Treatise on differential calculus by Isaac Todhunter

By Isaac Todhunter

This Elibron Classics publication is a facsimile reprint of a 1864 version through Macmillan and Co., Cambridge and London.

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4 A morphism : p --j p' between two Drinfeld Amodule over F is q5 E F { r } such that +pa = pa+ for all a E A. A non-zero morphism is called an isogeny. Invertible morphism is isomorphism. A simple count of degrees shows that isogenies can occur only among Drinfeld modules of the same rank and characteristic. As t generates A over IFq, a general Drinfeld A-module p over F , of rank r over F can be fully described by pt = firi, where fi E F , f r is non-zero and fo = ~ ( t )It. is of generic characteristic, if fo is transcendental over IF,.

This extension arises naturally in rank one Drinfeld modules theory and the ideals of A become principal in H A , as we will see in Chapter 3. See also [Ros87]. Class field theory in terms of idele groups: In this approach, at the expense of dealing with a slightly more complicated objects, we do not have to talk about the modulus and can handle infinite extensions and local-global passage more smoothly. Let K be a global field and let AK, ZK and CK = I K / K * denote its adele ring, idele group and idele class group respectively.

A non-zero morphism is called an isogeny. Invertible morphism is isomorphism. A simple count of degrees shows that isogenies can occur only among Drinfeld modules of the same rank and characteristic. As t generates A over IFq, a general Drinfeld A-module p over F , of rank r over F can be fully described by pt = firi, where fi E F , f r is non-zero and fo = ~ ( t )It. 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.

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