On Gekeler's Conjecture for Function Fields

✍ Scribed by Bruno Anglès

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 130 KB
• Volume: 87
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2587

✦ Synopsis

Let P be a monic irreducible polynomial in F q [T ] such that d=deg P is even. We have obtained (B. AngleÁ s, 1999, J. Number Theory 79, 258 283), when q is odd, a class number congruence modulo P for the ideal class number of F q [T, -P] which is similar to the famous Ankeny Artin Chowla formula. As a consequence, we have that this latter class number is divisible by the characteristic of F q if and only if the Bernoulli Carlitz number B((q d &1)Â2) is divisible by P. This result is a special case of Gekeler's conjecture. In these notes, we give a class number congruence for the ideal class number of any totally real subfield F of the Pth cyclotomic function field (Theorem 2). As expected, this formula involves the Bernoulli Carlitz numbers. It also appears in this formula which we call the regulator modulo P of F : R F . In the case where F=F q (T, -P), then R F 0 (mod P). Unfortunately, this is not the case for general F. In Section 3, we show that if R F 0 (mod P), then Gekeler's conjecture is true for the field F (Theorem 4). 2001 Academic Press 4 P =[z # Q ac P , [P] C (z)=0].