Automorphic forms with integral Fourier coefficients

We classify newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. Under the extended Riemann hypothesis for odd real Dirichlet characters, these newforms are finite in number. We produce tables for weights 3 and 4, where finiteness holds unconditionally.

The problem of summation of Fourier series in finite form is formulated in the weak sense, which allows one to consider this problem uniformly both for classically convergent and for divergent series. For series with polynomial Fourier coefficients \(a_n, b_n \in \mathbb{R}[n]\), it is proved that the sum of a Fourier series can be represented as a linear combination of 1, \(\delta(x)\), \(\cot \frac{x}{2}\) and their derivatives. It is shown that this representation can be found in a finite number of steps. For series with rational Fourier coefficients \(a_n, b_n \in \mathbb{R}(n)\), it is shown that the sum of such a series is always a solution of a linear differential equation with constant coefficients whose right-hand side is a linear combination of 1, \(\delta(x)\), \(\cot \frac{x}{2}\) and their derivatives. Thus, the issue of summing a Fourier series with rational coefficients is reduced to the classical problem of the theory of integration in elementary functions.

An algebraic description of basic discrete symmetries (space reversal P, time reversal T, and their combination PT) is studied. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphism groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dąbrowski groups) of the orthogonal group and eight types of the real Clifford algebras is defined with the use of isomorphisms between the automorphism groups and finite groups. Over the field of complex numbers there is a correspondence between two nonisomorphic double coverings of the complex orthogonal group and two types of complex Clifford algebras. It is shown that these correspondences associate with a well-known Atiyah–Bott–Shapiro periodicity. Generalized Brauer–Wall groups are introduced on the extended sets of the Clifford algebras. The structure of the inequality between the two Clifford–Lipschitz groups with mutually opposite signatures is elucidated. The physically important case of the two different double coverings of the Lorentz groups is considered in details.

The Cauchy problem for the Schrodinger equation with a monomial nonlinear term is considered. Under a growth condition on the nonlinear term, it is shown that low energy, exponentially decaying initial data yield a global solution which is analytic in its space variables and whose domain of analyticity expands in time. It is also shown that low energy analytic initial data on a symmetric tube domain lead to a global solution which is analytic on the same tube domain for all time.

Let f be a newform of weight k ≥ 3 with Fourier coefficients in a number field K . We show that the universal deformation ring of the mod λ Galois representation associated to f is unobstructed, and thus isomorphic to a power series ring in three variables over the Witt vectors, for all but finitely many primes λ of K . We give an explicit bound on such λ for the 6 known cusp forms of level 1, trivial character, and rational Fourier coefficients. We also prove a somewhat weaker result for weight 2.

Special values of L-functions by a Siegel–Weil–Kudla–Rallis formula

Generalised Kummer constructions and Weil restrictions

In the case of the upper half-plane, or the unit disc in the theory of one complex variable, there is a simple classical way of constructing the Poisson kernel from the Szegb kernel function. This construction was successfully used in the more general cases of the four classical types of bounded symmetric domains by L. K. Hua [9], and in the case of wedge domains (i.e., tube domains over convex cones) by E. M. Stein, G. Weiss, and M. Weiss [18], and the author [11]. Recently S. G. Gindikin [4] determined the Szegb kernel function for a wider class of generalizations of the classical upper half-plane than wedge domains, the so-called Siegel domains of type ii introduced by I. I. PjatetskiiSapiro [16]. The purpose of the present paper is to study the Poisson kernel obtained by the classical construction for these domains. First, in ? 2, we consider the case of a general Siegel domain of type II. The main result here is Theorem 2.4 which states that the Poisson kernel is an approximate identity. This theorem contains as a special case the analogous theorem of [18], while its proof given here is considerably simpler than the special proofs in [18] and [11]. As applications of Theorem 2.4, we establish a few simple facts about HI-spaces on D. We do not pursue this investigation very far; the present paper is rather meant to lay the ground work for a future study of finer questions of analysis on generalized half-planes. In ? 3, we consider the case of a Siegel domain of type II which is symmetric in the sense of E. Cartan. It is known [16], [12] that every bounded symmetric domain has a realization of this kind. We prove that in this case the Poisson kernel is harmonic in the sense of the theory of symmetric spaces. This identifies our Poisson kernel with a kernel studied in more general contexts by R. Hermann [7] and H. Furstenberg [3]. In the case of symmetric domains, our results go farther than those in [3] and [7] in two directions. We obtain fairly strong results on the boundary behaviour of the Poisson integral (?? 2 and 4) and we find easily computable explicit formulas for the Poisson kernel (?5). In ? 4, we consider bounded symmetric domains in the canonical Harish-

We introduce Hecke operators on de Rham cohomology of compact oriented manifolds. When the manifold is a quotient of a Hermitian symmetric domain, we prove that certain types of such operators are compatible with the usual Hecke operators on automorphic forms.

