We give a theoretical lower bound for the slope of a Siegel modular cusp form that is as least as good as Eichler's lower bound. In degrees $n=5,6$ and 7 we show that our new bound is strictly better. In the process we find the forms of smallest dyadic trace on the perfect core for ranks $n \le 8$. In degrees $n=5,6$ and 7 we settle the value of the generalized Hermite constant $\gamma_n'$ introduced by Bergé and Martinet and find all dual-critical pairs.

An alternative proof is given of a result, originally due to Guido Mislin, giving necessary and sufficient conditions for the inclusion of a subgroup to induce an isomorphism in mod p cohomology.

Higher algebraic K-theoretic analogues of Hasse's norm theorem in class field theory are proved. The result gives a partial affirmative answer to a question posed by Bak and Rehmann.

Let $J$ be a Jacobi real symmetric matrix on $l_{2}$ with zero diagonal and non-diagonal entries of the form $\{1+p_{n}\}$. If $p_{n-1}\pm p_{n}=O(n^{-\alpha})$ with some $\alpha>2/3$, then the existence of bounded solutions of $Ju=\lambda u$ is proved for almost every $\lambda\in(-2,2)$ with the WKB-type asymptotic behavior.

An inequality is established involving colengths of the tight closure of ideals of systems of parameters in local rings with some mild conditions. As an application, a proof is given of a result due to Goto and Nakamura (first conjectured by Watanabe and Yoshida), which states that the Hilbert–Samuel multiplicity of a parameter ideal is greater than or equal to the colength of the tight closure of the ideal. The result is also further refined.

The notions of controlled truncations for operators in the Roe algebras $C^*(X)$ of a coarse space $(X, \cal{E})$ with uniformly locally finite coarse structure, and rank distributions on $(X, \cal{E})$ are introduced. It is shown that the controlled propagation operators in an ideal $I$ of $C^*(X)$ are exactly the controlled truncations of elements in $I$. It follows that the lattice of the ideals of $C^*(X)$ in which controlled propagation operators are dense is isomorphic to the lattice of all rank distributions on $(X, \cal{E})$. If $X$ is a discrete metric space with Yu's property A, then the ideal structure of the Roe algebra $C^*(X)$ is completely determined by the rank distributions on $X$.

The authors present a technique of deriving basic hypergeometric identities from specializations using fewer parameters, by using the classical Cauchy identity on the expansion of the power of $x$ in terms of the $q$-binomial coefficients. This method is referred to as ‘Cauchy augmentation’. Despite its simple appearance, the Cauchy identity plays a key role in parameter augmentation. For example, one can reach the $q$-Gauss summation formula from the Euler identity by using the Cauchy augmentation twice. This idea also applies to Jackson's $_2\phi_1$ to $_3\phi_1$ transformation formula. Moreover, a transformation formula analogous to Jackson's formula is obtained.

Let $\Cal F$ be a holomorphic foliation (possibly with singularities) on a non-singular manifold $M$, and let $V$ be a complex analytic subset of $M$. Usual residue theorems along $V$ in the theory of complex foliations require that $V$ be tangent to the foliation (that is, a union of leaves and singular points of $V$ and $\Cal F$); this is the case for instance for the blow-up of a non-dicritical isolated singularity. In this paper, residue theorems are introduced along subvarieties that are not necessarily tangent to the foliation, including the blow-up of the dicritical situation.

We present a Borel reduction from a subset shift equivalence relation of a countable group to a subgroup conjugacy relation of a free product. The technique gives a much shorter proof of an earlier result of Thomas and Velickovic.

It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems cannot hold in general. This raises the question: “What information can we obtain from the existence of non-trivial Killing vector fields (or, respectively, harmonic one-forms)?” This paper gives answers to this problem; the results obtained are optimal.

A complete characterization of the categorical quotients of $({\mathbb P}^1)^n$ by the diagonal action of $\SL(2, {\mathbb C})$ with respect to any polarization is given by M. Polito, in ‘$\SL(2, {\mathbb C})$-quotients de $({\mathbb P}^1)^n$’, C. R. Acad. Sci. Paris Sér. I 321 (1995) 1577–1582. In this paper, these categorical quotients are obtained by certain linear systems on ${\mathbb P}^{n-3}$?>, depending on the given polarization.

