Special values of \(L\)-functions in characteristic \(p\)
The paper under review gives an interpretation of the special value of the \(L\)-function associated to a shtuka over a curve defined over \({\mathbb F}_q\), in terms of extension groups in the category of shtukas, much in the flavor of Bloch-Kato conjectures. Results in the same context have been previously obtained by \textit{G. Anderson} and \textit{D. Thakur} [Ann. Math. (2) 132, No. 1, 159--191 (1990; Zbl 0713.11082)] and [J. Number Theory 60, No.1, 165--209 (1996; Zbl 0868.11031)]. Unlike Anderson, the author works within the framework of shtukas, rather than \(t\)-motives. Associated to an \(\ell\)-adic lisse sheaf \(\mathcal{F}\) with coefficients in \(\mathbb{Z}_\ell\) on a curve defined over \(\mathbb{F}_q\) , \textit{K. Kato} [Arithmetic algebraic geometry. Lect. Notes Math. 1553, 50--163 (1993; Zbl 0815.11051)] shows that the appropriately defined special value \(L^*(X,F,T)\) relies on two trivializations (one of which given by his zeta-elements, the other by height-pairings) of \(\det(H^{\bullet}_c(X,\mathcal{F})\otimes{\mathbb Q}_p)^{-1}\). The author attempts on a similar goal, now the \(\ell\)-adic sheaf replaced by an \(F\)-crystal (in the sense of [\textit{G. Böckle} and \textit{R. Pink}, Cohomological theory of crystals over function fields. EMS Tracts in Mathematics 9. Zürich: European Mathematical Society (EMS). (2009; Zbl 1186.14002)] and \textit{M. Emerton} and \textit{M. Kisin} [Geometric aspects of Dwork theory. Vol. I, II. Berlin: Walter de Gruyter. 677--700 (2004; Zbl 1104.14012)]) coming from a shtuka. The author explains that there are three major differences as compared to the situation Kato places himself in loc. cit., \S3.5: (a) The special value may be transcendental over the field of definition \(\mathbb{F}_q(C)\) of the shtuka. (b) The relevant \(L\)-function may be expressed as the characteristic polynomial of an operator \(u\) on a certain space (which may be infinite dimensional), such that the kernel and the cokernel of \(1-u\) are groups of extensions in the category of shtukas. (c) One needs Fontaine's theory in equal characteristics. The main tools that go into the proofs are \textit{G. Anderson}'s [J. Number Theory 80, No. 2, 291--303 (2000; Zbl 0977.11036)] trace formula (as generalized by \textit{G. Böckle} and \textit{R. Pink} [Zbl 1186.14002]) and a light dose of \(p\)-adic Hodge Theory in equal characteristics, borrowed from the recent preprint of \textit{A. Genestier} and the author [Théorie de Fontaine en égales caractéristiques (2008), to appear in Ann. Sci. Éc. Norm. Supér. (4), see also \url{http://www.math.jussieu.fr/~vlafforg/fontaine.pdf}].
scientific article; zbMATH DE number 5598293
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