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We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end, we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.

Let H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let , where p, q> 0, α = (α1,…,αn) with αj > -1, j =1,..., n, be the class of all measurable functions f defined on Dn such thatwhere Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on . We provide a characterization for a function f being in . Using the characterization we prove the following result: Let p> 1, then the Cesàro operator is bounded on the space .

In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors’ definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors’ theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to SL2 of the integers of the number field. This gives rise to a theta inversion formula, to which the authors’ Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.

We correct Proposition 3.12 and Lemma 3.13 of the paper published in Vol. 2013, No.13, pp.3006-3052. The corrections do not affect the other statements of the paper. In this note, we correct a flow in the statement of Proposition 3.12 of [1] which also leads to a modification in the statement of Lemma 3.13 of [1]. We recall that in this proposition one considers morphisms of schemes X ?→π Y ?→q S, where q is proper, flat, with equidimensional fibers of dimension n and π is finite, flat and surjective. Imposing certain conditions on the fibers it is claimed that the loci of s € S fulfilling these conditions are open subsets of S. A missing condition should be added and the correct version of Parts (g) and (h) of Proposition 3.12 should be as follows: (g) Ys has no embedded components and the discriminant scheme of π s : Xs → Ys is of pure codimension one and smooth; (h) Ys has no embedded components and the discriminant scheme of π s : Xs → Ys is of codimension one, irreducible and generically reduced.

Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq 5$ and has complex multiplication by the full ring of integers $\mathcal{O}_K$ of $K$. In this paper, we construct $p$-adic analogues of the Eisenstein-Kronecker series for such elliptic curve as Coleman functions on the elliptic curve. We then prove $p$-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function., v2. The current version is the synthesis of {\S}1-{\S}3 of the first version of this article with the content of arXiv:0807.4008 "The Kronecker limit formulas via the distribution relation." {\S}4,{\S}5 of the first version of this paper will be treated in a subsequent article

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module., 54 pages. v4: new title, paper completely reorganized; final version, to appear in Duke Math Journal

The object of this paper is to initiate a study of the Fourier coefficients of a weight $1$ harmonic Maass form and relate them to the complex Galois representation associated to a weight $1$ newform, which is the form’s image under a certain differential operator. In this paper, our focus will be on weight $1$ dihedral newforms of prime level $p\equiv3(\operatorname{mod}{4})$ . In this case we give properties of the Fourier coefficients that are similar to (and sometimes reduce to) cases of Stark’s conjectures on derivatives of $L$ -functions. We also give a new modular interpretation of certain products of differences of singular moduli studied by Gross and Zagier. Finally, we provide some numerical evidence that the Fourier coefficients of a mock-modular form whose shadow is exotic are similarly related to the associated complex Galois representation.

Let $k$ be a field finitely generated over $\mathbb{Q}$ , and let $X$ be a curve over $k$ . Fix a prime $\ell$ . A representation $\rho:\pi_{1}(X)\rightarrow \operatorname{GL}_{m}(\mathbb{Z}_{\ell})$ is said to be geometrically Lie perfect if any open subgroup of $\rho(\pi_{1}(X_{\overline{k}}))$ has finite abelianization. Let $G$ denote the image of $\rho$ . Any closed point $x$ on $X$ induces a splitting $x:\Gamma_{\kappa(x)}:=\pi_{1}(\operatorname{Spec}(\kappa(x)))\rightarrow\pi_{1}(X_{\kappa(x)})$ of the restriction epimorphism $\pi_{1}(X_{\kappa(x)})\rightarrow \Gamma_{\kappa(x)}$ (here, $\kappa(x)$ denotes the residue field of $X$ at $x$ ) so one can define the closed subgroup $G_{x}:=\rho\circ x(\Gamma_{\kappa(x)})\subset G$ . The main result of this paper is the following uniform open image theorem. Under the above assumptions, for any geometrically Lie perfect representation $\rho:\pi_{1}(X)\rightarrow\operatorname{GL}_{m}(\mathbb{Z}_{\ell})$ and any integer $d\geq 1$ , the set $X_{\rho,d}$ of all closed points $x\in X$ such that $G_{x}$ is not open in $G$ and $[\kappa(x):k]\leq d$ is finite and there exists an integer $B_{\rho,d}\geq 1$ such that $[G:G_{x}]\leqB_{\rho,d}$ for any closed point $x\in X\smallsetminus X_{\rho,d}$ with $[\kappa(x):k]\leq d$ . ¶ A key ingredient of our proof is that, for any integer $\gamma\geq 1$ , there exists an integer $\nu=\nu(\gamma)\geq 1$ such that, given any projective system $\cdots\rightarrow Y_{n+1}\rightarrow Y_{n}\rightarrow\cdots\rightarrow Y_{0}$ of curves (over an algebraically closed field of characteristic $0$ ) with the same gonality $\gamma$ and with $Y_{n+1}\rightarrow Y_{n}$ a Galois cover of degree greater than $1$ , one can construct a projective system of genus $0$ curves $\cdots\rightarrowB_{n+1}\rightarrow B_{n}\rightarrow \cdots\rightarrow B_{\nu}$ and degree $\gamma$ morphisms $f_{n}:Y_{n}\rightarrow B_{n}$ , $n\geq \nu$ , such that $Y_{n+1}$ is birational to $B_{n+1}\times_{B_{n},f_{n}}Y_{n}$ , $n\geq \nu$ . This, together with the case for $d=1$ (which is the main result of part I of this paper), gives the proof for general $d$ . ¶ Our method also yields the following unconditional variant of our main result. With the above assumptions on $k$ and $X$ , for any $\ell$ -adic representation $\rho:\pi_{1}(X)\rightarrow\operatorname{GL}_{m}(\mathbb{Z}_{\ell})$ and integer $d\geq 1$ , the set of all closed points $x\in X$ such that $G_{x}$ is of codimension at least $3$ in $G$ and $[\kappa(x):k]\leq d$ is finite.

