By Dinesh S. Thakur
This booklet offers an exposition of functionality box mathematics with emphasis on contemporary advancements referring to Drinfeld modules, the mathematics of unique values of transcendental features (such as zeta and gamma services and their interpolations), diophantine approximation and comparable fascinating open difficulties. whereas it covers many themes taken care of in ‘Basic buildings of functionality box mathematics’ by means of David Goss, it enhances that ebook with the inclusion of modern advancements in addition to the therapy of recent issues equivalent to diophantine approximation, hypergeometric services, modular kinds, transcendence, automata and solitons. there's additionally new paintings on multizeta values and log-algebraicity. the writer has incorporated a variety of worked-out examples. Many open difficulties, which could function strong thesis difficulties, are mentioned.
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Extra info for Function Field Arithmetic
The algebraic closure of F ( ( t ) )in this case consists of ‘generalized fractional power series’ of the form CiEsfiti, where S c Q is a well-ordered subset such that for some m, all the elements of mS have denominators power of p (rather than denominator one, in Newton’s case) and satisfying further complicated combinatorial conditions, described by Kedlaya in [KedOl], so that they are algebraic over F ( ( t ) ) . 4. Class field theory and Galois group also for descriptions of the algebraic closure of F ( ( t ) ) for , perfect F and for completion of the algebraic closure etc.
We do not have close cousins in the classical case for the higher rank Drinfeld modules. We start by motivating the simplest case, that of Carlitz module by analytic approach, introducing analogs of the exponential function. Then we quickly indicate how to get good analogs of e, 27r2, roots of unity, factorial, binomial functions, and zeta functions, Bernoulli numbers, modular forms etc. We will study these in more detail in the later chapters. Then, after developing the Drinfeld modules for general A , we will see applications to the objects of arithmetic interest.
As lZ*I = 2, whereas IFq[t]* I= 1F;l = q - 1, this is an analog of quadratic reciprocity for lFq[t]:The n-th power residue symbol (alp), E F; is defined as usual. ) We have (a/P), = a(Norm(P)-l)/"modPas usual, for prime P not dividing a. Now let P and P' be irreducible monic polynomials of degree d and d' respectively. Write P ( t ) = - a q i ) and P'(t) = d' (t - a'qz'). nt(t Fqd no Now a E a(a)mod(t - a ) . By considering the symbol over IFq [t]and [t],we see that so that where the last equality follows by symmetry by just counting the number of switches of signs.
Function Field Arithmetic by Dinesh S. Thakur