学术报告（胡创强 9.2）

摘要： Drinfeld modular curves are used to construct sequences of curves with many rational points over any non-prime field. The specific structure of Drinfeld modular curve plays an important role in the field of coding. Indeed, constructing a linear error correction code with a sufficiently long code length is a fundamental problem in coding theory. In the 1980s, V.D.Goppa used the algebraic curve over finite fields to construct a special linear error correction code, which is now called the algebraic geometry code. The parameters (code length, dimension, minimum Hamming distance) of this type of linear code mainly depend on the geometric properties of the corresponding algebraic curve, namely, the number of rational points and genus. It is proved theoretically that there is a family of asymptotically optimal linear error-correction codes whose parameters attain the Drinfeld–Vladut bound. Surprisingly, in 1982, Tsfasman, Vl˘adut¸, and Zink proved the existence of an asymptotically optimal long linear code with relative parameters which exceeds the Gilbert–Varshamov bound within a certain range. This work shows a vital link between Ihara’s quantity and the realm of coding theory. In practical applications, we need to know the explicit construction of such algebraic geometry codes, and it boils down to finding a family of asymptotically good function field sequences (called tower) which are measured by the Ihara’s constant. In 2000, based on his procedure for constructing explicit towers of modular curves, Elkies deduced explicit equations of rank-2 Drinfeld modular curves which coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. In 2015, Bassa, Beelen, Garcia, and Stichtenoth constructed a celebrated (recursive and good) tower (BBGS tower for short) of curves and outlined a modular interpretation of the defining equations. In this talk, we aim to construct a sequence of Drinfeld modular curves which are organized in an elegant manner — an hierarchical topology tree which we call the T-torsion tree. We believe that our novel approach by the T-torsion tree not only promotes the classic torsion sequence structure, but also further integrates the internal connections of different torsion structures.