By T. N. Shorey

This can be a built-in presentation of the speculation of exponential diophantine equations. The authors current, in a transparent and unified style, purposes to exponential diophantine equations and linear recurrence sequences of the Gelfond-Baker concept of linear types in logarithms of algebraic numbers. themes coated contain the Thue equations, the generalised hyperelliptic equation, and the Fermat and Catalan equations. the required preliminaries are given within the first 3 chapters. every one bankruptcy ends with a bit giving info of similar effects.

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2, in particular in Theorem 2, we recalled the definition and existence of a fine moduli space Yr (n) for Drinfeld A-modules of rank r and characteristic prime to n that carry a level n-structure. From now on, we only consider the case r = 2, and therefore omit the superscript r whenever r = 2. 2, by M(n) we denote the τ -sheaf corresponding to the universal Drinfeld A-module on Y(n), and by gn : Y(n) → Spec A(n) its characteristic. The first observation we will need in the following is due to Drinfeld.

If we extend it to a free module and choose τ = 0 on the complement, we find that M(n)(k−2) is of pullback type. Also it is not difficult to see that the crystal R 0 gn! M(n)(k−2) is zero. Since, moreover, g¯ n is smooth and proper of relative dimension 1, Proposition 5 yields the following. Proposition 11. The crystal R i gn! M(n)(k−2) is zero for i = 1. The crystal S (k) (n) = R 1 gn! M(n)(k−2) is of pullback type and hence flat. 32 Gebhard Böckle In [Bö04], jointly with R. Pink we computed some explicit examples of such motives for A = Fq [t] and n = (t).

Is there a conjecture à la Birch and Swinnerton–Dyer (BSD) for Amotives? A naive analogue of BSD cannot hold, since it is known due to a result of Poonen (cf. [Po95]) that the naive analogue of the Mordell–Weil group for a Drinfeld Amodule over a field L as in Section 2 is of infinite A-rank. In [An96] in certain cases, a finite rank A-module has been constructed, that could serve as a starting point to investigate such a conjecture. Arithmetic over Function Fields 27 Question 3. Is there a Riemann hypothesis or a (substitute for a) functional equation?

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