Good towers of function fields

Abstract

SummaryAlgebraic curves are used in many different areas, including error-correcting codes. In such applications, it is important that the algebraic curve C meets some requirements. The curve must be defined over a finite field F q with q elements, and then the curve also should have many points over this field. There are limits on how many points N (C) an algebraic curve C defined over a finite field can have.

Good families of Drinfeld modular curves

“…Moreover, from [6, V.4,VII.5] we see that if δ is odd and all prime ideals p i occurring in the decomposition of n have even degree, among all the points of x 0 (n) that are lying above a given elliptic point of x(1) there are exactly 2 s that are ramified in the covering x(n)/x 0 (n) (with ramification index q + 1). The latter statement is equivalent to saying that these 2 s points of x 0 (n) have ramification index 1 in x 0 (n)/x (1). Counting the number of points of x 0 (n) lying above the supersingular points of x(1) now is direct and yields the stated lower bound on N 1 (x 0 (n)).…”

“…By computing the precise formula for the genus and the number of rational points on reductions of F q [T ]-Drinfeld modular curves X 0 (n), Gekeler [8] showed that for a series (n k ) k∈N of polynomials of A coprime with an irreducible polynomial P ∈ A, and whose degrees tend to infinity, the family of Drinfeld modular curves X 0 (n k )/F P attains the Drinfeld-Vladut bound when considered over F (2) P . In case n k = T k and P = T − 1, explicit equations for the modular curves X 0 (T k ) were given in [2], while some more general examples (including defining equations in generic A-characteristic 0) were given in [1]. For A = F q [T ] and δ = 1 the situation has therefore to a large extent been investigated both theoretically and explicitly.…”

“…We now turn our attention again to the Galois cover x(n)/x (1). It was shown in [6] that the only ramification in this cover occurs above the so-called elliptic points (with ramification index q + 1) and the cusps of x(1).…”

“…Moreover, as mentioned before, the number of cusps on x(1) equals δh(F ). The elliptic points were studied in [6, V.4,VII.5]: The number of elliptic points on x(1) is 0 if δ is even and P (−1) if δ if odd, each with ramification index q + 1 in the cover x(n)/x (1).…”

“…Compared to our approach this means that the "pyramid" in Figure 3 starts at X 0 (p 2 ), but otherwise the recursive description is similar: The points on the curve X 0 (T k ) are identified with points in X 0 (T 2 ) × · · · × X 0 (T 2 ), while each of the component curves X 0 (T 2 ) can be described using a single parameter v i . For more details see [2,1]. 6 An new explicit example of an optimal Drinfeld modular tower…”