Function Field Arithmetic by Dinesh S. Thakur

By Dinesh S. Thakur

This ebook presents an exposition of functionality box mathematics with emphasis on fresh advancements touching on Drinfeld modules, the mathematics of unique values of transcendental capabilities (such as zeta and gamma capabilities and their interpolations), diophantine approximation and similar attention-grabbing open difficulties. whereas it covers many subject matters handled in ‘Basic buildings of functionality box mathematics’ via David Goss, it enhances that ebook with the inclusion of modern advancements in addition to the therapy of latest issues corresponding to diophantine approximation, hypergeometric capabilities, modular varieties, transcendence, automata and solitons. there's additionally new paintings on multizeta values and log-algebraicity. the writer has integrated a variety of worked-out examples. Many open difficulties, that can function sturdy thesis difficulties, are mentioned.

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Is of generic characteristic, if fo is transcendental over IF,. If fo = 0, it is of characteristic t. More generally, if g~ is the minimal polynomial of fo over JFq, then it is xi=, Drinfeld modules 38 of characteristic p. So, for example, if F = A/@ and L is the canonical map, we get p of characteristic p. Note that Ct = t r is isomorphic to pt = t f1r by the isomorphism f:’(q-l), 1 ) which in general is defined only over an extension of F q ( t , f l ) . So there is only one isomorphism class of rank one A-modules of generic characteristic in this case.

We will study its interpolat2e(z) + + + + Drinfeld modules 34 tions, functional equations, special values later. > of binomial coefficients. 14 a new analog of differentiation restoring the differential equation property! Classically, the Bernoulli numbers, which occur in power sums and zeta values at both negative and positive integers and whose divisibilities give valuable information on class groups of cyclotomic fields, are defined by the generating function z / ( e Z - 1) = C B,z"/n!. ,once we have a good factorial as mentioned above.

If we take the usual definition of the Hilbert class field as the maximal abelian everywhere unramified extension, then it is of infinite degree over the base, because of the constant field extensions. If we decree that the field of constants should not increase, then there are h such extensions Ki, all with Galois group isomorphic to the class group Pico(K). If L is degree h constant field extension, then LKi = LK1- - - Kh, for any i. And this degree h2 invariantly defined extension is suggested [AT68, pa.

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