By Gerard B. M. van der Geer, BJJ Moonen, René Schoof

Ever because the analogy among quantity fields and serve as fields was once found, it's been a resource of suggestion for brand new principles, and an extended heritage has no longer whatsoever detracted from the attraction of the subject.As a deeper figuring out of this analogy can have super results, the quest for a unified process has develop into a type of Holy Grail. the arriving of Arakelov's new geometry that attempts to place the archimedean locations on a par with the finite ones gave a brand new impetus and ended in miraculous luck in Faltings' fingers. there are many additional examples the place rules or ideas from the extra geometrically-oriented international of functionality fields have ended in new insights within the extra arithmetically-oriented international of quantity fields, or vice versa.These invited articles through top researchers within the box discover a variety of points of the parallel worlds of functionality fields and quantity fields. issues variety from Arakelov geometry, the hunt for a conception of types over the sector with one point, through Eisenstein sequence to Drinfeld modules, and t-motives.This quantity is geared toward a large viewers of graduate scholars, mathematicians, and researchers attracted to geometry and mathematics and their connections.

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**Sample text**

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?