On Drinfeld modular curves with many rational points over finite fields
Given a finite field \({\mathbb F}_q\) and a curve \(X/{\mathbb F}_q\) (where a ``curve'' is a smooth projective geometrically connected algebraic curve over \({\mathbb F}_q\)), the number \(N = \# X(F_q)\) of rational points is roughly \(q+1\), where the difference \(|N-(q+1)|\) is subject to various bounds due to A. Weil and others. These depend on \(q\) and the genus \(g\) of \(X\), and are in general not sharp. For several reasons stemming from combinatorics, but also from genuine arithmetical interest, one would like to know the precise bounds. In particular, there is a lively contest (the results of which are documented and updated in \url{http://www.wins.uva.nl/~geer}) in constructing curves that, for \((q,g)\) given, realize the maximum number of rational points. These constructions are arithmetical in nature, and in many cases make use of modular curves of some sort: classical elliptic, Shimura, or Drinfeld modular curves. The reason is twofold: modular curves are well understood and their points easy to describe, and the underlying arithmetic produces enough of the symmetry required. Pursuing the strategy described below, the author of the present paper is able to find a series of new record curves for \(q = 4\) and 9. The main idea is as follows: Take a Drinfeld modular curve \(X_0(n)\) of Hecke type, reduced at a suitable place \((p)\) of degree 1 such that the reduction has a large number of rational points over the residue field extension \(L\) of degree 2 (which as an abstract field is isomorphic with \({\mathbb F}_{q^2}\)). The relevant invariants of \(Y:= X_0(n)/L\) (genus, number of supersingular points, and of cusps) are given through complicated but explicit formulas, which involve only the splitting type of \(n\). Then choose an Atkin-Lehner involution \(w\) (which corresponds to a divisor \(n'\) of \(n\)) with the number of fixed points as large as possible. (The last condition depends on ``good'' choices of \(n\) and \(n'\).) Now the number \(N\) of rational points of \(X:= w\setminus Y\) is at least \(1/2\) of the corresponding number of \(Y\), whereas the genus \(g(X)\) will be less than \(\frac 12g(Y)\) in view of the Hurwitz formula. Thus \(X\) has a ``better'' ratio \(N/g\) than \(Y\), and is a candidate for a record curve. These considerations are especially difficult in the case of \(\text{char}({\mathbb F}_q) = 2\), where complications from two sides appear: the determination of the number of fixed points of \(w\) (which involves calculations in a certain quaternion algebra) and the determination of higher ramification at the fixed points.
scientific article; zbMATH DE number 1891456