On a tower of Ihara and its limit

A modular interpretation of various cubic towers

“…Given in this form, it is clear that for q = 2, the tower A reduces to the tower in [9]. In [14,6] it was shown that the tower in [5] also can be defined by the reducible equation…”

Good towers of function fields

“…Moreover, Polynomial (5.7) defines the dual tower of Caro-Garcia [CG12] whose ramification was already clarified. With this information we state the ramification structure of Tower C as follows.…”

“…Lemma 5.1 (see [CG12]). The ramification locus of Tower C contains exactly three places of C 0 ; namely (α 0 = −1), (α 0 = 0), and (α 0 = ∞).…”

“…In fact Tower C/F = (C i ) i≥0 and Tower BeGS/F = (B i ) i≥1 of Bezerra, Garcia and Stichtenoth [BGS05b] are essentially the same. More precisely, it is shown in [CG12] that C i = B i for all i ≥ 1. Hence the exact genus of the function fields in Tower C can be given as follow (see [BGS05b]).…”

A new tower with good $p$-rank meeting Zink’s bound

“…In the remainder of the preliminary section, we collect and cite several facts from [5,13,6, 1] on a tower of function fields defined over a cubic field F q 3 . This tower will be useful in later sections for computing the genera of the function fields in our new tower.…”