L-Functions and Tamagawa Numbers of Motives

The notion of a motif was first defined and studied by A. Grothendieck, and this paper is an attempt to understand some of the implications of his ideas for arithmetic. We will formulate a conjecture on the values at integer points of L-functions associated to motives. Conjectures due to Deligne and Beilinson express these values “modulo Q* multiples” in terms of archimedean period or regulator integrals. Our aim is to remove the Q* ambiguity by defining what are in fact Tamagawa numbers for motives. The essential technical tool for this is the Fontaine-Messing theory of p-adic cohomology. As evidence for our Tamagawa number conjecture, we show that it is compatible with isogeny, and we include strong results due to one of us (Kato) for the Riemann zeta function and for elliptic curves with complex multiplication.

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