Representation by ternary quadratic forms

Abstract

The problem of determining when an integral quadratic form represents every positive integer has received much attention in recent years, culminating in the 15 and 290 Theorems of Bhargava–Conway–Schneeberger and Bhargava–Hanke. For ternary quadratic forms, there are always local obstructions, but one may ask whether there are ternary quadratic forms which represent every locally represented integer. Indeed, such forms exist and are called regular, and Jagy, Kaplansky, and Schiemann proved that there are at most 913; however, only 899 of these are actually known to be regular. We consider the remaining 14 forms, and establish the regularity of each under the Generalized Riemann Hypothesis, following the method pioneered by Ono and Soundararajan. Moreover, we consider the exceptional arithmetic consequences if a large, locally represented integer is not globally represented by a ternary quadratic form, proving that some Dirichlet ‐function would necessarily have a Siegel zero or that some quadratic twist of an elliptic curve would have an unusually large Tate–Shafarevich group.