The Fredholm properties of Toeplitz operators $T_a$ on Hardy spaces $H^p$ ($1<p<\infty$) with continuous symbols $a$ are well understood. We consider $T_a$ acting on $H^1$, where the operator is bounded provided that $a$ belongs to the class of symbols given by Janson and Stegenga's result on the pointwise multipliers on $H^1$. A necessary and sufficient condition for $T_a$ to be a Fredholm operator is given when $a$ is continuous and satisfies a mild additional condition (much weaker than Hölder continuity). A formula for the index of $T_a$ is also derived. In addition, we study the case of matrix-valued symbols and Toeplitz operators on $\rm{BMO}_A$.

Sharp decay estimates are provided in this paper for spherical averages of a certain multilinear extension operator on $L^{2}(\mathbb{S}^{n-1})\times \ldots\times L^{2}(\mathbb{S}^{n-1})$.

Let $p{>}2$ be a prime, and let $k$ be a field of characteristic zero, linearly disjoint from the $p$th cyclotomic extension of $\Q$. Given a projective Galois representation $\varrho \colon \Gal(\kbar/k) {\To} \PGL_2(\F_p)$ with cyclotomic determinant, two twists $X_\varrho(p)$ and $ X'_\varrho(p)$ of a certain rational model of the modular curve $X(p)$ can be attached to it. The $k$-rational points of these twists classify the elliptic curves $E/k$ such that ${\rhobar}_{E, p}{=}\varrho$, where ${\rhobar}_{E, p}$ denotes the projective Galois representation associated with the $p$-torsion module $E[p]$. The octahedral ($p{=}3$) and icosahedral ($p{=}5$) genus-zero cases are discussed in further detail.

We study the behaviour of non-negative solutions to the equation

formula here

This equation is an ‘explicitly’ soluble special case of a class of non-local evolution equations, of which the behaviour of solutions has been studied by Uchiyama [9]. In this paper, certain fine properties of solutions such as a local relaxation within their supports are obtained for the present special case. The asymptotic forms (for large time) of the solutions whose initial measures have compact supports are identified within the error of the order O(t−3/2). It is also shown that the comparison principle does not hold. With a simple change of variables, the equation ([midast ]) is transformed to the equation arising in Dyson's approach to the Wigner semi-circle law of eigenvalues of random matrices, and the present results have immediate consequences on the latter.

In a beautiful result, Herrero (D. A. Herrero, ‘Normal limits of nilpotent operators’, Indiana Univ. Math. J. 23 (1973/74) 1097–1108) showed that a normal operator on $l^2$ lies in the closure of the set of nilpotent operators if and only if its spectrum is connected and contains zero. In the quest for an automatic continuity result for algebra homomorphisms between ${\rm C}^*$-algebras, Dales showed that, if a discontinuous algebra homomorphism $\theta :{\cal A} \rightarrow {\cal U}$ exists between ${\rm C}^*$-algebras ${\cal A}$ and ${\cal U}$, and if $\theta({\cal A})$ is dense in ${\cal U}$, then there is a ${\rm C}^*$-algebra ${\cal U}_2$ with a dense subalgebra ${\cal N} \subset {\cal U}_2$ such that every $x \in {\cal N}$ is quasinilpotent (see p. 685 of H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs 24, Oxford University Press, 2001). (A discontinuous homomorphism $\theta_2 :{\cal A}_2 \rightarrow {\cal U}_2$ can be defined with the same basic properties as $\theta$, but the revised target space ${\cal U}_2$ has a dense subalgebra consisting of quasinilpotent elements.) As remarked by Dales, no such ${\rm C}^*$-algebra was then known; but here we present one. Indeed, using the full power of Herrero's result, one may arrange that every $x \in {\cal N}$ is nilpotent. The ${\rm C}^*$-algebra is constructed in a ‘neat’ way; it is most naturally constructed as a non-separable, concrete ${\rm C}^*$-algebra of operators on a separable Hilbert space ${\cal K}$; but one can arrange that the algebra ${\cal U}$ itself be separable if desired.

We study the conditions for truncated symmetric products of manifolds to be manifolds. In particular, we show that suitably defined spaces of systems of real roots of real polynomials are homeomorphic to real projective spaces.

Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\backslash G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion that involves root systems.