In 2007, Shimoda, in connection with a long-standing question of Sally, asked whether a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most 2. In this paper, having reduced the conjecture to the case of dimension 3, if the ring is regular and local of dimension 3, we explicitly describe a family of prime ideals of height 2 minimally generated by three elements. Weakening the hypothesis of regularity, we find that, to achieve the same end, we need to add extra hypotheses, such as completeness, infiniteness of the residue field, and the multiplicity of the ring being at most 3. In the second part of the paper, we turn our attention to the category of standard graded algebras. A geometrical approach via a double use of a Bertini theorem, together with a result of Simis, Ulrich, and Vasconcelos, allows us to obtain a definitive answer in this setting. Finally, by adapting work of Miller on prime Bourbaki ideals in local rings, we detail some more technical results concerning the existence in standard graded algebras of homogeneous prime ideals with an (as it were) excessive number of generators.

In this paper we study the Weil-Petersson geometry of $\overline{\mathcal{M}_{g,n}}$, the compactified moduli space of Riemann surfaces with genus g and n marked points. The main goal of this paper is to understand the growth of the diameter of $\overline{\mathcal{M}_{g,n}}$ as a function of $g$ and $n$. We show that this diameter grows as $\sqrt{n}$ in $n$, and is bounded above by $C \sqrt{g}\log g$ in $g$ for some constant $C$. We also give a lower bound on the growth in $g$ of the diameter of $\overline{\mathcal{M}_{g,n}}$ in terms of an auxiliary function that measures the extent to which the thick part of moduli space admits radial coordinates., 26 pages, 7 figures

We deal with finitely generated modules over an artin algebra. In his Philadelphia Notes, M.Auslander showed that any homomorphism is right determined by a module C, but a formula for C which he wrote down has to be modified. The paper includes now complete and direct proofs of the main results concerning right determiners of morphisms. We discuss the role of indecomposable projective direct summands of a minimal right determiner and provide a detailed analysis of those morphisms which are right determined by a module without any non-zero projective direct summand., The paper has been revised and expanded. The terminology has been changed as follows: "essential kernel" is replaced by "intrinsic kernel", "determinator" is replaced by "determiner". Sections 3, 4 and 5 are new

This paper establishes new estimates for linear Schroedinger equations in R^3 with time-dependent potentials. Some of the results are new even in the time-independent case and all are shown to hold for potentials in scaling-critical, translation-invariant spaces. The proof of the time-independent results uses a novel method based on an abstract version of Wiener's Theorem., 49 pages; this is an expanded and improved version of the older paper

A natural notion for the signature $C_{\pm}({\mathcal V}(p))$ of the curvature of the zero set ${\mathcal V}(p)$ of a non-commutative polynomial $p$ is introduced. The main result of this paper is the bound \[ \operatorname{deg} p \leq2 C_\pm \bigl({\mathcal V}(p) \bigr) + 2. \] It is obtained under some irreducibility and nonsingularity conditions, and shows that the signature of the curvature of the zero set of $p$ dominates its degree. ¶ The condition $C_+({\mathcal V}(p))=0$ means that the non-commutative variety ${\mathcal V}(p)$ has positive curvature. In this case, the preceding inequality implies that the degree of $p$ is at most two. Non-commutative varieties ${\mathcal V}(p)$ with positive curvature were introduced in Indiana Univ. Math. J. 56 (2007) 1189-1231). There a slightly weaker irreducibility hypothesis plus a number of additional hypotheses yielded a weaker result on $p$. The approach here is quite different; it is cleaner, and allows for the treatment of arbitrary signatures. ¶ In J. Anal. Math. 108 (2009) 19-59), the degree of a non-commutative polynomial $p$ was bounded by twice the signature of its Hessian plus two. In this paper, we introduce a modified version of this non-commutative Hessian of $p$ which turns out to be very appropriate for analyzing the variety ${\mathcal V}(p)$.

