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- W2017219920 abstract "Let P be a monic irreducible polynomial in Fq[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 Fq[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 Fq if and only if the Bernoulli–Carlitz number B((qd−1)/2) 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 : RF. In the case where F=Fq(T, P ), then RF≢0 (mod P). Unfortunately, this is not the case for general F. In Section 3, we show that if RF≢0 (mod P), then Gekeler's conjecture is true for the field F (Theorem 4)." @default.
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- W2017219920 date "2001-04-01" @default.
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- W2017219920 title "On Gekeler's Conjecture for Function Fields" @default.
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- W2017219920 doi "https://doi.org/10.1006/jnth.2000.2587" @default.
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