The birational composition of arbitrary quadratic form with binary quadratic form

Kaplansky’s Ternary Quadratic Form

This paper proves that if N is a nonnegative eligible integer, coprime to 7, which is not of the form x2+y2+7z2, thenN is square-free. The proof is modelled on that of a similar theorem by Ono and Soundararajan, in which relations between the number of representations of an integer np2 by two quadratic forms in the same genus, the pth coefficient of an L-function of a suitable elliptic curve, a...

Ramanujan’s Ternary Quadratic Form

do not seem to obey any simple law.” Following I. Kaplansky, we call a non-negative integer N eligible for a ternary form f(x, y, z) if there are no congruence conditions prohibiting f from representing N. By the classical theory of quadratic forms, it is well known that any given genus of positive definite ternary quadratic forms represents every eligible integer. Consequently if a genus consi...

¿ Functions of a Quadratic Form

Let Q be a positive definite integral quadratic form in ti variables, with the additional property that the adjoint form Q' is also integral. Using the functional equation of the Epstein zeta function, we obtain a symmetric functional equation of the ¿-function of Q with a primitive character to mod q (additive or multiplicative) defined by £io(2(x))C(x)-ï. Re(s) > nl2> where the summation exte...

Composition of Binary Quadratic Forms

Quadratic Form Expansions for Unitaries

We introduce techniques to analyze unitary operations in terms of quadratic form expansions, a form similar to a sum over paths in the computational basis when the phase contributed by each path is described by a quadratic form over R. We show how to relate such a form to an entangled resource akin to that of the one-way measurement model of quantum computing. Using this, we describe various co...