For a Euclidean Jordan algebra V with the corresponding symmetric cone $\Omega$, we consider the semigroup $\Gamma\sb{\Omega}$ of elements in the automorphism group $G(T\sb{\Omega})$ of the tube domain $V$ + $i\Omega$ which can be extended to $\Omega$ and maps $\Omega$ into itself. A study of this semigroup was first worked out by Koufany in connection to Jordan algebra theory and Lie theory of semigroups. In this work we give a new proof of Koufany's results and generalize up to infinite dimensional Jordan algebras, so called $JB$-algebras. One of the nice examples of the semigroup $\Gamma\sb{\Omega}$ is from the Jordan algebra $Sym(n,\IR$) of symmetric matrices. However, $V\sb{\sigma}$ the set of all self-adjoint operators on $\IR\sp{n}$ with respect to a non-degenerated symmetric bilinear form $\sigma$, is a non-Euclidean Jordan algebra with a cone $\Omega\sb{\sigma}$ which is isomorphic to $\Omega$ of the symmetric cone of $Sym(n,\IR)$. We get an isomorphism of the automorphism groups between two tube domains which also induces an isomorphism between two Lie semigroups. The Lorentzian cone, which is one of the irreducible symmetric cones, is an essential tool in the study of semigroups in Mobius and Lorentzian geometry. J. D. Lawson studied the Mobius and Lorentzian semigroups with an Ol'shanskii decomposition even in the infinite dimensional cases. We study these semigroups via a Jordan algebra theory.

The basic objects in supergeometry are the superspaces, “spaces on which some of the coordinates commute and the rest anticommute.” A precise definition of superspaces and their mappings is given in terms of sheaves. This chapter is devoted to the exposition of the basic facts about superspaces. In § 1 the fundamental definitions are introduced, especially the concept of a supermanifold, a space with local systems of independent coordinate functions. In § 2 it is explained how supermanifolds and superfibrations are classified up to isomorphism in the smooth category. The obstructions to the validity of the analogous theorems for complex manifolds are cohomological invariants. In § 3 flag superspaces are introduced; these are important in their own right and also provide good illustrations of the general theory. In § 4 are gathered the obvious supergeometric definitions in the theory of integrability, including the Frobenius theorem and connections. In § 5 objects specific to supergeometry are introduced—integral forms. The complex of integral forms is constructed using a canonical right connection on the sheaf of Berezin volume forms of the supermanifold; the idea of a right connection on an arbitrary vector superbundle is also defined and studied here. Integration on supermanifolds is the subject of §§ 6–9. The fundamental idea here is the Berezin integral of a volume form with compact support on a supermanifold; this is defined in § 6. Various derived constructions which permit one to define integrals over immersed supermanifolds of any dimension are described in §§ 7 and 9. The Stokes formula for integral forms is proved in § 8. Finally, in § 10 there is a list of finite-dimensional simple Lie superalgebras over the field of complex numbers.

A symmetric domain is a bounded domain in C N which becomes a symmetric Riemannian space with respect to its Bergman metric. For a symmetric domain 𝒟, we denote by G the connected component of the group of all analytic automorphisms of 𝒟 (or, what is the same, that of the group of all isometries of 𝒟 onto itself) with its natural topology and by K the isotropy subgroup of G at an (arbitrary) point z 0 ∈ 𝒟. Then, as is well-known ([1], [2]), G is a (connected) semi-simple Lie group of non-compact type with center reduced to the identity, K is a maximal compact subgroup of G and 𝒟 is identified with the coset-space G/K. A symmetric domain 𝒟 is decomposed uniquely into the direct product 𝒟 1 × ⋯ × 𝒟 S of irreducible symmetric domains 𝒟 i (i. e. the domains which cannot be decomposed any further) corresponding to the direct decomposition G 1 × ⋯ × G s of G into simple components. Irreducible symmetric domains have been classified completely by E. Cartas [1] into four main series (I), (II), (III) and (IV) of classical domains (see no 3 below) and two exceptional domains corresponding to the simple Lie groups of type (E 6) and (E 7).

Arbres, arborescences et racines carrées symétriques

Using methods applied by Atiyah in equivariant K-theory, Bredon obtained exact sequences for the relative cohomologies (with rational coefficients) of the equivariant skeletons of (sufficiently nice) T-spaces, T=(S^1)^n, with free equivariant cohomology over the cohomology of BT. Here we characterise those finite T-CW complexes with connected isotropy groups for which an analogous result holds with integral coefficients.

Symmetry reduction for the Kadomtsev-Petviashvili equation using its infinite dimensional invariance group