Reduction theory over quadratic imaginary fields

Hurwitz developed a reduction theory for real binary quadratic forms of positive discriminant based on least-remainder continued fractions. For each quadratic imaginary field k, we develop a similar theory for complex binary quadratic forms of nonzero discriminant. This uses a Markov partition for the geodesic flow over the quotient of hyperbolic 3-space by the Bianchi group B_k. When k has a Euclidean algorithm, our theory is based on least-remainder continued fractions.     

The fermat equation over quadratic fields

Kummer's method of proof is applied to the Fermat equation over quadratic fields. The concept of an m-regular prime, p, is introduced and it is shown that for certain values of m, the Fermat equation with exponent p has no nontrivial solutions over the field Q(√m).     

Frobenius Galois groups over quadratic fields

There exists a quadratic fieldQ(√D) over which every Frobenius group is realizable as a Galois group.     

Multiplicative arithmetic of finite quadratic forms over Dedekind rings

Letq(X) be a quadratic form in an even numberm of variables with coefficients in a Dedekind ringK. Let us assume that the setsR(q,a) = {N∈K ^m ;q(N) = a} of representations of elementsa ofK by the formq are finite. Then certain multiplicative relations are obtained by elementary means between the setsR(q,a) andR(q,ab), whereb is a product of prime elementsρ ofK with finite coefficientsK/ρK. The relations imply similar multiplicative relations between the numbers of elements of the setsR(q,a), which formerly could be obtained only in some special cases like the case whenK = ℤ is the ring of rational integers and only by means of the theory of Hecke operators on the spaces of theta-series. As an application, an almost elementary proof of the Siegel theorem on the mean number of representations of integers by integral positive quadratic forms of determinant 1 is given. Dedicated to the memory of Professor K G Ramanathan     

2-universal -lattices over real quadratic fields

Let be a real quadratic field with m a square-free positive rational integer, and be the ring of integers in F. An -lattice L on a totally positive definite quadratic space V over F is called r-universal if L represents all totally positive definite -lattices l with rank r over . We prove that there exists no 2-universal -lattice over F with rank less than 6, and there exists a 2-universal -lattice over F with rank 6 if and only if m=2, 5. Moreover there exists only one 2-universal -lattice with rank 6, up to isometry, over .     

Unimodular quadratic forms over global function fields

The isometry problem is studied for unimodular quadratic forms over the Hasse domains of global function fields. Over the polynomial ring k[x] the problem reduces to classification of forms over k; but examples are provided showing that in general no such reduction occurs, even when the underlying ring is Euclidean. Connections with the structure of the ideal class group are given, and a complete invariant for the isometry class is found in the ternary isotropic case.     

Normal bases of ray class fields over imaginary quadratic fields

We develop a criterion for a normal basis (Theorem 2.4), and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than ${\mathbb{Q}(\sqrt{-1})}$ and ${\mathbb{Q}(\sqrt{-3})}$ (Theorem 4.2). This result would be an answer for the Lang-Schertz conjecture on a ray class field with modulus generated by an integer (≥2) (Remark 4.3).     

Finiteness properties of arithmetic groups over function fields

We determine when an arithmetic subgroup of a reductive group defined over a global function field is of type FP _∞ by comparing its large-scale geometry to the large-scale geometry of lattices in real semisimple Lie groups.     

Universal quadratic forms and indecomposables over biquadratic fields

The aim of this article is to study (additively) indecomposable algebraic integers of biquadratic number fields K and universal totally positive quadratic forms with coefficients in . There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field K. Furthermore, estimates are proven which enable algorithmization of the method of escalation over K. These are used to prove, over two particular biquadratic number fields and , a lower bound on the number of variables of a universal quadratic forms.