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Space filling curves over finite fields. (English) Zbl 1016.11022

Math. Res. Lett. 6, No. 5-6, 613-624 (1999); Corrections 8, 689-691 (2001).
From the text: In this note, we construct curves over finite fields which have, in a certain sense, a “lot” of points, and give some applications to the zeta-functions of curves and Abelian varieties over finite fields. In fact, we found the basic construction of curves \(\mathbb{Z}^n\) which go through every rational point, as part of an unsuccessful attempt to find curves of growing genus over a fixed finite field with lots of points in the sense of the Drinfeld-Vladut bound.
Theorem. Let \(k\) be a finite field, \(X/k\) smooth and quasi-projective and geometrically connected, of dimension \(n\geq 1\). Let \(E/k\) be a finite extension. There exists a smooth, geometrically connected curve \(C_0/k\), and an immersion \(\pi:C_0\to X\) which is bijective on \(E\)-valued points.
(From the correction:) “O. Gabber has kindly pointed out to me that the proof of Lemma 5 of the original paper is wrong. The error occurs in the last paragraph of the proof. The effect of correcting this error is that in Lemmas 4, 5, and 6, and in Corollary 7, what is asserted to hold for \(r\) sufficiently large holds only for \(r\) sufficiently large and sufficiently divisible. Indeed, Gabber has constructed examples to show that Lemma 6 and Corollary 7 can be false without this extra proviso. In the corrections, we also modify the statement of Lemma 5, so that its new, weaker conclusion applies in a more general setting”.

MSC:

11G20 Curves over finite and local fields
14G15 Finite ground fields in algebraic geometry
11G10 Abelian varieties of dimension \(> 1\)
11G25 Varieties over finite and local fields
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