with E. Klassen and Y. Kopeliovic, Realizing alternating groups as monodromy groups of genus one covers, PAMS 129 (2000), 111–119. Precisely: The full family of 3-branch point covers of genus 1 curves have nonconstant maps to the moduli space of genus 1 curves. