On tame towers over finite fields

Abstract

We discuss the asymptotic behaviour of the genus and the number of rational places in towers of function fields over a finite field.Both authors were partially supported by GMD/CNPq and the first author also by PRONEX K41.96.0883.00 (Brazil).Brought to you by | The University of York Authenticated Download Date | 7/7/15 8:01 AM 2 ! .The proof of Property (1) is due to H. G. Rü ck; it is given in an appendix of this paper. It was noticed by M. Zieve that Property (1) also follows from Property (2) and the fact that the roots of HðX Þ are in F p 2 . Garcia, Stichtenoth and Rü ck, Tame towers over finite fields 54 Brought to you by | The University of York Authenticated Download Date | 7/7/15 8:01 AM

On the tensor rank of the multiplication in the finite fields

“…Let us note that on the one hand this bound improves the bound μ p 2 (n) 2n(1 + 2 p−3 ) with p 5 found by Ballet-Chaumine in [7] from a tower of Garcia-Stichtenoth-Rück [13], for the case where q = p; on the other hand, this bound significantly improves the general case where q = p m with m 1. Corollary 3.1.…”

“…But let us note that in the cases of the extensions of an arbitrary finite field F q where q = p 5 and q = p m > 16 with m an odd integer, the upper bounds and asymptotic upper bounds obtained in [7] and [4][5][6] are better than those obtained in Corollaries 3.1 and 3.2. This is due to the use of the descent over F q of the definition field of the towers of Garcia-Stichtenoth [12] and Garcia-Stichtenoth-Rück [13] defined over F q 2 . In fact, we can improve these bounds by using Theorem 2.1 with the towers of Garcia-Stichtenoth [12] and Garcia-Stichtenoth-Rück [13] defined over F q 2 because we are able to descend the definition field of these towers over F q as in [4][5][6][7].…”

“…This is due to the use of the descent over F q of the definition field of the towers of Garcia-Stichtenoth [12] and Garcia-Stichtenoth-Rück [13] defined over F q 2 . In fact, we can improve these bounds by using Theorem 2.1 with the towers of Garcia-Stichtenoth [12] and Garcia-Stichtenoth-Rück [13] defined over F q 2 because we are able to descend the definition field of these towers over F q as in [4][5][6][7]. However, it seems more interesting firstly to try descending the definition field of the modular curves used in this paper for the case of an arbitrary finite field F q , which we are not able to do at present.…”

Good towers of function fields

“…Generally, tame towers have the advantage that the genus computation is simple. In [GSR03], by studying the asymptotic behaviour of the number of rational places in tame towers, Garcia, Stichtenoth and Rück produced several good towers of Fermat type and of quadratic extensions. In [BB05] Beelen and Bouw explained the optimal tower in [GSR03] by considering the Picard-Fuchs differential equations in characteristic p and applied their study to towers of modular curves to find new asymptotically good towers.…”

“…class field towers (see among others [Ser83,NX01]), 2. modular towers (see among others [Iha81,Elk98,Elk01,TVZ82]), and 3. explicit towers (see among others [GS95,GS96b,GSR03,BGS05b]). …”