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In this article we construct a Galois and Hecke equivariant morphism connecting the first cohomology group on Faltings’ site of a formal strict neighborhood of the ordinary locus in a formal modular curve of level prime to p, with coefficients in the analytic distributions of a certain analytic weight k on the p-adic Tate module of the universal elliptic curve to the overconvergent modular forms of weight \(k+2\). We prove that this morphism is an isomorphism on the finite slope parts.

After 40 years of operation with the original design, Pan Pacific Copper determined it was necessary to rebuild the Saganoseki Flash Smelting Furnace to continue safe operation. The original design employed a rigid steel frame, which, through hearth growth, led to severe distortion of the frame. Contributing to the continued growth of the hearth were thermal cycles that occurred during the government mandated annual shutdowns. Hatch designed a unique sprung bound, pivoting binding frame to maximize crucible size within the existing furnace footprint, while integrating the PPC designed cooling jackets. The bound system maintains tight brick joints, while the new conductive hearth design with integrated bottom cooling produces a protective freeze layer to accommodate higher furnace throughput. Minimization of furnace downtime for the rebuild was achieved through effective construction planning, highly trained contractors, and through an efficient start-up and ramp-up to full production.

Given a totally real field F and a prime integer p which is unramified in F, we construct p-adic families of overconvergent Hilbert modular forms (of non-necessarily parallel weight) as sections of, so called, overconvergent Hilbert modular sheaves. We prove that the classical Hilbert modular forms of integral weights are overconvergent in our sense. We compare our notion with Katz’s definition of p-adic Hilbert modular forms. For F = ℚ, we prove that our notion of (families of) overconvergent elliptic modular forms coincides with those of R. Coleman and V. Pilloni.

This article was delivered as the 2010 Shann Memorial Lecture at the University of Western Australia on 17 August 2010. It charts a brief history of the development of finance, before focusing on the regulatory response and issues arising from the financial turbulence of recent years.

Cet article est une exploration constructive des rapports entre les symboles modulaires classiques et les symboles modulaires p-adiques surconvergents. Plus precisement, nous donnons une preuve constructive d'un theoreme de controle (Theoreme 1.1) du deuxieme auteur [19] ; ce theoreme demontre l'existence et l'unicite des « liftings propres » des symboles propres modulaires classiques de pente non-critique. Comme application, nous decrivons un algorithme en temps polynomial pour le calcul explicite des fonctions L p-adiques associees dans ce cas-la. Dans le cas de pente critique, le theoreme de controle echoue toujours a produire des « liftings propres » (voir Theoreme 5.14 et [16] pour un succedane), mais l'algorithme « reussit » neanmoins a produire des fonctions L p-adiques. Dans les deux dernieres sections, nous presentons des donnees numeriques pour plusieurs exemples de pente critique et examinons le polygone de Newton des fonctions L p-adiques associees.

Given a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $, we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$, to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $-coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$-adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope $\leq h$.

Let E be an elliptic curve which is defined over Q and has stable reduction modulo a given prime p. Assuming that E is modular, one can associate to E a p-adic L-function Lp(E, s). (See [-Mz-SwD, A-V, Vi, Mz-T-T] for its construction in various cases.) This function is defined by a certain interpolation property and is analytic for seZp. In this paper, we will assume that E has split multiplicative reduction at p. Under this assumption the interpolation property implies that Lp(E, 1)=0. We will prove a formula for Ep(E, 1) which was discovered experimentally by Mazur, Tate, and Teitelbaum [Mz-T-T]. By Tate's p-adic urfiformization theory, there is a p-adic integer q~:epZp (which we refer to as the Tate period for E) and a p-adic analytic isomorphism

This paper examines the experience of nine industrial countries with monetary targeting. The paper suggests that monetary targets were adopted as a tactical response to a particular economic situation, not as monetary rules. Other objectives were given precedence over targets when thought desirable. Most countries changed the targeted aggregate, and two dropped targets altogether. While inflation fell in most countries, the extent to which this was due to the pursuit of monetary targets is unclear. The place of monetary aggregates in many countries now appears to be as one among a number of indicators considered by the authorities in the setting of monetary policy.

