Good recursive towers over prime fields exist
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Mathematische Annalen
Good recursive towers over prime fields exist Alp Bassa1
· Christophe Ritzenthaler2
Received: 2 April 2020 / Revised: 23 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We give a construction and equations for good recursive towers over any finite field Fq with q = 2 and 3. Mathematics Subject Classification 11G20 · 11T71 · 14H25 · 14G05 · 14G15
1 Introduction Let Fq be a finite field with q = p n elements, where p is a prime and n ≥ 1 an integer. A central quantity in the theory of curves over finite fields of large genus is the Ihara constant A(q), which is defined as A(q) := lim sup g→∞
Nq (g) , g
where Nq (g) = max # X (Fq ) with X running over all smooth projective absolutely irreducible curves defined over Fq of genus g(X ) = g > 0. Drinfel’d–Vl˘adu¸t [16] obtained the inequality √ (1.1) A(q) ≤ q − 1, which is still the only known upper bound.
Communicated by Wei Zhang. The authors acknowledge support by the PHC Bosphorus 39652NB-TÜB˙ITAK 117F274 and thank the Nesin Mathematical Village for its inspiring environment.
B
Alp Bassa [email protected] Christophe Ritzenthaler [email protected]
1
Department of Mathematics, Bo˘gaziçi University, Bebek, 34342 Istanbul, Turkey
2
Institut de recherche mathématique de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France
123
A. Bassa, C. Ritzenthaler
When the exponent n is even, Ihara [9] used reductions of Shimura curves to show that equality holds in (1.1). This result was also obtained by Tsfasman–Vl˘adu¸t–Zink [13] for n = 2 and 4 using reductions of elliptic modular curves. For any q, using class field theory, Serre [11] showed that A(q) > c log(q) for some constant c > 0 1 [10, Theorem 5.2.9]). In particular independent of q (one can take for instance c = 96 A(q) > 0 for all q. The exact value of A(q) is however unknown when q is not a square. Garcia and Stichtenoth [6] marked a major turning-point in the theory by introducing the notion of recursive towers T = (X m )m∈N . These towers are described by two morphisms f , g : C0 ⇒ C−1 where C0 and C−1 are curves defined over Fq . This defines recursively (Cm )m≥0 by the fiber product C0
Cm+1 πm
Cm
g
fm
C−1
where f m (P0 , . . . , Pm ) = f (Pm ) and πm (P0 , . . . , Pm+1 ) = (P0 , . . . , Pm ). In other terms one has Cm = {(P0 , . . . , Pm ) ∈ C0m+1 : g(Pi ) = f (Pi−1 ), 1 ≤ i ≤ m}. One then considers the normalization X m of Cm and we will still denote the induced cover by πm : X m+1 → X m . Although the curves X m are smooth, it is not automatic that they are absolutely irreducible or that their genus goes to infinity. If it is so, T = (X m )m∈N with the morphisms πm is called a tower. It is a good tower if the limit of the tower λ(T ) := lim sup
# X m (Fq ) g(X m )
is positive and an optimal tower if it reaches the Drinfel’d–Vl˘adu¸t bound. For n even, Garcia and Stichtenoth exhibited explicit examples of optimal towers. Compared to constructions using class field towers or modular curves, r
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