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)

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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 X be a connected scheme. Then one can associate (after Grothendieck) to X its algebraic fundamental group π1(X). This group π1(X) is a profinite group which is uniquely determined (up to inner automorphisms) by the property that the category of finite, discrete sets equipped with a continuous π1(X)-action is equivalent to the category of finite etale coverings of X. Moreover, the assignment X → π1(X) is a functor from the category of connected schemes (and morphisms of schemes) to the category of profinite topological groups and continuous outer homomorphisms (i.e., continuous homomorphisms of topological groups, where we identify any two homomorphisms that can be obtained from one another by composition with an inner automorphism).

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The local pro-p anabelian geometry of curves

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The Geometry of the Compactification of the Hurwitz Scheme

This paper forms the first part of a three-part series in which we treat various topics in absolute anabelian geometry from the point of view of developing abstract algorithms ,o r"software", that may be applied to abstract profinite groups that "just happen" to arise as (quotients of) ´ fundamental groups from algebraic geometry. One central theme of the present paper is the issue of understand- ing the gap between relative, "semi-absolute", and absolute anabelian geometry. We begin by studying various abstract combinatorial prop- erties of profinite groups that typically arise as absolute Galois groups or arithmetic/geometric fundamental groups in the anabelian geome- try of quite general varieties in arbitrary dimension over number fields, mixed-characteristic local fields, or finite fields. These considerations, combined with the classical theory of Albanese varieties, allow us to derive an absolute anabelian algorithm for constructing the quotient of an arithmetic fundamental group determined by the absolute Galois group of the base field in the case of quite general varieties of arbitrary dimension. Next, we take a more detailed look at certain p-adic Hodge- theoretic aspects of the absolute Galois groups of mixed-characteristic local fields. This allows us, for instance, to derive, from a certain result communicated orally to the author by A. Tamagawa, a "semi- absolute" Hom-version — whose absolute analogue is false! — of the anabelian conjecture for hyperbolic curves over mixed-characteristic lo- cal fields. Finally, we generalize to the case of varieties of arbitrary dimension over arbitrary sub-p-adic fields certain techniques devel- oped by the author in previous papers over mixed-characteristic local fields for applying relative anabelian results to obtain "semi-absolute" group-theoretic contructions of thefundamental group of one hy- perbolic curve from thefundamental group of another closely related hyperbolic curve.

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Topics in Absolute Anabelian Geometry I: Generalities

In this paper, we develop the theory of “cuspidalizations” of the etale fundamental group of a proper hyperbolic curve over a finite or nonarchimedean mixed-characteristic local field. The ultimate goal of this theory is the group-theoretic reconstruction of the etale fundamental group of an arbitrary open subscheme of the curve from the etale fundamental group of the full proper curve. We then apply this theory to show that a certain absolute $p$-adic version of the Grothendieck Conjecture holds for hyperbolic curves “of Belyi type”. This includes, in particular, affine hyperbolic curves over a nonarchimedean mixed-characteristic local field which are defined over a number field and isogenous to a hyperbolic curve of genus zero. Also, we apply this theory to prove the analogue for proper hyperbolic curves over finite fields of the version of the Grothendieck Conjecture that was shown in [Tama].

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Absolute anabelian cuspidalizations of proper hyperbolic curves

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Shinichi Mochizuki

Papers

The local pro-p anabelian geometry of curves

The Geometry of the Compactification of the Hurwitz Scheme

Topics in Absolute Anabelian Geometry I: Generalities

Absolute anabelian cuspidalizations of proper hyperbolic curves

A Theory of Ordinary $p$-Adic Curves