A counterexample to Perret’s conjecture on infinite class field towers for global function fields. (English) Zbl 0943.11052
Let \(N_q(g)\) denote the maximum number of \(\mathbb{F}_q\)-rational points on a smooth, absolutely irreducible curve of genus \(g\) defined over \(\mathbb{F}_q\). Of basic importance in the asymptotic theory of such curves (or function fields) is
\[
A(q)=\limsup_{g\rightarrow\infty} {N_q(g)\over g}.
\]
M. Perret [J. Number Theory 38, 300-322 (1991; Zbl 0741.11044)] described a method to obtain better lower bounds for \(A(q)\) than a bound due to Serre, but his method depended on a conjecture concerning whether certain towers of function fields were in fact infinite towers. The authors show that Perret’s conjecture fails by giving a counterexample based on the theory of narrow ray class fields.
Reviewer: Robert F.Lax (Baton Rouge)
MSC:
11R58 | Arithmetic theory of algebraic function fields |
11R37 | Class field theory |
11G20 | Curves over finite and local fields |
14H05 | Algebraic functions and function fields in algebraic geometry |