is a modular form of weight $k$ on some
congruence subgroup $\Gamma'$ of $\Gamma$. The level of $\Gamma'$
depends upon $\beta$ and the level of $\Gamma$.

The reader is alerted that the principle is a theorem only when suitable restrictions
upon $k$ and the multiplier system of $f(z)$. In the application here
$f(z) = \vartheta(2z)^s$, the generating function for
$r_s(m)$, so $k=s/2$ and the multiplier system is expressed
in terms of the Jacobi symbol.