Abstract
Elkies conjectures that all recursively defined asymptotically optimal towers of function fields over finite fields with square cardinality arise from elliptic modular curves, Shimura curves, or Drinfeld modular curves by appropriate reduction. In this paper we review the recursive asymptotically optimal towers constructed so far, discuss the reasons behind Elkies’ conjecture, present numerical evidence of this conjecture, and sketch Elkies’ proof of modularity of the new families.
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