Good recursive towers over prime fields exist

A survey on recursive towers and Ihara's constant

“…Then considering differentials, we see that −x −q j i dx i = x q k −2 i−1 dx i−1 . This and equation (11) immediately imply that…”

“…Using this factor, one exactly recovers the first Garcia-Stichtenoth tower. Optimizing the bound from equation (11), one obtains equation (10).…”

“…While with the first two Garcia-Stichtenoth towers, the equations were not found using modular theory and only later a modular explanation was found, in [7], simultaneously, the equations were introduced as well as motivated from a modular point of view. In this subsection, we would like to briefly sketch this modular interpretation of equation (11). In order to do this, we need to fix a few notations:…”

“…Note that the supersingular Drinfeld module −τ m + 1 gives rise to the splitting locus of the tower H, just like the supersingular elliptic curves give rise to the many rational points on the classical modular curves X 0 (N ). More is said in [7] about equation (11) and related equations. In [23], the key-ingredients from [7,1] were worked out further and used to give a precise modular description for each of the function fields in tower H. Considering Drinfeld modules as in D m,j , but in much greater generality, was used to obtain curves with many rational points over F q m in [36].…”

“…Example 6.1. Consider the tower satisfying the recursion from equation (11). Then considering differentials, we see that −x −q j i dx i = x q k −2 i−1 dx i−1 .…”