Algebraic Geometry

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By Alexander Polishchuk

The purpose of this ebook is to provide a latest therapy of the speculation of theta features within the context of algebraic geometry. the newness of its strategy lies within the systematic use of the Fourier-Mukai remodel. the writer begins by means of discussing the classical concept of theta services from the viewpoint of the illustration idea of the Heisenberg crew (in which the standard Fourier remodel performs the renowned role). He then indicates that during the algebraic method of this concept, the Fourier–Mukai remodel can frequently be used to simplify the present proofs or to supply thoroughly new proofs of many very important theorems. Graduate scholars and researchers with powerful curiosity in algebraic geometry will locate a lot of curiosity during this quantity.

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27rZ (2) k(s)u-Sds is given by the formulae o 1 Jl { k(u;x,y) = ylog t A u log x2 ° if if if if u> y2 xy < u < y2 x 2 < u < xy u < x 2. Chapter 2 Artin L- Functions 54 Now consider the integral 1. JK = -2 7rZ r (- ;k (s)) k(s; x, y)ds. "K On the one hand, it is equal to (logYlx)2 - Lk(p;x,y) p where p runs over all zeroes of (K(S). Write p = (3 + i'y. 2]) NK(r; so) « 1 + r(log IdKI + nK log(lsol + 2)). Since it follows that L k(p; x, y) «x- 2c5 ~~1-c5 1 00 1 r2dNK(r; 1) c5 «x- 2c5 (8- 2 + 8- 1 log IdKI).

2, we have 7fD(X) = II~: Lix + O(IDI~x~nF logM(K/F)x). Proof We have 7fD(X) IDI. ICI LIX . ) -1Gf LIX = '" ~ ( 7fc(x) -1Gf c where the sum is taken over all conjugacy classes C contained in D. 2. Remark. 1 as O(IClx~nF logM(K/F)x). Thus Artin's conjecture allows us to replace ICI with ICI ~. 1 even without assuming Artin's conjecture. We give two such results below. Chapter 2 Artin L- Functions 50 Let D be a union of conjugacy classes in C and let H be a subgroup of C satisfying (i) Artin's conjecture is true for the irreducible characters of H (ii) H meets every class in D.

This has the following immediate corollary. If KIF is a Galois extension of odd degree and (K (s) has a zero of order:::; 3 at a point So then all Arlin L-functions of KIF are analytic at so. 2 of Stark. Of course, Stark's result makes no assumption on the Galois group of KIF. We give a brief outline of the proof. Assume the theorem is false, and take G to be a minimal counterexample for which Artin's conjecture fails, at a point s = So where the order of (K(S) is small as explained in the statement.

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