Quadratic Forms Representing All Odd Positive Integers

Abstract

Abstract. 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( x) = n has no integer solutions is O(D 2+ǫ ).

Integers represented by positive-definite quadratic forms and Petersson inner products

“…In 2014, the author [20] proved a stronger result in a more specific case. If r = 4 and det(Q) is a fundamental discriminant, then every locally represented integer n ≫ D(Q) 2+ǫ is represented.…”

“…If r = 4 and det(Q) is a fundamental discriminant, then every locally represented integer n ≫ D(Q) 2+ǫ is represented. The bound given in [20] is ineffective (and is related to the possible presence of an L-function with a Siegel zero) and not explicit. However, it is amenable to explicit computations and it was used to prove that x 2 +3y 2 +3yz +3yw +5z 2 +zw +34w 2 , which has D(Q) = N(Q) = 6780, represents all odd positive integers.…”

Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients

“…. Obviously, the operators φ p,jp , defined for an arbitrary modular form by the right hand sides of (3.1), (3.2), for distinct primes p commute, and as noticed Improvements on this are possible by [7,8] but have been made effective so far only in few cases, see [13]. At least if the conductor M χ of the character χ is small compared to M these don't give much for our present purpose because of the additional factors coming from oldforms which we computed above.…”

A “Four Integers” Theorem and a “Five Integers” Theorem

“…Rouse [21] remarks that at present there is no general algorithm for determining the integers represented by a positive ternary integral quadratic form. Assuming that the three forms in (2) do in fact represent all positive odd integers, Rouse [21] has shown that a positive integral quadratic form in any number of variables is (2, 1)-universal if and only if it represents the positive odd integers 1 to 451 inclusive.…”

“…Rouse's main result [21] was the minimal set of positive odd integers needed for (2, 1)-universality. [4].…”