An Elementary Approach to L-Functions mod p

✍ Scribed by G.W. Anderson

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 157 KB
• Volume: 80
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2452

✦ Synopsis

We give a brief presentation of basic facts concerning the relationship of L-functions mod p and the Cartier operator. We take the notion of the Cartier operator for granted but develop all other notions that we need from scratch. An idea taken from Tate's famous sum of residues equals zero'' paper plays a key role and gives the paper a distinctly functional analytic'' flavor. All the ideas in the paper have been ``in the air'' for some time; our contribution is simply to organize them and to express them succinctly. 2000 Academic Press 2.1. Gauges Let M be a finitely generated A-module. A function #: M Ä Z 0 ~[& ] will be called a gauge if there exist generators a 1 , ..., a k of A as an F q -algebra and generators m 1 , ..., m r of M as an A-module such that for all & # Z the set [m # M | #(m) &] is the set of m # M admitting a representation m= : r i=1 f i (a 1 , ..., a k ) m i