In this paper we study the local behavior of a solution to the Lam\'e system with \emph{Lipschitz} coefficients in dimension $n\ge 2$. Our main result is the bound on the vanishing order of a nontrivial solution, which immediately implies the strong unique continuation property. This paper solves the open problem of the strong uniqueness continuation property for the Lam\'e system with Lipschitz coefficients in any dimension.

This paper presents an overview of some of the applications of the martingale inequalities of D. L. Burkholder to $L^p$-bounds for singular integral operators, concentrating on the Hilbert transform, first and second order Riesz transforms, the Beurling–Ahlfors operator and other multipliers obtained by projections (conditional expectations) of transformations of stochastic integrals. While martingale inequalities can be used to prove the boundedness of a wider class of Calderón–Zygmund singular integrals, the aim of this paper is to show results which give optimal or near optimal bounds in the norms, hence our restriction to the above operators. ¶ Connections of Burkholder’s foundational work on sharp martingale inequalities to other areas of mathematics where either the results themselves or techniques to prove them have become of considerable interest in recent years, are discussed. These include the 1952 conjecture of C. B. Morrey on rank-one convex and quasiconvex functions with connections to problems in the calculus of variations and the 1982 conjecture of T. Iwaniec on the $L^p$-norm of the Beurling–Ahlfors operator with connections to problems in the theory of qasiconformal mappings. Open questions, problems and conjectures are listed throughout the paper and copious references are provided.

In this paper we show how to compute the �� norm , �� 0, using the dyadic grid. This result is a consequence of the description of the Hardy spaces H p (R N ) in terms of dyadic and special atoms. Recently, several novel methods for computing the BMO norm of a function f in two dimensions were discussed in (9). Given its importance, it is also of interest to explore the possibility of computing the norm of a BMO function, or more generally a function in the Lipschitz class �α, using the dyadic grid in R N. It turns out that the BMO question is closely related to that of approximating functions in the Hardy space H 1 (R N ) by the Haar system. The approximation in H 1 (R N ) by affine systems was proved in (2), but this result does not apply to the Haar system. Now, if H A (R) denotes the closure of the Haar system in H 1 (R), it is not hard to see that the distance d(f, H A ) of f ∈ H 1 (R) to H A is ∼ � R ∞ 0 f(x) dx �, see (1). Thus, neither dyadic atoms suffice to describe the Hardy spaces, nor the evaluation of thenorm in BMO can be reduced to a straightforward computation using the dyadic intervals. In this paper we address both of these issues. First, we give a characterization of the Hardy spaces H p (R N ) in terms of dyadic and special atoms, and then, by a duality argument, we show how to compute the norm in �α(R N ), α ≥ 0, using the dyadic grid. We begin by introducing some notations. Let J denote a family of cubes Q in R N , and Pd the collection of polynomials in R N of degree less than or equal to d. Given α ≥ 0, Q ∈ J, and a locally integrable function g, let pQ(g) denote the unique polynomial in P(α) such that (g − pQ(g)) χQ has vanishing moments up to order (α). For a locally square-integrable function g, we consider the maximal function M ♯,2 α,J g(x) given by

This paper is concerned with the problem of existence of invariant sublattices for a positive matrix or a positive operator on $L_p(\mu)$. Common invariant sublattices for certain semigroups of positive operators are constructed. The paper also provides extensions of Perron-Frobenius theorem.

