Abstract:
Let $f$ be an indefinite ternary quadratic form, and let $q$ be an integer such that $-q det(f)$ is not a square. Let $N(T,f,q)$ denote the number of integral solutions of the equation $f(x)=q$ where $x$ lies in the ball of radius $T$ centered at the origin. We are interested in the asymptotic behavior of $N(T,f,q)$ as $T$ tends to infinity. We deduce from the results of our joint paper with Z. Rudnick that $N(T,f,q)$ grows like cE(T,f,q)$ as $T$ tends to infinity, where $E(T,f,q)$ is the Hardy-Littlewood expectation (the product of local densities) and $0 \le c \le 2$. We give examples of $f$ and $q$ such that $c$ takes the values 0, 1, 2.

Abstract:
In this paper we generalize the idea of "essentially unique" representations by ternary quadratic forms. We employ the Siegel formula, along with the complete classification of imaginary quadratic fields of class number less than or equal to 8, to deduce the set of integers which are represented in essentially one way by a given form which is alone in its genus. We consider a variety of forms which illustrate how this method applies to any of the 794 ternary quadratic forms which are alone in their genus. As a consequence, we resolve some conjectures of Kaplansky regarding unique representation by the forms $x^2 +y^2 +3z^2$, $x^2 +3y^2 +3z^2$, and $x^2 +2y^2 +3z^2$.

Abstract:
In this paper, I use Siegel-Weil formula and Kudla matching principle to prove some interesting identities between representation number (of ternary quadratic space) and the degree of Heegner divisors.

Abstract:
An integral quadratic polynomial is called regular if it represents every integer that is represented by the polynomial itself over the reals and over the $p$-adic integers for every prime $p$. It is called complete if it is of the form $Q({\mathbf x} + {\mathbf v})$, where $Q$ is an integral quadratic form in the variables ${\mathbf x} = (x_1, \ldots, x_n)$ and ${\mathbf v}$ is a vector in ${\mathbb Q}^n$. Its conductor is defined to be the smallest positive integer $c$ such that $c{\mathbf v} \in {\mathbb Z}^n$. We prove that for a fixed positive integer $c$, there are only finitely many equivalence classes of positive primitive ternary regular complete quadratic polynomials with conductor $c$. This generalizes the analogous finiteness results for positive definite regular ternary quadratic forms by Watson and for ternary triangular forms by Chan and Oh.

Abstract:
Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds for the least positive integer k such that the equation Q=k is insoluble in integers, despite being soluble modulo every prime power.

Abstract:
We discuss an unusual phenomenon in (integral) positive ternary quadratic forms. We also describe an interesting pairing of genera of ternary forms.

Abstract:
We give details of a formerly known relation between ternary quadratic forms and quaternion orders through the even Clifford algebra. Based on this and classifications of ternary quadratic forms we give a completely explicit classification of quaternion orders in the padic case.

Abstract:
G.L. Watson \cite{watson1, watson2} introduced a set of transformations, called Watson transformations by most recent authors, in his study of the arithmetic of integral quadratic forms. These transformations change an integral quadratic form to another integral quadratic form with a smaller discriminants, but preserve many arithmetic properties at the same time. In this paper, we study the change of class numbers of positive definite ternary integral quadratic formula along a sequence of Watson transformations, thus providing a new and effective way to compute the class number of positive definite ternary integral quadratic forms. Explicit class number formulae for many genera of positive definite ternary integral quadratic forms are derived as illustrations of our method.