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

In the present paper, which forms the third part of a three-part series on an algorithmic approach to absolute anabelian geometry, we apply the ab- solute anabelian technique of Belyi cuspidalization developed in the second part, together with certain ideas contained in an earlier paper of the author concerning the category-theoretic representation of holomorphic structures via either the topologi- cal group SL2(R) or the use of "parallelograms, rectangles, and squares", to develop a certain global formalism for certain hyperbolic orbicurves related to a once- punctured elliptic curve over a number field. This formalism allows one to construct certain canonical rigid integral structures ,w hich we refer to aslog-shells, that are obtained by applying the logarithm at various primes of a number field. More- over, although each of these local logarithms is "far from being an isomorphism" both in the sense that it fails to respect the ring structures involved and in the sense (cf. Frobenius morphisms in positive characteristic!) that it has the effect of exhibiting the "mass" represented by its domain as a "somewhat smaller collection of mass" than the "mass" represented by its codomain, this global formalism allows one to treat the logarithm operation as a global operation on a number field which satisfies the property of being an "isomomorphism up to an appropriate renormal- ization operation", in a fashion that is reminiscent of the isomorphism induced on differentials by a Frobenius lifting, once one divides by p. More generally, if one thinks of number fields as corresponding to positive characteristic hyperbolic curves and of once-punctured elliptic curves on a number field as corresponding to nilpotent ordinary indigenous bundles on a positive characteristic hyperbolic curve, then many aspects of the theory developed in the present paper are reminiscent of (the positive characteristic portion of) p-adic Teichmuller theory.

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Topics in absolute anabelian geometry iii: global reconstruction algorithms

In this paper, we discuss various “general nonsense” aspects of the geometry of semi-graphs of profinite groups [cf. [Mzk3], Appendix], by applying the language of anabelioids introduced in [Mzk4]. After proving certain basic properties concerning various commensurators associated to a semi-graph of anabelioids, we show that the geometry of a semi-graph of anabelioids may be recovered from the category-theoretic structure of certain naturally associated categories — e.g., “temperoids” [in essence, the analogue of a Galois category for the “tempered funda

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Semi-graphs of Anabelioids

Foundations of P-Adic Teichmuller Theory

On the one hand, one knows (cf. the Remark in [4] following Theorem 4.2) that the Grothendieck Conjecture cannot hold in the naive sense (i.e., if one removes the condition of “compatibility with the filtrations” from the outer isomorphisms considered – see, e.g., [8]), so one must put some sort of condition on the outer isomorphisms of Galois groups that one considers. The condition discussed here is that they preserve the higher ramification groups. One can debate how natural a condition this is, but at least it gives some sort of idea of how “good” an outer isomorphism of Galois groups must be in order to arise “geometrically.”

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A Version of the Grothendieck Conjecture for p-Adic Local Fields

In this paper, we study the pro-Σ fundamental groups of configuration spaces, where Σ is either the set of all prime numbers or a set consisting of a single prime number. In particular, we show, via two somewhat distinct approaches, that, in many cases, the “fiber subgroups” of such fundamental groups arising from the various natural projections of a configuration space to lower-dimensional configuration spaces may be characterized group-theoretically.

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Shinichi Mochizuki

Papers

Topics in absolute anabelian geometry iii: global reconstruction algorithms

Semi-graphs of Anabelioids

Foundations of P-Adic Teichmuller Theory

A Version of the Grothendieck Conjecture for p-Adic Local Fields

The algebraic and anabelian geometry of configuration spaces