The class of mixed Tsirelson spaces is an important source of examples in the recent development of the structure theory of Banach spaces. The related class of modified mixed Tsirelson spaces has also been well studied. In the present paper, we investigate the problem of comparing isomorphically the mixed Tsirelson space T((Sn, θn) ∞=1) and its modified version TM((Sn, θn) ∞=1). It is shown that these spaces are not isomorphic for a large class of parameters (θn). 1 ≤ p < ∞. Figiel and Johnson (7) provided an analytic description, based on iteration, of the norm of the dual of Tsirelson's original space. Subse- quently, other examples of spaces were constructed with norms described it- eratively, notable among them were Tzafriri's spaces (20) and Schlumprecht's space(18). Gowers' and Maurey's solution to the unconditional basic se- quence problem (8) is a variation based on the same theme. It has emerged in recent years that, far from being isolated examples, Tsirelson's space and its variants from an important class of Banach spaces. Argyros and Deliyanni (2) were the first to provide a general framework for such spaces by defining the class of mixed Tsirelson spaces. Among the earliest vari- ants of Tsirelson's space was its modified version introduced by Johnson (9). Casazza and Odell (6) showed that Tsirelson's space is isomorphic to its modified version. This isomorphism was exploited to study the struc- ture of the space. The modification can be extended directly to the class of mixed Tsirelson spaces, forming the class of modified mixed Tsirelson spaces. It is thus of natural interest to determine if a mixed Tsirelson space is isomorphic to its modified version. This question has been considered by various authors, e.g., (3, 12), who provided answers in what may be con- sidered "extremal" cases. In the present paper, we show that for a large class of parameters, a mixed Tsirelson space and its modified version are not isomorphic.

In this paper we complete the results of our papers [2], [3] and show how to generate from the hypergeometric function F ⁡ 1 / 6 , 1 / 6 ; 1 ; ζ fundamental solutions for the classical Tricomi operator relative to any point in the elliptic, parabolic, or hyperbolic region of the operator.

To each generic tangle projection from the three-dimensional real vector space onto the plane, we associate a derived endofunctor on a graded parabolic version of the Bernstein-Gel'fand category $\mathcal{O}$. We show that this assignment is (up to shifts) invariant under tangle isotopies and Reidemeister moves and defines therefore invariants of tangles. The occurring functors are defined via so-called projective functors. The first part of the paper deals with the indecomposability of such functors and their connection with generalised Temperley-Lieb algebras which are known to have a realisation via decorated tangles. The second part of the paper describes a categorification of the Temperley-Lieb category and proves the main conjectures of [BFK]. Moreover, we describe a functor from the category of 2-cobordisms into a category of projective functors.

It is known that any locally compact group that is approximable by finite groups must be unimodular. However, this condition is not sufficient. For example, simple Lie groups are not approximable by finite ones. In this paper we consider the approximation of locally compact groups by more general finite algebraic systems. We prove that a locally compact group is approximable by finite semigroups iff it is approximable by finite groups. Thus, there exist some locally compact groups and even some compact groups that are not approximable by finite semigroups. We prove also that whenever a locally compact group is approximable by finite quasigroups (latin squares) it is unimodular. The converse theorem is also true: any unimodular group is approximable by finite quasigroups and even by finite loops. In this paper we prove this theorem only for discrete groups. For the case of non-discrete groups the proof is rather long and complicated and is given in a separate paper.

In this paper we study biholomorphic maps between Teichmüller spaces and the induced linear isometries between the corresponding tangent spaces. The first main result in this paper is the following classification theorem. If $M$ and $N$ are two Riemann surfaces that are not of exceptional type, and if there exists a biholomorphic map between the corresponding Teichmüller spaces Teich($M$) and Teich($N$), then $M$ and $N$ are quasiconformally related. Also, every such biholomorphic map is geometric. In particular, we have that every automorphism of the Teichmüller space Teich($M$) must be geometric. This result generalizes the previously known results (see [2], [5], [7]) and enables us to prove the well-known conjecture that states that the group of automorphisms of Teich($M$) is isomorphic to the mapping class group of $M$ whenever the surface $M$ is not of exceptional type. In order to prove the above results, we develop a method for studying linear isometries between $L^1$-type spaces. Our focus is on studying linear isometries between Banach spaces of integrable holomorphic quadratic differentials, which are supported on Riemann surfaces. Our main result in this direction (Theorem 1.1) states that if $M$ and $N$ are Riemann surfaces of nonexceptional type, then every linear isometry between $A^1(M)$ and $A^1(N)$ is geometric. That is, every such isometry is induced by a conformal map between $M$ and $N$.

We revisit Reinhold Baer's work on equivalent extensions, which can be considered as a forerunner of the authors' series of equivalence theorems. Our focus is on a paper entitled Extension Types of Abelian Groups published by Baer in 1949. In this paper, the main results were for a rather restrictive class of extensions called little extensions, but the notion of two extensions of A by B being equivalent given there are generally applicable. Our theme here is that Baer's vision and understanding of extensions placed him much ahead of the time in which he studied the subject in the 1930's and '40's.

