On Gekelerʹs Conjecture for Function Fields Original Research Article

Let P be a monic irreducible polynomial in imageq[T] such that d=deg P is even. We have obtained (B. Anglès, 1999, J. Number Theory79, 258–283), when q is odd, a class number congruence modulo P for the ideal class number of imageq[T, image] 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 imageq if and only if the Bernoulli–Carlitz number B((qd−1)/2) is divisible by P. This result is a special case of Gekelerʹs 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 : RF. In the case where F=imageq(T, image ), then RFnot identical with0 (mod P). Unfortunately, this is not the case for general F. In Section 3, we show that if RFnot identical with0 (mod P), then Gekelerʹs conjecture is true for the field F (Theorem 4).