Construction of arithmetic automorphic functions for special Clifford groups

We are interested in finding the relation between rings of endomorphisms of toroidal groups and algebraic number fields which are not totally real. For algebraic number fields K , we define real maximal algebraic number fields K 0 and 𝔬 K 0 -quasi-abelian varieties. We show that any simple 𝔬 K 0 -quasi-abelian variety is constructed by embeddings of a non-totally real algebraic number field K and it has finite dimensional cohomology groups.

We describe some constructions of orthonormal lattices in totally real subfields of cyclotomic fields, obtained by endowing their ring of integers with a trace form. We also describe constructions of quaternion division algebras over such fields. Orthonormal lattices and quaternion division algebras over totally real fields find use in wireless networks in ultra wideband communication, and we describe the application.

In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some systems of generalized Sylvester matrix equations using the ranks of their coefficient matrices. The results of this paper are new and available over the real number field, the complex number field, and the quaternion algebra.

Inspired by Stark's analytic proof of class number finiteness of a ring of integers in an algebraic number field, we give a new proof of the finiteness of the number of classes of ideals in a maximal order of a totally definite quaternion algebra over a totally real number field. Our proof makes use of Epstein zeta function properties. This approach leads to alternative proofs of Eichler's mass formula and even parity of the number of ramified primes in the quaternion algebra.

The Brauer-Speiser theorem asserts that the Schur index of a real-valued complex irreducible character of a finite group is either $1$ or $2$. In this paper we present a brief proof of this result. From this it follows that the $K$-central nontrivial division algebra components of group algebras over a real algebraic number field $K$ are quaternions.

The coincidence site lattice (CSL) problem and its generalization to ℤ-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class number 1 over real algebraic number fields.

In this paper, we solve the problem of constructing a commutative algebra of quaternions and octonions. A proof of the theorem is given that the commutativity of quaternions can be ensured by specifying a set of sign coefficients of the directions of reference of the angles between the radius vectors in the coordinate planes of the vector part of the coordinate system of the quaternion space. The method proposed in the development of quaternions possessing the commutative properties of multiplication is used further to construct a commutative octonion algebra. The results obtained on improving the algebra of quaternions and octonions can be used in the development of new hypercomplex numbers with division over the field of real numbers, and can also find application for solving a number of scientific and technical problems in the areas of field theory, physical electronics, robotics, and digital processing of multidimensional signals.

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$ -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

On the modular forms of weight 1/2 over algebraic number fields

Let {Km}m≥4 be the family of non-normal totally real cubic number fields as- sociated with the Q-irreducible cubic polynomials Pm(x) = x 3 − mx 2 − (m + 1)x − 1, m ≥ 4. We determine all these Km's with class numbers hm ≤ 3: there are 14 such Km's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents em of the ideal class groups of these Km go to infinity with m and we determine all these Km's with ideal class groups of exponents em ≤ 3: there are 6 such Km with ideal class groups of exponent 2, and 6 such Km with ideal class groups of exponent 3.

We define Jacobi forms over a totally real algebraic number field K and construct examples by first embedding the group and the space into the symplectic group and the symplectic upper half space respectively. Then symplectic modular forms are created and Jacobi forms arise by taking the appropriate Fourier coefficients. Also some known relations of Jacobi forms to vector valued modular forms over rational numbers are extended to totally real fields.

Let F denote a totally real number field, and let K/F denote a totally imaginary quadratic extension. We fix an automorphic cuspidal representation π of GL2 (F ), and a finite order Hecke character χ of K . Thus χ is a representation of GL1 (K ). Under certain hypotheses, it is known that the central critical value L(π ⊗ χ, 1/2) is algebraic up to a known transcendental factor. Explicit formulae for this value have been given by a number of authors, notably Gross, Waldspurger, and Zhang. Essentially, the work of Gross and Zhang shows that this value is given by the height of a certain CM divisor on a suitable space, while the work of Waldspurger gives a criterion for nonvanishing of this value in terms of a certain linear functional arising from representation theory, and a formula in terms of torus integrals on a quaternion algebra. Our goal in this article is to explicate the connections between these works, and to provide a bridge between the general representation-theoretic framework described by Gross (see his article [Gro] in this volume) and the theorems of Zhang [Zha01a] and Waldspurger [Wal85]. We want to point out that the formula we will discuss has numerous applications to arithmetic and Iwasawa theory (see [BD96] and its various sequels). We will therefore attempt to formulate the representation-theoretic results in terms that are familiar to number theorists. We will not however discuss any arithmetic applications directly – the reader will find some of these applications elsewhere in this volume. Needless to say, the present work is mostly expository. The ideas are largely drawn from the papers [Gro87], [Gro], [Wal85], [Zha01a]. However, the organization here is perhaps novel. Our main contribution is given in Theorem 6.4. While the ingredients in this theo-

We completely determine the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in totally real 2-regular number fields. The result is almost periodic with period 8. Moreover, the 2-regular case is precisely the class of totally real number fields that have homotopy cartesian “Bökstedt square”, relating the K-theory of the 2-integers to that of the fields of real and complex numbers and finite fields. We also identify the homotopy fibers of the forgetful and hyperbolic maps relating hermitian and algebraic K-theory. The result is then exactly periodic of period 8 in the orthogonal case. In both the orthogonal and symplectic cases, we prove a 2-primary hermitian homotopy limit conjecture for these rings.

A characterization of number fields by p-closed extensions

Let m, n and p be integers with and . We showed in previous papers that if , then typical ranks of -tensors over the real number field are p and p+1 if and only if there exists a non-singular bilinear map . We also showed that the ‘if’ part is also valid in the case where . In this paper, we consider the case where and show that the typical ranks of -tensors over the real number field are p and p+1 in several cases including the case where there is no non-singular bilinear map . In particular, we show that the ‘only if’ part of the above mentioned fact is not valid for the case .