(Both the published version and the arXiv version are available online.)
We construct an infinite sequence of curves C1, C2, ... over Fq,
together with nonconstant separable rational
maps hi : Ci→Ci-1,
such that the following hold:
(1): The genus gi of Ci
satisfies gi→∞ as
i→∞;
(2): We have gi ≤ c di,
where c is an absolute constant and di
is the degree of the composite map Ci→C1;
(3): For each i and each closed point P
on Ci, there is some j>i
for which P ramifies in Cj→Ci.
We also construct a sequence of curves and maps satisfying (1), (3), and
(2'): The number Ni of
Fq-rational points on Ci
is greater than an absolute positive constant times di.
These two sequences disprove conjectures posed by Stichtenoth in 2003.
The motivation for Stichtenoth's conjectures is as follows.
According to Weil's
Riemann hypothesis for curves, a genus-g curve over Fq has at most O(g+1) points, where the
implied constant depends on q. For fixed q,
there are only a handful of known methods for constructing towers of curves
over Fq which satisfy both (1) and
(4): Ni/gi does not approach zero
as i→∞.
It is straightforward to show that (gi-1)/di is nondecreasing while
Ni/di is nonincreasing.
Thus, if (1) holds then (4) is equivalent to the combination of (2) and (2').
Stichtenoth noted that, in all known examples satisfying (1) and (2),
only finitely many closed points of C1
ramify in the tower; but our example satisfying (3) fails this spectacularly,
as every closed point of C1 ramifies.
Stichtenoth also noted that, in all known examples satisfying (1) and (2'),
there is some Fq-rational point of some
Ci which splits completely in every
Cj→Ci with j>i;
but again, our example satisfying (3) cannot have this property.
Our construction suggests that there may be further methods of constructing towers of curves satisfying (1) and (4), so it is not clear what properties one should expect such towers to have.
Michael Zieve: home page publication list