Steven H. Weintraub

Moduli spaces of Abelian surfaces : compactification, degenerations, and theta functions

The first third of the monograph focuses on the construction and description of toroidal compactifications of moduli spaces, which specializes to the Igusa compactification in the principally polarized case. In the next section, the emphasis is on the construction of degenerate Abelian surfaces, usi

A mild generalization of Eisenstein’s criterion

We state and prove a mild generalization of Eisenstein’s Criterion for a polynomial to be irreducible, correcting an error that Eisenstein made himself. Eisenstein originally stated and proved the irreducibility criterion we now name after him in [2]. Both his statement and proof are virtually identical to how we would formulate them today. In that paper Eisenstein was actually concerned with the lemniscate, where the relevant question was irreducibility of polynomials with coefficients in the Gaussian integers, rather than in the ordinary integers, but, as he observed, the statement and proof are identical in either case. Indeed, in [2], he applied his citerion to show that, for a prime p, the p-th cyclotomic polynomial Φp(x) = (xp −1)/(x−1) is irreducible. He used the same trick we still use today, observing that his criterion applies to the polynomial Φp(x + 1). The first proof of the irreducibility of Φp(x) had been given by Gauss [4, Article 341], with a simpler proof having been given by Kronecker [5], but Eisenstein’s proof was simpler still. Also, as Eisenstein observed, Gauss’s and Kronecker’s proofs used particular properties of p-th roots of 1, and so only could be applied to Φp(x), while his criterion applies far more generally. (Actually, Schonemann had given an irreducibility criterion in [6] that is easily seen to be equivalent to Eisenstein’s criterion, and had used it to prove the irreducibility of Φp(x), but this had evidently been overlooked by Eisenstein; for a discussion of this see [1].) Eisenstein then went on to remark that the proof of his criterion goes through to show the following more general result: Let f (x) = anx + . . .+a0 be any primitive polynomial with integer coefficients and suppose there is a prime p such that p does not divide an, p divides ai for i = 0, . . . ,n− 1, and for some k with 0 ≤ k ≤ n− 1, p2 does not divide ak. Then f (x) is irreducible (in Z[x]). However, this claim is false, as we see from the following factorization, valid for any k ≥ 1 and any m ≥ 0: (xk + p)(xk+m +(p2 − p)xm + p) = x2k+m + p2xk+m +(p3 − p2)xm + pxk + p2. The point of this note is to establish a correct result along these lines. Theorem 1. Let f (x) = anx + . . . + a0 ∈ Z[x] be a polynomial and suppose there is a prime p such that p does not divide an, p divides ai for i = 0, . . . ,n− 1, and for some k with 0 ≤ k ≤ n−1, p2 does not divide ak. Let k0 be the smallest such value of k. If f (x) = g(x)h(x), a factorization in Z[x], then min(deg(g(x)),deg(h(x))) ≤ k0. In particular, for a primitive polynomial f (x), if k0 = 0 then f (x) is irreducible, and if k0 = 1 and f (x) does not have a root in Q, then f (x) is irreducible. Proof. Suppose we have a factorization f (x) = g(x)h(x). Let g(x) have degree d0 and h(x) have degree e0. Let d be the smallest power of x whose coefficient in g(x) is not divisible by 2000 Mathematics Subject Classification. 12E05, Secondary 01A55.

Jordan Canonical Form:Theory and Practice

Jordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra. The JCF of a linear transformation, or of a matrix, encodes all of the structural information about that linear transformation, or matrix. This book is a careful development of JCF. After beginning with background material, we introduce Jordan Canonical Form and related notions: eigenvalues, (generalized) eigenvectors, and the characteristic and minimum polynomials. We decide the question of diagonalizability, and prove the Cayley-Hamilton theorem. Then we present a careful and complete proof of the fundamental theorem: Let V be a finite-dimensional vector space over the field of complex numbers C, and let T : V → V be a linear transformation. Then T has a Jordan Canonical Form. This theorem has an equivalent statement in terms of matrices: Let A be a square matrix with complex entries. Then A is similar to a matrix J in Jordan Canonical Form, i.e., there is an invertible matrix P and a matrix J in Jordan Canonical Form with A = PJP-1. We further present an algorithm to find P and J, assuming that one can factor the characteristic polynomial of A. In developing this algorithm we introduce the eigenstructure picture (ESP) of a matrix, a pictorial representation that makes JCF clear. The ESP of A determines J, and a refinement, the labeled eigenstructure picture (lESP) of A, determines P as well. We illustrate this algorithm with copious examples, and provide numerous exercises for the reader.

Several Proofs of the Irreducibility of the Cyclotomic Polynomials

Abstract We present a number of classical proofs of the irreducibility of the nth cyclotomic polynomial ϕn(x). For n prime we present proofs due to Gauss (1801), in both the original and a simplified form, Kronecker (1845), and Schönemann/Eisenstein (1846/1850). For general n, we present proofs due to Dedekind (1857), Landau (1929), and Schur (1929).

