In this paper we establish a formula for the average of representation numbers of ternary quadratic forms in a spinor genus over a totally real number field.

We construct lattices with quadratic structure over the integers in quadratic number fields having the property that the rank of the quadratic structure is constant and equal to the rank of the lattice in all reductions modulo maximal ideals. We characterize the case in which such lattices are free. The construction gives a representative of the genus of such lattices as an orthogonal sum of “s...

In 1989, Shanks introduced the NUCOMP algorithm [10] for computing the reduced composite of two positive definite binary quadratic forms of discriminant ∆. Essentially by applying reduction before composing the two forms, the intermediate operands are reduced from size O(∆) to O(∆) in most cases and at worst to O(∆). Shanks made use of this to extend the capabilities of his hand-held calculator...

Let F be a totally real field and χ an abelian totally odd character of F . In 1988, Gross stated a p-adic analogue of Stark’s conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross’s conjecture when F is a real quadratic field and χ is a narrow ring class ...