First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Transcendental number theory, by Alan Baker. Pp. x, 147. £4·90. 1975. SBN 0 521 20461 5 (Cambridge University Press)

/pdf/compactifying-the-space-of-stable-maps-10odrhuhej.pdf

Compactifying the space of stable maps

We study the structure of the stacks of twisted stable maps to the classifying stack of a finite group G—which we call the stack of twisted G-covers, or twisted G-bundles. For a suitable group Gwe show that the substack corresponding to admissible G-covers is a smooth projective fine moduli space. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Twisted Bundles and Admissible Covers

Complete discrete valuation fields Extensions of discrete valuation fields The norm map Local class field theory I Local class field theory II The group of units of local number fields Explicit formulas for the Hilbert symbol Explicit formulas for the Hilbert pairing on formal groups The Milnor $K$-groups of a local field Bibliography List of notations Index

Local Fields and Their Extensions

In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion of the tangent vector to an arc in a manifold is used in §8.2 and §8.3, and the notion of a vector field is introduced in an auxiliary capacity in §8.3. The Campbell-Hausdorff formula is introduced only to establish the First and Second Principles of §8.1 below; if you are willing to take those on faith the formula (and exercises dealing with it) can be skimmed. Exercises 8.27–8.29 give alternative descriptions of the Lie algebra associated to a Lie group, but can be skipped for now.

Lie Algebras and Lie Groups

Let Σ be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers. In this paper, we prove that various objects that arise from the geometry of the configuration space of a hyperbolic curve over an algebraically closed field of characteristic zero may be reconstructed group-theoretically from the pro-Σ fundamental group of the configuration space. Let X be a hyperbolic curve of type (g, r) over a field k of characteristic zero. Thus, X is obtained by removing from a proper smooth curve of genus g over k a closed subscheme [i.e., the “divisor of cusps”] of X whose structure morphism to Spec(k) is finite étale of degree r; 2g−2+r > 0. Write Xn for the n-th configuration space associated to X, i.e., the complement of the various diagonal divisors in the fiber product over k of n copies of X. Then, when k is algebraically closed, we show that the triple (n, g, r) and the generalized fiber subgroups — i.e., the subgroups that arise from the various natural morphisms Xn → Xm [m < n], which we refer to as generalized projection morphisms — of the pro-Σ fundamental group Πn of Xn may be reconstructed group-theoretically from Πn whenever n ≥ 2. This result generalizes results obtained previously by the first and third authors and A. Tamagawa to the case of arbitrary hyperbolic curves [i.e., without restrictions on (g, r)]. As an application, in the case where (g, r) = (0, 3) and n ≥ 2, we conclude that there exists a direct product decomposition Out(Πn) = GT Σ ×Sn+3 — where we write “Out(−)” for the group of outer automorphisms [i.e., without any auxiliary restrictions!] of the profinite group in parentheses and GT (respectively, Sn+3) for the pro-Σ Grothendieck-Teichmüller group (respectively, symmetric group on n+3 letters). This direct product decomposition may be applied to obtain a simplified purely grouptheoretic equivalent definition — i.e., as the centralizer in Out(Πn) of the union of the centers of the open subgroups of Out(Πn) — of GT . One of the key notions underlying the theory of the present paper is the notion of a pro-Σ log-full subgroup — which may be regarded as a sort of higher-dimensional analogue of the notion of a pro-Σ cuspidal inertia subgroup of a surface group — of Πn. In the final section of the present paper, we show that, when X and k satisfy certain conditions concerning “weights”, the pro-l log-full subgroups may be reconstructed group-theoretically from the natural outer action of the absolute Galois group of k on the geometric pro-l fundamental group of Xn. 2010 Mathematics Subject Classification. Primary 14H30; Secondary 14H10.

Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups

In this paper, we study the pro-Σ anabelian geometry of hyperbolic curves, where Σ is a nonempty set of prime numbers, over Galois groups of “solvably closed extensions” of number fields — i.e., infinite extensions of number fields which have no nontrivial abelian extensions. The main results of this paper are, in essence, immediate corollaries of the following three ingredients: (a) classical results concerning the structure of Galois groups of number fields; (b) an anabelian result of Uchida concerning Galois groups of solvably closed extensions of number fields; (c) a previous result of the author concerning the pro-Σ anabelian geometry of hyperbolic curves over nonarchimedean local fields.

/pdf/global-solvably-closed-anabelian-geometry-1ors00tz3h.pdf

Global solvably closed anabelian geometry

Dans cet article, nous presentons une theorie concernant l'uniformisation et les espaces de modules des courbes hyperboliques p-adiques. D'une part, cette theorie etend aux places non archimediennes les uniformisations de Fuchs et Bers et les espaces de modules des courbes hyperboliques complexes. Pour cette raison, nous designerons souvent cette theorie sous le nom de theorie de Teichmuller p-adique. D'autre part, cette theorie peut etre vue comme un analogue hyperbolique de la theorie de Serre-Tate pour les varietes abeliennes ordinaires et leurs espaces de modules. L'objet au centre de la theorie de Teichmuller p-adique est le champ des modules des nilcurves. Ce champ est un recouvrement plat du champ des modules de courbes hyperboliques en caracteristique p. Il parametre les courbes hyperboliques munies de donnees auxiliaires d'uniformisation en caracteristique p. La geometrie de ce champ de modules peut s'analyser de maniere combinatoire au voisinage de l'infini. D'autre part, une analyse globale de sa geometrie mene a une demonstration de l'irreductibilite du champ des modules de courbes hyperboliques via des methodes de caracteristique p. Diverses parties de ce champ des nilcurves admettent des relevements canoniques au-dessus desquels on obtient des coordonnees canoniques et des representations galoisiennes canoniques. Ces coordonnees canoniques sont l'analogue, pour les courbes hyperboliques, des coordonnees canoniques dans la theorie de Serre-Tate et l'analogue p-adique des coordonnees de Bers dans la theorie de Teichmuller. De plus, les representations galoisiennes qui apparaissent eclairent d'un jour nouveau l'action exterieure du groupe de Galois d'un corps local sur le complete profini du groupe de Teichmuller.

/pdf/an-introduction-to-p-adic-teichmuller-theory-ajmc9hwk4d.pdf

An introduction to p-adic Teichmüller theory

§0. Introduction §1. The Scheme-Theoretic Fourier Transform §2. Discrete Gaussians and Gauss Sums §3. Review of Degree Computations in [HAT] §4. The Fourier Transform of an Algebraic Theta Function: The Case of an Etale Lagrangian Subgroup §5. The Fourier Transform of an Algebraic Theta Function: The Case of a Lagrangian Subgroup with Nontrivial Multiplicative Part §6. Generic Properties of the Norm §7. Some Elementary Computational Lemmas §8. The Various Contributions at Infinity §9. The Main Theorem §10. The Theta Convolution

/pdf/the-scheme-theoretic-theta-convolution-tp3n0nhwyb.pdf

The Scheme-Theoretic Theta Convolution

A hyperbolic curve is an algebraic curve obtained by removing r points from a smooth, proper curve of genus g, where g and r are nonnegative integers such that 2g−2+r > 0. If X is a hyperbolic curve over the field of complex numbers C, then X gives rise in a natural way to a Riemann surface X . As one knows from complex analysis, the most fundamental fact concerning such a Riemann surface (due to Köbe) is that it may be uniformized by the upper half-plane, i.e.,

/pdf/the-intrinsic-hodge-theory-of-p-adic-hyperbolic-curves-4qa9gw8lsj.pdf

Shinichi Mochizuki

Papers

Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups

Global solvably closed anabelian geometry

An introduction to p-adic Teichmüller theory

The Scheme-Theoretic Theta Convolution

The Intrinsic Hodge Theory of $p$-adic Hyperbolic Curves