Semi-free Zp-actions on highly-connected manifolds

O. Introduction In this paper we study semi-free cyclic group actions on highly-connected manifolds whose dimensions are a multiple of 4. The actions we consider are required to induce the identity map on homology, and to have fixed-point sets either a union of isolated points or a highly-connected submanifold. We consider both smooth and piecewise-linear actions. There are two basic problems in the study of such actions- when they exist and what they look like. We attack these problems in three stages: First, we construct specific examples of manifolds with actions on them. Second we show that a salient characteristic of these actions holds generally, so that any action resembles one we construct. Third, we show how to modify specific actions in order to obtain actions in great generality. More specifically, we proceed as follows: In Chapter I we construct actions on highly-connected manifolds via an equivariant plumbing technique. That is, we first construct bundles with actions on them, and then plumb them together respecting the actions. In order to use this technique to construct actions with isolated fixed points, we must first find a basis for the middle-dimensional homology with respect to which the cupproduct form has a matrix of a certain type. An algebraic argument shows one always exists. Its existence also helps in constructing actions with positivedimensional fixed-point sets. By construction, these actions have invariant spheres forming a basis for the middle-dimensional homology. In Chapter II we show that such invariant spheres always exist for cyclic group actions which have isolated fixed points and act trivially on homology. (Although the specific calculations are for oddorder groups, the same proof works for arbitrary cyclic groups.) Thus the actions we construct differ from those in the general case only in that we begin with linear actions on normal bundles to these spheres, rather than on arbitrary PL actions on regular neighborhoods of these spheres. Chapter III is devoted to showing how to modify actions (such as, but by no means only, those constructed in Chapter I) to obtain new actions on new manifolds. Here the group actions we consider are those of cyclic groups of

An Interesting Algebraic Variety

In this article I would like to describe the structure of the Igusa compactification of the Siegel modular space of degree two and level two. This space is of particular interest for several reasons. For one thing, it lies in the intersection of a number of different areas of mathematics--Riemann surfaces, theta functions a nd automorphic forms, algebraic geometry, and differential topology. For another, it is a special case of several general constructions--t hose of the theory of compactifications, geometric invariant theory, resolution of singularities, and symplectic geometry. Further, there are a number of different and fruitful methodsalgebraic, analytic, geometric, and topological--with which to analyze this space. (Not to be neglected is the chance to impress the reader with an object whose name has 13 words, thereby being 6 1/2 times as good as the Fermat conjecture.) The point of this article is that this space has a beautiful and intricate geometric structure, which I shall attempt to describe well enough for the reader to appreciate. I will begin by describing elliptic curves (thereby, like Prousts madeleine, taking the reader back to the halcyon days of graduate school) and then discuss our situation, which is a generalization thereof. Recall that an elliptic curve (as in figure IIa) is C/L, the complex plane C modulo a lattice L. (A lattice is the set of integral linear combinations of two non-collinear vectors based at some point of the complex plane.) Without changing the elliptic curve, we may normalize the lattice by requiring that the vectors be based at the origin, the first vector be the standard unit vector in the x-direction, and the second vector have positive y-coordinate, so that the lattice is specified by the endpoint of this vector, i.e. by a point x in S 1 = {zeC[Imz>O}

On Legendre’s Work on the Law of Quadratic Reciprocity

Abstract Legendre was the first to state the law of quadratic reciprocity in the form in which we know it and he was able to prove it in some but not all cases, with the first complete proof being given by Gauss. In this paper we trace the evolution of Legendres work on quadratic reciprocity in his four great works on number theory.

Invariants of branched covering from the work of Serre and Mumford

We study invariants of Mumford and Serre in the context of topological branched covering spaces, and obtain formulae relating these invariants to the Arf invariant. 1991 Mathematics Subject Classification: 57M12; 57R15. §0. Introduction In recent years, several authors, including P.E. Gönner and R. Perus, Gallagher, J.R Serre, V. Snaith, R. Jardine and A. Fröhlich ([CP], [G], [Sl], [Sn], [J], [F]), have studied the Hasse-Witt invariant of the quadratic form -» Tr£jK(jc) associated to a finite separable extension J? of a field K. In particular, in [S 1] Serre gave a formula for this invariant in terms of the second Stiefel-Whitney class of a Galois representation. Subsequently, bis result was extended by [Sn], [F] from the perspectives of Brauer groups and algebraic /T-theory. In [S 2], Serre provided a similar formula in the setting of branched covering maps between Riemann surfaces. Since then, Esnault, Kahn, and Viehweg have provided a common generalization of Serres results in the framework of Dedekind schemes [EKV]. The object of this paper is to both understand and extend Serres results in [S 2] from a topological viewpoint. Our setting is a branched covering map : -» between two oriented manifolds of arbitrary dimension. As Serre did, we require that 1 The first author is partially supported by NSF Grant DMS-89-03302 2 The second author is partially supported by NSF Grant DMS-88-03552, LEQSF (Louisiana Educational Quality Support Fund) grant (1981-84)-RD-A-14, and the Sonderforschungsbereich für Geometrie and Analysis, Universität Göttingen 536 R. Lee, S. H. Weintraub all ramification indices be odd. In this setting, there are combinatorially defined Stiefel-Whitney classes w^G, n). On the other band, using differential topology, we have the direct image b ndle η+(ΜΎ\χ) where MY\X is the KO-theory orientation class (see §2) and hence Stiefel-Whitney classes ^(71^) = ^(η^(ΜΎ\χ)). For the first Stiefel-Whitney classes, we have the equality

The Irreducibility of the Cyclotomic Polynomials

The irreducibility of the cyclotomic polynomials is a fundamental result in algebraic number theory that has been proved many times, by many different authors, in varying degrees of generality and using a variety of approaches and methods of proof. We examine these in the spirit of our inquiry here.

Generalized Homology Theory

We begin by presenting the famous Eilenberg-Steenrod axioms for homology. We then proceed by drawing (many) useful consequences from these axioms. At the end of this chapter we introduce cohomology.