and every morphism x : SpecFgn X, the trace of the arithmetic Frobe-
nius J ^ ( —l)z tr Jg|/^(Rx-M) is absolutely convergent, no matter how we
embed Q^ into C. We show that the triangulated categories D+(Xsm,v4)
are stable under the two operations R/* and R / \ The question of stability
under /* remains open.
In the final section (Section 4) we show that our formalism of convergent
complexes is rich enough to prove the general Lefschetz Trace Formula for
the arithmetic Frobenius on algebraic stacks. Our main result is given in
Definition 5.33 and Theorem 5.38. As Corollary 5.39 we get the result we
conjectured in [3]. To finish, we give a rather interesting example, due to P.
Deligne. We show how our trace formula applied to the stack SDti of curves
of genus one, may be interpreted as a type of Selberg Trace Formula. It
gives the sum J2k - ^ tr Tp\Sk+2i where Tp is the
Hecke operator on the
space of cusp forms of weight k + 2, in terms of elliptic curves over the finite
field ¥p.
Notations and Conventions. Our reference for algebraic stacks is [16].
We always assume all algebraic stacks to be locally noetherian, in particular
quasi-separated. Stacks will usually be denoted by German letters X, 2)
etc., whereas for spaces we use Roman letters X , Y etc. A gerbe X/X is
called neutral, if it has a section, i.e. if it is isomorphic to B(G/X), for some
relative algebraic space of groups G/X. For a group scheme G, we denote
by its connected component of the identity. Presentations X » X of an
algebraic stack X are always smooth, and of finite type if X is of finite type.
For an algebraic stack X we denote by |X| the set of isomorphism classes
of points of X. We consider |X| as a topological space with respect to the
Zariski topology.
If I is a category, we denote by oh I the objects, by 8.1 the arrows in /
and by J
o p
the dual category. A box in a commutative diagram denotes a
cartesian diagram.
The natural numbers start with zero. N = {0,1,2,...} .
Acknowledgment. I would like to thank Prof. P. Deligne for encouraging
me to undertake this work and for pointing out the example of Section 4 to
Kai Behrend
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