Reinhold Baer’s visits to the University of Chicago were memorable events. His enthusiasm was infectious, his wide knowledge of so many things was fully appreciated, and his lectures were inspiring. It is perhaps not widely known that his influence on John Thompson was crucial in John’s student years. I have vivid memories of the fine times I had with him, his charming wife Marianna, and their son Klaus, a distinguished Egyptologist on the Chicago faculty. This paper is not directly connected with any of the areas in which he worked (perhaps not all readers will be aware that his early work was in the field of topology). However, Markov chose the Mathematische Annalen for his ground breaking papers, and I think the German mathematical community appreciated the importance of these papers. The enthusiasm that Frobenius showed was impressive. So I believe that Reinhold would have thought it appropriate for this paper to be dedicated to him. I. K.

We say that a group is {\em almost abelian} if every commutator is central and squares to the identity. Now let $G$ be the Galois group of the algebraic closure of the field $\QQ$ of rational numbers in the field of complex numbers. Let $G^{\ab+\epsilon}$ be the quotient of $G$ universal for homomorphisms to almost abelian profinite groups and let $\QQ^{\ab+\epsilon}/\QQ$ be the corresponding Galois extension. We prove that $\QQ^{\ab+\epsilon}$ is generated by the roots of unity, the fourth roots of the (rational) prime numbers and the square roots of certain sine-monomials. The inspiration for the paper came from recent studies of algebraic $\Gamma$-monomials by P.~Das and by S.~Seo. This paper has appeared as Duke Math. J. 114 (2002) 439-475.

Associated with each body $K$ in Euclidean $n$-space $\mathbb {R}\sp n$ is an ellipsoid $\Gamma\sb 2K$ called the Legendre ellipsoid of $K$. It can be defined as the unique ellipsoid centered at the body's center of mass such that the ellipsoid's moment of inertia about any axis passing through the center of mass is the same as that of the body. ¶ In an earlier paper the authors showed that corresponding to each convex body $K\subset\mathbb {R}\sp n$ is a new ellipsoid $\Gamma\sb {-2}K$ that is in some sense dual to the Legendre ellipsoid. The Legendre ellipsoid is an object of the dual Brunn-Minkowski theory, while the new ellipsoid $\Gamma\sb {-2}K$ is the corresponding object of the Brunn-Minkowski theory. ¶ The present paper has two aims. The first is to show that the domain of $\Gamma\sb {-2}$ can be extended to star-shaped sets. The second is to prove that the following relationship exists between the two ellipsoids: If $K$ is a star-shaped set, then $\Gamma\sb {-2}K\subset\Gamma\sb 2K$ ¶ with equality if and only if $K$ is an ellipsoid centered at the origin. This inclusion is the geometric analogue of one of the basic inequalities of information theory–the Cramer-Rao inequality.

We introduce a framework for a nonlinear potential theory without a kernel on a reexiv e, strictly convex and smooth Banach space of functions. Nonlinear potentials are dened as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic func- tional. The framework allows a development of other main concepts of nonlin- ear potential theory such as capacities, equilibrium potentials and measures of nite energy. x1. Introduction The goal of this paper is to present a fairly general setting which allows a development of basic concepts of nonlinear potential theory. This setting provides a unied approach to several aspects of nonlinear potential theory with kernel, as well as some kernel free potential theory. The concepts that can be developed include capacities of sets and functions, nonlinear poten- tials, equilibrium potentials, reduced functions, balayage, and measures of nite energy. The framework of our approach is a reexiv e, strictly convex and smooth Banach space of functions satisfying two additional hypotheses. Nonlinear potential theory in function spaces has been the subject of re- search in several papers during seventies (e.g., (7), (12), (19)). The goal was to extend the Dirichlet space theory to the nonlinear setting. This was achieved under various hypotheses. The common hypothesis was that the underlying function space is a Banach space with a vector lattice structure. Almost at the same time, a dieren t type of nonlinear potential the- ory began to take shape in the works of Fuglede, Meyers, and Havin and

In this paper we consider the almost sure convergence of a series of contractions (of an arbitrary Hilbert space) with random weights. The paper is a continuation of a previous work [PSW], in which only convergence in operator norm was investigated. We obtain conditions ensuring the existence of universal sets on which these series are converging almost everywhere, for any contraction. The paper is also a continuation of the paper [SW], in which an analogous problem concerning ergodic averages was considered, as well as the paper [S], which deals with a variant of the problem. The proofs of our results rely on uniform estimates of random polynomials which were established in a recent paper by the second author and proved by means of metric entropy methods.

