Invariants of some algebraic curves related to Drinfeld modular curves
This paper is about properties of Drinfeld modular curves, classifying Drinfeld modules defined on \(\mathbb{F}_q[T]\) (where \(\mathbb{F}_q\) is a finite field), especially formulae for the genera, formulae for the number of cusps, descriptions of function fields\dots.NEWLINENEWLINENEWLINEThe main and important result, which takes place in the long story of the comparison over a finite field between genus and number of rational points of a curve, is the following: let \((N_k)_k\) be a sequence of elements of \(\mathbb{F}_q[T]\) coprime with \(T\), such that \(\text{deg} N_k\) tends to \(\infty\), let \(X_0(N_k)_{/\mathbb{F}_q}\) be the reduction modulo \(T\) of the Drinfeld modular curve \(X_0(N_k)\) (this is a smooth curve over \(\mathbb{F}_q = \mathbb{F}_q[T]/(T))\). Then the sequence \((X_0(N_k)_{/\mathbb{F}_q})_k\) is asymptotically optimal over \(\mathbb{F}_{q^2}\) (viewed as a quadratic extension of \(\mathbb{F}_q[T]/(T))\), meaning that, when \(k\) tends to \(\infty\) NEWLINE\[NEWLINE\frac{\#(X_0(N_k)_{/\mathbb{F}_q}(\mathbb{F}_{q^2}))}{\text{genus of }X_0(N_k)}\to q-1.NEWLINE\]
scientific article; zbMATH DE number 1678871
scientific article