We extend the work of Ash and Stevens [Ash-Stevens 97] on p-adic analytic families of p-ordinary arithmetic cohomology classes for GL(N,Q) by introducing and investigating the concept of p-adic rigidity of arithmetic Hecke eigenclasses. An arithmetic eigenclass is said to be "rigid" if (modulo twisting) it does not admit a nontrivial p-adic deformation containing a Zariski dense set of arithmetic specializations. This paper develops tools for explicit investigation into the structure of eigenvarieties for GL(N). We use these tools to prove that known examples of non-sefldual cohomological cuspforms for GL(3) are rigid. Moreover, we conjecture that for GL(3), rigidity is equivalent to non-selfduality., 23 pages

The role of the financial sector in the business cycle has long been a subject of study, but has moved somewhat more to the fore in recent years. In part, the renewed interest has been motivated by the apparent increase in the frequency of asset price events, typically facilitated by the extension of credit, and their heightened importance for economic performance in the industrial economies. In addition, the work on the revised Basel capital accord has focused attention on the possibility that regulatory arrangements might exacerbate the cyclical behavior of banks. The fickle nature of cross-border capital flows, including in Asia in the late 1990s, brings an international dimension to the issues.

This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.

The principal aim of this article is to sketch the proof of the following famous assertion.

This paper outlines the inflation objective for monetary policy in Australia, which we describe as seeking to achieve a broad central tendency for inflation of between 2 and 3 per cent over the long run – a “thick point” – rather than a narrow target band. It also provides a more detailed rationale for this objective. In doing so, the paper discusses the issues relevant in determining the appropriate mean inflation rate at which policy should aim, the degree of variation of inflation around that central point, and how policy should respond to shocks. A simple model of the economy is presented which attempts to address these issues in a consistent framework.

The structure of the cuspidal divisor class group is investigated by relating this structure to arithmetic properties of special values of L-functions of weight two Eisenstein series. A new proof of a theorem of Kubert (Proposition 3.1) concerning the group of modular units is derived as a consequence of the method. The key lemma is a nonvanishing result (Theorem 2.1) for values of the "L-function" attached to a one-dimensional cohomology class over the relevant-congruence subgroup. Proposition 4.7 provides data regarding Eisenstein series and associated subgroups of the cuspidal divisor class group which the author hopes will simplify future calculations in the cuspidal group. Let Xr/Q be Shimura's canonical model over Q of the complete modular curve associated to the congruence group r c SL2(Z), where r is one of the groups r(N), r1(N), or ro(N) The cuspidal divisor class group Cr C Pic°(Xr) plays an important role in the theory of the arithmetic of the modular curve Xr, for example in Mazur's proof of Ogg's conjecture [13] and in the proof by Mazur and Wiles of the main conjecture of Iwasawa theory [14]. Subgroups of the cuspidal group lead to the isogenies used in the Eisenstein descent theory. These subgroups can also be used to give congruence formulas for the universal special values of the L-function of Xr which are compatible with the descent theoretic results and the conjecture of Birch and Swinnerton-Dyer [3, 5, 12, 19]. The structure of the cuspidal group has been investigated extensively by Kubert and Lang [6, 7]. Their approach is based on a study of the group of modular units. In the present paper we propose another approach based on a study of integrality and divisibility properties of special values of L-functions of weight two Eisenstein series. Typically such an L-function is a product of two Dirichlet L-functions whose special values are closely related to the arithmetic of cyclotomic fields. Using a theorem of L. tllashington [20] on the non-p-part of the class number in cyclotomic Zp-extensions we describe a method of computing subgroups of Cr which should be useful for the descent theory. The corresponding congruence formulas for universal special values of L-functions are an immediate consequence of the method. Since Eisenstein series occur in many settings our point of view offers the prospect of generalization. Such a generalization to Hilbert-Blumenthal varieties may be an Received by the editors September 4, 1984. 1980 Mathematics Subject Classification. Primary llF67, llG16; Secondary llFll, llF33. l Partially supported by NSF Grant MCS 82-01762. t1985 American Mathematical Society 0002-9947/8S $1.00 + $.25 per page

