-
Quadratic Forms Representing All Odd Positive Integers
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 136, Number 6, December 2014
- pp. 1693-1745
- 10.1353/ajm.2014.0041
- Article
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We consider the problem of classifying all positive-definite
integer-valued
quadratic forms that represent all positive odd integers. Kaplansky
considered
this problem for ternary forms, giving a list of 23 candidates, and
proving
that 19 of those represent all positive odds. (Jagy later dealt with a
20th
candidate.) Assuming that the remaining three forms represent all
positive
odds, we prove that an arbitrary, positive-definite quadratic form
represents
all positive odds if and only if it represents the odd numbers from 1 up
to
451. This result is analogous to Bhargava and Hanke's celebrated
290-theorem.
In addition, we prove that these three remaining ternaries represent all
positive odd integers, assuming the Generalized Riemann Hypothesis. This
result is made possible by a new analytic method for bounding the cusp
constants of integer-valued quaternary quadratic forms $Q$ with
fundamental
discriminant. This method is based on the analytic properties of
Rankin-Selberg
$L$-functions, and we use it to prove that if $Q$ is a quaternary form
with
fundamental discriminant, the largest locally represented integer $n$ for
which
$Q(\vec{x}) = n$ has no integer solutions is $O(D^{2 + \epsilon})$.