INTRODUCTION

5

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

pth

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 G° 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

me.

Kai Behrend