We classify isolated terminal cyclic quotient singularities in dimension three, and isolated Gorenstein terminal cyclic quotient singularities in dimension four. In addition, we give a new proof of a combinatorial lemma of G. K. White using Bernoulli functions. Let X be a smooth algebraic variety over C, and let wx be the canonical bundle of X. For each n > O, if Ii(X, WOn) =# 0, there is a natural pluricanonical map On: X -PI'(X, wOn)*. An algebraic variety is of general type if on is a birational map for n sufficiently large. For a variety of general type, the pluricanonical images On(X) are the most natural birational modets of X to study. Canonical singularities are the singularities which may occur in the pluricanonical models of varieties of general type. In dimension 1, the pluricanonical models are smooth so there are no canonical singularities; in dimension 2 the canonical singularities coincide with the classical rational double points. One characterization of the rational double points is as quotient singularities: if G is any finite subgroup of S1(2, C), then the quotient C2/G has a rational double point, and every rational double point is analytically isomorphic to such a quotient singularity. Reid and Shepherd-Barron [10], and independently Tai [14], have given a condition for quotient singularities to be canonical in arbitrary dimensions (although not all canonical singularities are quotient singularities in dimensions greater than two). Terminal singularities are a class of canonical singularities which play an important role in birational geometry (as evidenced by recent work of Mori [8], Reid [12], and Tsunoda [15]). In this note we study cyclic quotient singularities which are terminal. In dimension three we explicitly describe all isolated terminal cyclic quotient singularities, while in dimension four, we describe isolated terminal cyclic quotient singularities which are also Gorenstein. The description uses a combinatorial lemma due to G. K. White [17]; we have given a new proof of this lemma (Corollary 1.4 below) using Bernoulli functions. Received by the editors March 31, 1983. 1980 Mathematics Subject Classification. Primary 14B05; Secondary IOA40, 14L30, 32B30, 52A25.

In the present paper we shall give evidence to support the claim (Conjecture I below and (1.3)) that every elliptic curve A/o which can be parametrized by modular functions admits a canonical modular parametrization whose properties can be related to intrinsic properties of A. In particular, we will see how such a parametrizat ion can be used to prove some rather pleasant integrality properties of Stickelberger elements ad p-adic L-functions attached to A. In addition, if Conjecture I is true then we can give an intrinsic characterization of the isomorphism class of a special elliptic curve in the Q-isogeny class of A distinguished by modular considerations. For most of the paper we have opted for the concrete approach and defined modular parametrizations in terms of X I (N) (Definition 1.1). However, to justify our view of these parametrizations as being canonical, we begin here with a more intrinsic definition. Recall that Shimura ([19], Chap. 6; see w 1 of this paper) has defined a compatible system of canonical models of modular curves {Xs, SeS~}, where 5 p is a certain collection of open subgroups of the group GL(2, Az) over the finite adeles A I of Q. We define the adelic upper half-plane to be the pro-variety )~=lL_m Xs and give )~ the Q-structure induced by the s field of modular functions whose q-expansions at the 0-cusp have coefficients in Q. A modular parametrization of A is a Q-morphism ~: ) ( ~ A which sends

In the remaining pages we display three sets of tables of algebraic parts of special values of L-functions, $$A\left( X \right) = \frac{{\pi \left( {\mathop X\limits^ -} \right)\;\;L\left( {f,X,l} \right)}}{{\Omega _f^{{\mathop{\rm sgn}} \left( X \right)}}}$$ Here χ denotes a primitive quadratic character of conductor mχ. In the first two sets of tables mχ is taken to be positive or negative depending on whether χ(−1) = sgn χ is plus or minus one. The modular form f ranges through the weight two parabolic eigenforms for the following modular curves: 1. X0 (N), N prime ≤ 43; 2. Genus one curves X0 (N), N = 14, 15, 20, 21, 24, 27, 32, 36, 49; 3. X1 (13). The complex number \(\Omega _f^{{\mathop{\rm sgn}} \chi }\) is an appropriate period of f(z)dz on the corresponding modular curves.

Mazur and Swinnerton-Dyer [30] have shown how to pass from a TD-eigenfunction on (ℚ/ℤ)p,Δ to a p-adic distribution on ℤ p,Δ * . Applying this procedure to the universal modular symbol on a modular curve, X, Mazur [26] defines a p-adic L-function LP (X0(N), χ, s) associated to an Eisenstein prime P for Г0(N), N prime.

Let Γ be a group of type (N1, N2) and E ∈ E 2 (Γ) be a J -eigenfunction. This chapter is devoted to describing the subgroup CE of the cuspidal group and the speical values of the associated cohomology class φE ∈ H1(X;A(E)).

In this chapter we develop the tools needed to describe the subgroup of H1(X(Γ);ℚ/ℤ) corresponding to the cuspidal group C(Γ).