Annales de l’institut Fourier (AIF)

Tzvetkov, Nikolay

Invariant measures for the defocusing Nonlinear Schrödinger equation

Parkinson, James

Isotropic random walks  on affine buildings

Howe, Everett W.

Lauter, Kristin E.

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L’INSTITUT FOURIER
Everett W. HOWE & Kristin E. LAUTER
Corrigendum to: Improved upper bounds for the number of points on
curves over finite fields
Tome 57, no 3 (2007), p. 1019-1021.
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Ann. Inst. Fourier, Grenoble
57, 3 (2007) 1019-1021
CORRIGENDUM TO:
IMPROVED UPPER BOUNDS FOR THE NUMBER
OF POINTS ON CURVES OVER FINITE FIELDS
Tome 53 (2003), no 6, p. 1677-1737
by Everett W. HOWE & Kristin E. LAUTER
1. Introduction
In Section 7.2 of our paper [3] there is a mistake in an argument about a
standard form for triple covers of elliptic curves in characteristic 3. In this
corrigendum we identify the error and make a corrected statement about
the standard form for such triple covers. The goal of [3, §7] was to show
that two particular polynomials do not occur as Weil polynomials of curves
over a finite field. The error we made invalidates our arguments that these
polynomials do not occur. We have new arguments that can replace the
invalid ones. Sketches of these new arguments can be found in the second
appendix to [2]; in a forthcoming paper we will give the full details of the
new techniques, and use them to further improve some of the upper bounds
in the van der Geer-van der Vlugt tables of curves with many points [1].
2. The error, and a corrected statement
We use the notation and conventions of [3] without further explanation.
Keywords: Curve, rational point, zeta function, Weil bound, Serre bound, Oesterlé
bound.
Math. classification: 11G20, 14G05, 14G10, 14G15.
1020 Everett W. HOWE & Kristin E. LAUTER
Recall that the goal of [3, §7.1] was to find a standard form for triple
covers of elliptic curves over finite fields of characteristic 3. In that section,
we showed that every such triple cover of an elliptic curve E can be written
in the form z3−fz = g, where f and g are functions on E satisfying certain
conditions. Specifically, let us say a pair (f, g) is well-conditioned at a point
P of E if one of the following conditions holds: either
(1) the order ordP g of g at P is not a multiple of 3, or
(2) we have 2 ordP g > 3 ordP f .
We showed in [3, §7.1] that every triple cover of E has a model z3−fz = g
such that f has no poles outside ∞ and no multiple zeros anywhere, and
such that (f, g) is well-conditioned at every finite pole of g. The model could
be made to satisfy the further requirement that (f, g) be well-conditioned
at ∞, unless f is constant and g has a triple pole at ∞.
The error in [3] occurs in §7.2, starting at the second full paragraph
on page 1717. The problem lies in the statement that for all P we have
either 2 ordP g > 3 ordP f or ordP g 6≡ 0 mod 3, except when P = ∞ and
ordP g = −3. In particular, the erroneous statement claims that the model
can be chosen so that (f, g) is well-conditioned at all finite P , not just at
the poles of g. The erroneous statement is in fact true for those finite points
P for which ordP f = 0, because for these points either P is a pole of g or
we have 2 ordP g > 0 = 3 ordP f . However, the statement can fail to hold
for points P for which ordP f = 1.
What is true is that for every P 6= ∞ for which ordP f > 0, there is a
constant cP ∈ k such (f, g + c3P − cP f) is well-conditioned at P . Note that
over k the triple cover z3 − fz = g + c3P − cP f is isomorphic to the triple
cover z3−fz = g. To take account of this change, the final paragraph of [3,
§7.2] should be replaced with the following:
If ordP f is odd, let cP be an element of k such that either
2 ordP (g + c3P − cP f) > 3 ordP f or ordP (g + c3P − cP f) 6≡
0 mod 3. Let gP = g + c3P − cP f . Then the contribution to
the different at P is{
1 if 3 ordP f − 2 ordP gP 6 0;
2 + 3 ordP f − 2 ordP gP if 3 ordP f − 2 ordP gP > 0.
In particular, when ordP f is odd the contribution at P to
the different is odd.
The contribution to the different at P thus depends on f and g in a
more complicated manner than we had thought, and several of the cases
ANNALES DE L’INSTITUT FOURIER
CORRIGENDUM: IMPROVED UPPER BOUNDS FOR CURVES 1021
we consider in §7.3 and §7.4 of [3] cannot be eliminated as easily as we
argued in those sections.
BIBLIOGRAPHY
[1] G. van der Geer & M. van der Vlugt, “Tables of curves with many points”, Math.
Comp. 69 (2000), p. 797-810, Updates at http://www.science.uva.nl/~geer/.
[2] E. W. Howe & K. E. Lauter, “Improved upper bounds for the number of points
on curves over finite fields”, http://arxiv.org/pdf/math.NT/0207101.
[3] E. W. Howe & K. E. Lauter, “Improved upper bounds for the number of points
on curves over finite fields”, Ann. Inst. Fourier (Grenoble) 53 (2003), p. 1677-1737.
Everett W. HOWE
Center for Communications Research,
4320 Westerra Court,
San Diego, CA 92121-1967, USA.
however@alumni.caltech.edu
Kristin E. LAUTER
Microsoft Research
One Microsoft Way,
Redmond, WA 98052, USA.
klauter@microsoft.com
TOME 57 (2007), FASCICULE 3


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