On almost strong approximation for algebraic groups

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer N N , first we give the classification of positive definite ternary quadratic forms of level 4 N 4N explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level 4 N 4N , which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level 4 N 4N . As applications, we derive an explicit base of Eisenstein series space of modular forms of weight 3 / 2 3/2 and level 4 N 4N , and give new proofs of some interesting identities involving representation number of ternary quadratic forms.

A fundamental result in the arithmetic of integral quadratic forms is the strong approximation theorem for the spin group Spin of a non-degenerate inde®nite quadratic form in three or more variables. (In practice, one invokes the corresponding version for the kernel O0 of the spinor norm function.) This result is due to the works of Eichler and Kneser [Ei], [Kn] in the 1950s, and has ever since played a central role in the proofs of a number of important theorems for both inde®nite as well as de®nite quadratic forms. On the other hand, it is well-known that such a strong approximation cannot possibly hold for i† the group SO of a inde®nite quadratic form as it is not simply connected, and ii† for the group Spin of a de®nite quadratic form as Spin1 is compact. The purpose of this paper is to show that there exists a version of an ``almost'' strong approximation theorem for de®nite quadratic forms, one forOm and one forO ‡ m m 3† and show that they too have serious applications in the arithmetic theory; e.g., in the asymptotic distributions of classes in a genus and in the representation theory of de®nite forms. It is anticipated that such almost strong approximations would also extend to other classical groups and will be treated in a future article. We give a brief outline of the contents of this paper. Terminology and notations are generally those from [ OM]. For simplicity as well as convenience we shall consider here only positive de®nite integral Z-lattices. A key step in the proof of the strong approximation theorem for O0 is the following result on inde®nite representations with approximation property: Invent. math. 129, 471±487 (1997)

For each positive integer n n , let g Z ( n ) g_\mathbb {Z}(n) be the smallest integer such that if an integral quadratic form in n n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z ( n ) g_\mathbb {Z}(n) squares of integral linear forms. We show that every positive definite integral quadratic form is equivalent to what we call a balanced Hermite–Korkin–Zolotarev-reduced form and use it to show that the growth of g Z ( n ) g_\mathbb {Z}(n) is at most an exponential of n \sqrt {n} . Our result improves the best known upper bound on g Z ( n ) g_\mathbb {Z}(n) which is on the order of an exponential of n n . We also define an analogous number g O ∗ ( n ) g_{\mathcal O}^*(n) for writing Hermitian forms over the ring of integers O \mathcal O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1 1 , we show that the growth of g O ∗ ( n ) g_{\mathcal O}^*(n) is at most an exponential of n \sqrt {n} . We also improve on results of both Conway and Sloane and Kim and Oh on s s -integrable lattices.

SUMMARY This paper is concerned with the characterization of the variance covariance matrix for a normal family in which certain quadratic forms are independent and follow cχ 2 distributions. It is also proved that the necessary and sufficient condition for any quadratic or linear form to be distributed independently of a sum of a finite number of non-negative definite quadratic forms is that it is distributed independently of each one of them.

We study the generalized Whittaker models for G = GSp(2;R) associated with indefinite binary quadratic forms when they arise from two standard representations of G: (i) a generalized principal series rep- resentation induced from the non-Siegel maximal parabolic subgroup and (ii) a (limit of) large discrete series representation. We prove the uniqueness of such models with moderate growth property. Moreover we express the values of the corresponding generalized Whittaker functions on a one-parameter subgroup of G in terms of the Meijer G-functions.

Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures

Consider strong approximation for algebraic varieties defined over a number field k k . Let S S be a finite set of places of k k containing all archimedean places. Let E E be an elliptic curve of positive Mordell–Weil rank and let A A be an abelian variety of positive dimension and of finite Mordell–Weil group. For an arbitrary finite set T \mathfrak {T} of torsion points of E × A E\times A , denote by X X its complement. Supposing the finiteness of ⨿ ⨿ ( E × A ) {\amalg \kern -.25pc\amalg } (E\times A) , we prove that X X satisfies strong approximation with Brauer–Manin obstruction off S S if and only if the projection of T \mathfrak {T} to A A contains no k k -rational points.

Ternary quadratic forms over number fields with small class number

Positive definite binary hermitian forms with finitely many exceptions

Arithmetic purity of strong approximation for homogeneous spaces

The following is an expanded version of the notes for a survey lecture on the theta correspondence given at Trichy in January, 1996. The aim of this lecture was to give an idea of this theory and some of the directions in which the theory has been developing. Perhaps some of the material contained in sections 2,3 and 9 is new. The theory of theta correspondence gives one of the few general methods of constructing automorphic forms of groups over number fields, or admissible representations of groups over local fields. The method has its origin in the classical construction of theta functions which are modular forms–perhaps of half integral weight on the upper half plane. Here is a special case of this: Let q be a positive definite quadratic form on a lattice L ⊆ IR, q : L→ Z . Let p be a homogeneous polynomial on IR which is harmonic with respect to q, i.e. ∆qp = 0 where ∆q denotes the Laplacian with respect to q (∆q is the unique homogeneous differential operator of order 2 invariant under the orthogonal group of q). Then,

A quadratic $\mathbb {Z}$-lattice L of rank n is denned to be k-regular for a positive integer $k \leq n$ if L globally represents all quadratic $\mathbb {Z}$-lattices of rank k which are represented everywhere locally by L. It is shown that there exist only finitely many isometry classes of primitive positive definite quadratic $\mathbb {Z}$-lattices of rank 4 which are 2-regular.

Classes of indefinite quadratic forms in a genus are in correspondence with the Galois group of an abelian extension called the spinor class field (Estes and Hsia, Japanese J. Math. 16, 341–350 (1990)). Hsia has proved (Hsia et al., J. Reine Angew. Math. 494, 129–140 (1998)) the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. This far, the corresponding result for skew-hermitian forms was known only in some special cases, e.g., when the ideal (2) is square free over the base field. In this work we prove the existence of representation fields for quaternionic skew-hermitian forms in complete generality.

Let q q be a power of an odd prime, and F q [ T ] \mathbb {F}_q[T] be the ring of polynomials over a finite field F q \mathbb {F}_q of q q elements. A quadratic form f f over F q [ T ] \mathbb {F}_q[T] is said to be regular if f f globally represents all polynomials that are represented by the genus of f f . In this paper, we study definite regular quadratic forms over F q [ T ] \mathbb {F}_q[T] . It is shown that for a fixed q q , there are only finitely many equivalence classes of regular definite primitive quadratic forms over F q [ T ] \mathbb {F}_q[T] , regardless of the number of variables. Characterizations of those which are universal are also given.

First we give a formula of spherical functions on certain spherical homogeneous spaces. Then, applying it, we complete the theory of the spherical functions on the space X of nondegenerate unramified hermitian forms on a p-adic number field. More precisely, we give an explicit expression for the spherical functions, prove theorems on the spherical Fourier transforms on the space of Schwartz-Bruhat functions on X, and parametrize of all spherical functions on X. Finally, as an application, we give explicit expressions of local densities of representations of hermitian forms.