In this paper, we prove an isogeny criterion for abelian varieties that involves conditions on the formal groups of the varieties (see Theorem 1.1). In the particular case of abelian varieties over ℚ with real multiplication, we easily deduce from our criterion a new proof of the Tate conjecture which is independent of G. Faltings's work [11], as well as a bound for the minimal degree of an isogeny between two isogenous abelian varieties, as in the paper of D. Masser and G. Wustholz [17]. To this end, we use C. Deninger and E. Nart's result giving the link between the L-functions and the formal groups of such varieties (see [9]). Our method generalizes D. and G. Chudnovsky's transcendental proof of the isogeny theorem for elliptic curves over ℚ [6, Prop. 2.3] to the case of abelian varieties, with a systematic use of the Arakelov formalism of J.-B. Bost (see [1]).

The subject matter of this paper is an old one with a rich history, beginning with the work of Gauss and Eisenstein, maturing at the hands of Smith and Minkowski, and culminating in the fundamental results of Siegel. More precisely, if L is a lattice over Z (for simplicity), equipped with an integral quadratic form Q, the celebrated Smith-Minkowski-Siegel mass formula expresses the total mass of (L,Q), which is a weighted class number of the genus of (L,Q), as a product of local factors. These local factors are known as the local densities of (L,Q). Subsequent work of Kneser, Tamagawa and Weil resulted in an elegant formulation of the subject in terms of Tamagawa measures. In particular, the local density at a non-archimedean place p can be expressed as the integral of a certain volume form ωld over AutZp(L,Q), which is an open compact subgroup of AutQp(L,Q). The question that remains is whether one can find an explicit formula for the local density. Through the work of Pall (for p 6= 2) and Watson (for p = 2), such an explicit formula for the local density is in fact known for an arbitrary lattice over Zp (see [P] and [Wa]). The formula is obviously structured, though [CS] seems to be the first to comment on this. Unfortunately, the known proof (as given in [P] and [K]) does not explain this structure and involves complicated recursions. On the other hand, Conway and Sloane [CS, §13] have given a heuristic explanation of the formula. In this paper, we will give a simple and conceptual proof of the local density formula, for p 6= 2. The view point taken here is similar to that of our earlier work [GHY], and the proof is based on the observation that there exists a smooth affine group scheme G over Zp with generic fiber AutQp(L,Q), which satisfies G(Zp) = AutZp(L,Q). This follows from general results of smoothening [BLR], as we explain in Section 3. For the purpose of obtaining an explicit formula, it is necessary to have an explicit construction of G. The main contribution of this paper is to give such an explicit construction of G (in Section 5), and to determine its special fiber (in Section 6). Finally, by comparing ωld and the canonical volume form ωcan of G, we obtain the explicit formula for the local density in Section 7. The smooth group schemes constructed in this paper should also be of independent interest.

The main object considered in this paper is the trace function, defined as a suitably normalized trace of a product of intertwining operators for the Drinfeld-Jimbo quantum group, multiplied by the exponential of an element of the Cartan subalgebra. This function depends of two parameters -- the element of the Cartan subalgebra, and the highest weight of the Verma module in which the trace is taken. The main results of the paper are that the trace function satisfies two systems of difference equations with respect to the first parameter (the quantum Knizhnik-Zamolodchikov-Bernard and Macdonald-Ruijsenaars equations), and that it is symmetric with respect to the two parameters. In particular, this implies that for each of the above two systems of equations there is the dual system with respect to the second parameter, which is also satisfied by the trace function. The paper establishes a connection between the I.Frenkel-Reshetikhin theory of quantum conformal blocks, the work of Felder-Mukhin-Tarasov-Varchenko on the quantum KZB and Ruijsenaars equations, the work of Etingof-I.Frenkel- Kirillov Jr.-Styrkas on traces of intetwining operators, and the Macdonald- Cherednik theory. The methods of the paper are based on the theory of dynamical twists and R-matrices., Comment: 38 pages, amstex; some misprints and small errors were corrected in the new version

