O'Reilly logo

Convolution and Equidistribution by Nicholas M. Katz

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

CHAPTER 7

The Main Theorem

Lemma 7.1. Let G/k be a form of Gm, and N in Parith ι-pure of weight zero and arithmetically semisimple. The quotient group Garith,N /Ggeom,N is a group of multiplicative type, in which a Zariski dense subgroup is generated by the image of any single Frobenius conjugacy class Frobk,χ. If the quotient is finite, say of order n, then it is canonically Z/nZ, and the image in this quotient of any Frobenius conjugacy class FrobE,χ is deg(E/k) mod n.

Proof. Representations of the quotient Garith,N /Ggeom,N are objects in <N>arith which are geometrically trivial, i.e., those objects V ⊗ δ1, for V some completely reducible representation of Gal(k/k), which lie in <N>arith. Such an object is a finite direct sum of one-dimensional ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required