Effective bounds on class number and estimation for any step of towers of algebraic function fields over finite fields (Q2802992)

Let \(F=F/k\) be an algebraic function field of genus \(g=g(F)\geq 2\) defined over the finite field \(k={\mathbb F}_q\) of order \(q\), let \(h=h(F)\) denote the class number of \(F\). The goal of this paper is twofold: (i) To improve bounds on \(h\); (ii) Estimate the class number of all steps of certain towers of algebraic function fields over finite fields. For (i) the starting point is the exact formula \(S(F)=hR(F)\, (*)\) stated in [\textit{G. Lachaud} and \textit{M. Martin-Deschamps}, Acta Arith. 56, No. 4, 329--340 (1990; Zbl 0727.14019)], where \(S(F)\) depends on the numbers \(A_n(F)\) of effective divisors of degree \(n\) of \(F\) with \(0\leq n\leq g-1\) while \(R(F)\) depends on the reciprocal roots \((\pi,\bar\pi_i)\) of the enumerator of the zeta-function of \(F\). In general, one does not expect to compute all the data involved in \((*)\) and in fact one is only able to deduce accurate estimates for \(h\) such as \(h\geq q^{g-1}(q-1)^2/(q+1)(g+1)\) (loc. cit.). In this paper quite involved formulas give bounds on \(h\) which in some cases improve on among all the known bounds.NEWLINENEWLINEA tower \(\mathcal F\) over \(k\) is a sequence, \((F_i)_{i\geq 0}\) of function fields over \(k\) such that \(k\) is algebraically closed in \(F_i\) for each \(i\), \(F_i\subsetneq F_{i+1}\) with \(F_{i+1}|F_i\) finite and separable, and \(g(F_i)\to \infty\) as \(i\to\infty\). Under the hypothesis \(\beta_m(\mathcal F):=\lim{i\to\infty} B_m(F_i)/g(F_i)>0\), for some \(m\), where \(B_m(F_i)\) is the number of places of degree \(m\) of \(F_i|k\), the authors study the following parameters of each step \(F_i|k\) of \(\mathcal F\): the genus, the number of places of certain degree, and the class number. The results are illustrated by taking as a building block the tower in [\textit{A. Garcia} and \textit{H. Stichtenoth}, Invent. Math. 121, No. 1, 211--222 (1995; Zbl 0822.11078)], or the so called recursively defined towers in the sense that \(F_{i+1}=F_i(x_{i+1})\) with \(f\in k[X,Y]\), \(f(x_i,x_{i+1}=0\).