This paper is concerned with a general theorem on the number of nonreal zeros of transcendental functions. J. Fourier formulated the theorem in his work Analyse des equations determineesin 1831, but he did not give a proof. Roughly speaking, the theorem states that if a real entire function f( x)can be expressed as a product of linear factors, then we can count the nonreal zeros of f( x)by observing the behavior of the derivatives of f( x)on the real axis alone. As we shall see in the sequel, this theorem completely justifies his former argument, by which he tried to prove that the function J0(2 √ x) has only real zeros. It seems that no complete proof of the theorem is known, and no general theorem has been published that justifies the argument. Later, in 1930, G. Polya published a paper entitled Some problems connected with Fourier’s work on transcendental equations[P3]. In this paper, Polya conjectured two hypothetical theorems that are closely related to Fourier’s unproved theorem. In fact, he conjectured three, but he proved that two of them are equivalent to each other. The first hypothetical theorem is a modernized formulation of the theorem, and it justifies Fourier’s argument completely. The second conjecture was proved in 1990, but it is impossible to justify the argument using the conjecture alone. In the present paper, we prove Polya’s formulation of the theorem (his first conjecture) as well as its extensions, give a very simple and direct proof of the second conjecture mentioned above, and exhibit some applications of the results. In particular, we completely justify Fourier’s argument by our general theorems. Acknowledgments. Professor Fefferman has encouraged and helped us to publish this paper. The authors truly thank him for this. The authors also thank Professor Csordas for his kind interest in the results and his valuable suggestions on the proof of the Polya-Wiman conjecture. 1. Historical introduction. In this section, we briefly explain our results as well as their background. A real entire function is an entire function that assumes only real values on the real axis. Fourier’s unproven theorem asserts that we can know the number of nonreal zeros of such real entire functions by counting their critical points, which are defined as follows: Let f( x)be a real analytic function defined in an open

1. Introduction and main results. In this paper we investigate the relations between the rate of decay of solutions of Schrodinger equations, continuity properties of spectral measures of the corresponding operators, and dynamical properties of the corresponding quantum systems. The first main result of this paper shows that, in great generality, certain upper bounds on the rate of growth of L 2 norms of generalized eigenfunctions over expanding balls imply certain minimal singularity of the spectral measures. Consider an operator H � V defined by the differential expression H � V =− � + V( x) �

The twisted tensor L-function of f , which we denote by G(s, f), is a certain Dirichlet series associated to a quadratic extension of number fields K/F , and a cuspidal automorphic function f over K. It was introduced in [1] by Asai, following previous work of Shimura, in the case when f is a Hilbert modular cusp form over a real quadratic extension K of Q. In the past twenty odd years, this L-function has been considered more generally: for instance [11] and [12] deal with quadratic extensions of totally real fields, [17] with imaginary quadratic extensions of Q, and [3], [4] and [14] with general quadratic extensions of number fields. All these papers have been primarily concerned with establishing analytic properties of G(s, f) analogous to those in [1], such as meromorphic continuation to the entire complex plane, location and finiteness of the number of poles, and functional equation. The aim of this paper is to prove a rationality result for G(s, f) in the imaginary quadratic setting. If K is an imaginary quadratic field, and f a cusp form associated to K, we establish that there is a ‘period’ Ωj(f) such that

Let A be a unital $C^*$-algebra, given together with a specified state $\phi:A \to C$. Consider two selfadjoint elements a,b of A, which are free with respect to $\phi$ (in the sense of the free probability theory of Voiculescu). Let us denote $c:=i(ab-ba)$, where the i in front of the commutator is introduced to make c selfadjoint. In this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b. Some properties of the corresponding operation on probability measures are also discussed. The methods we use are combinatorial, based on the description of freeness in terms of non-crossing partitions; an important ingredient is the notion of R-diagonal pair, introduced and studied in our previous paper funct-an/9604012., Comment: LaTeX, 38 pages with 2 figures

The first part of the paper is concerned, among other things, with a characterization of filter regular sequences in terms of modules of generalized fractions. This characterization leads to a description, in terms of generalized fractions, of the structure of an arbitrary local cohomology module of a finitely generated module over a notherian ring. In the second part of the paper, we establish homomorphisms between the homology modules of a Koszul complex and the homology modules of a certain complex of modules of generalized fractions. Using these homomorphisms, we obtain a characterization of unconditioned strong d-sequences.

In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}. Quasi-$k$-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity., 21 pages, AMSTeX. Significantly changed version of paper originally Titled "On minimal isometric embeddings"

In this paper, we study the shuttle operator for a second order linear elliptic operator P on a noncompact manifold X. Zhao has introduced and studied the shuttle operator and its relation to the theory of positive solutions in the case of small perturbations of the Laplacian in IR, n ≥ 3. Zhao was motivated by works of Chung and Varadhan which consider one-dimensional Schrodinger operators. The main purpose of the paper is to extend the above studies. We prove that, in the general case, the spectral radius of the shuttle operator is strictly less, equal, or strictly greater than 1 if and only if the operator P is respectively subcritical, critical, or supercritical in X. We demonstrate the usefulness of the characterization for proving theorems. Our approach is purely analytic. ∗This research was partially supported by THE ISRAEL SCIENCE FOUNDATION administered by THE ISRAEL ACADEMY OF SCIENCES AND HUMANITIES and by the Fund for the Promotion of Research at the Technion

This paper is the second part of our work on differential Galois theory as we promised in [U3]. Differential Galois theory has a long history since Lie tried to apply the idea of Abel and Galois to differential equations in the 19th century (cf. [U3], Introduction). When we consider Galois theory of differential equation, we have to separate the finite dimensional theory from the infinite dimensional theory. As Kolchin theory shows, the first is constructed on a rigorous foundation. The latter, however, seems inachieved despite of several important contributions of Drach, Vessiot,…. We propose in this paper a differential Galois theory of infinite dimension in a rigorous and transparent framework. We explain the idea of the classical authors by one of the simplest examples and point out the problems.

