On indefinite binary quadratic forms

On Indefinite Binary Quadratic Forms and Quadratic Ideals

We consider some properties of indefinite binary quadratic forms F (x, y) = ax +bxy−y of discriminant ∆ = b +4a, and quadratic ideals I = [a, b−√∆ ]. AMS Mathematics Subject Classification (2000): 11E04, 11E12, 11E16

Neighbors of Indefinite Binary Quadratic Forms

In this paper, we derive some algebraic identities on right and left neighbors R(F ) and L(F ) of an indefinite binary quadratic form F = F (x, y) = ax + bxy + cy of discriminant Δ = b − 4ac. We prove that the proper cycle of F can be given by using its consecutive left neighbors. Also we construct a connection between right and left neighbors of F . Keywords—Quadratic form, indefinite form, cy...

A Level N Reduction Theory of Indefinite Binary Quadratic Forms

In this paper we study a geometric coding algorithm for indefinite binary quadratic forms Q for the congruence subgroup Γ(N), with respect to the usual fundamental domain FN , where N is assumed prime. The cycles Q1, . . . , Qn that this algorithm produces are such that the the corresponding paths γ1, . . . , γn in the Riemann surface X0(N)(C) have a nice behaviour around the elliptic points of...

On successive minima of indefinite quadratic forms

On Perfect Binary Quadratic Forms

A quadratic form f is said to be perfect if its values at points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all perfect binary integer quadratic forms. If there is an integer bilinear map s such that f(s(x, y)) = f(x)f(y) for all vectors x and y from the integer 2-dimensional lattice, then the form f is perfect. We give an explicit descri...