By Qing Liu

The articles during this quantity are accelerated types of lectures introduced on the Graduate summer time institution and on the Mentoring software for ladies in arithmetic held on the Institute for complex Study/Park urban arithmetic Institute. The subject of this system used to be mathematics algebraic geometry. the alternative of lecture issues used to be seriously prompted through the hot astounding paintings of Wiles on modular elliptic curves and Fermat's final Theorem. the most emphasis of the articles within the quantity is on elliptic curves, Galois representations, and modular kinds. One lecture sequence deals an advent to those items. The others speak about chosen contemporary effects, present study, and open difficulties and conjectures. The e-book will be an appropriate textual content for a complicated graduate subject matters path in mathematics algebraic geometry.

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Suppose p is a rank one generic characteristic Drinfeld A-module, with pa: = x x1r x2r2 and py = y y1r y2r2. Then the comparison of coefficients of r4 in the equation corresponding to y2 = x - x2, we see that (y2/x2)lo = -1, and hence the coefficients can not be in A or K . ) We have H A = K ( c ) with c2 = -1 and q = 1 x cy is the fundamental unit for HA. We write 7 = 1 x - cy for the conjugate, and leave it to the reader to verify that px = x yrp- Vr2, py = y (1 x)r/r cVr2 (from [Hay79]) now defines a rank one Drinfeld A-module of generic characteristic over 0 ~In fact, ~ we .

3) Conversely, given a closed subgroup of CK containing the connected component, there is a unique abelian extension with the property above with respect t o it. (4) For a local field K , and for a finite abelian extension L" of K,, the Artin map (which can be obtained by composing the global A r t i n map of (1) with the canonical inclusion of K,* in the ideles) induces isomorphism of K , * ~ N o ~ $ ~ ( L "onto ) * Gal(L"/K,) and f o r every open subgroup N of finite index in K,*, there exists a unique abelian extension L" of K , such that N o r m K ( L " ) * = N .

0 Madan [Mad69; Mad701 proved that if F is a finite Galois extension of a function field K , then h K divides h F : After reduction step reducing to a simple cyclic extension of degree prime to characteristic, we use the decomposition of the zeta functions into product of L-functions and then use the class number formula and integrality of L-functions etc. in the first proof and class field theory for the second. Since J(lFq) is a subgroup of J(lFqn),the divisibility follows for constant field extensions.