In this paper we give an application of equivariant moduli spaces to the study of smooth group actions on certain 4-manifolds. A rich source of examples for such actions is the collection of algebraic surfaces (compact and nonsingular) together with their groups of algebraic automorphisms. From this collection, further examples of smooth but generally nonalgebraic actions can be constructed by an equivariant connected sum along an orbit of isolated points. Given a smooth oriented 4-manifold X which is diffeomorphic to a connected sum of algebraic surfaces, we can ask: (i) which (finite) groups can act smoothly on X preserving the orientation, and (ii) how closely does a smooth action on X resemble some equivariant connected sum of algebraic actions on the algebraic surface factors of X? For the purposes of this paper we will restrict our attention to the simplest case, namely X p2(C) #...# p2(C), a connected sum of n copies of the complex projective plane (arranged so thatX is simply connected). Furthermore, ASSUMPTION. All actions will be assumed to induce the identity on H,(X, Z).

In recent papers [10, 11, 12, 13, 14], we defined some new ρ-invariants for any rational prime ρ congruent to 1 mod 4 and D-invariants for any positive square-free integer D such that the fundamental unit εD of real quadratic field Q(√D) satisfies NεD = –1, and studied relationships among these new invariants and already known invariants.One of our main purposes in this paper is to generalize these D-invariants to invariants valid for all square-free positive integers containing D with NεD = 1. Another is to provide an improvement of the theorem in [14] related closely to class number one problem of real quadratic fields. Namely, we provide, in a sense, a most appreciable estimation of the fundamental unit to be able to apply, as usual (cf. [3, 4, 5, 9, 12, 13]), Tatuzawa’s lower bound of L(l, XD) (Cf[7]) for estimating the class number of Q(√D) from below by using Dirichlet’s classical class number formula.

In the present paper, we show that an infinite dimensional vector whose components are Fourier coefficients of an automorphic form is characterized as an infinite dimensional vector which is annihilated by an infinite matrix constructed by the values of a Bessel function. Results and methods are all simple and concrete.Although the idea in the present paper is applicable to more general cases, our investigation will be restricted to the case of automorphic forms of weight 0, i.e., automorphic functions, with respect to SL(2, Z) on the upper half plane, in order to explain the main idea distinctly.

According to the Langlands functoriality conjecture, broadened to the setting of spherical varieties (of which reductive groups are special cases), a map between $L$ -groups of spherical varieties should give rise to a functorial transfer of their local and automorphic spectra. The “beyond endoscopy” proposal predicts that this transfer will be realized as a comparison between limiting forms of the (relative) trace formulas of these spaces. In this paper, we establish the local transfer for the identity map between $L$ -groups, for spherical affine homogeneous spaces $X=H\backslash G$ whose dual group is $\operatorname{SL}_{2}$ or $\operatorname{PGL}_{2}$ (with $G$ and $H$ split). More precisely, we construct a transfer operator between orbital integrals for the $(X\times X)/G$ -relative trace formula, and orbital integrals for the Kuznetsov formula of $\operatorname{PGL}_{2}$ or $\operatorname{SL}_{2}$ . Besides the $L$ -group, another invariant attached to $X$ is a certain $L$ -value, and the space of test measures for the Kuznetsov formula is enlarged to accommodate the given $L$ -value. The transfer operator is given explicitly in terms of Fourier convolutions, making it suitable for a global comparison of trace formulas by the Poisson summation formula, hence for a uniform proof, in rank $1$ , of the relations between periods of automorphic forms and special values of $L$ -functions.

The purpose of this paper is the study of some basic properties of universal induced characters and their applications to the representation theory of the classical groups (for the definition of a universal induced character, see § 3).The starting point was the paper [F] by E. Formanek on matrix invariants. In his paper [F], Formanek has investigated the Hilbert series for the ring of matrix invariants from the point of view of the representation theory of the general linear group and the symmetric group. In this paper we shall study polynomial concomitants of a group from the same point of view.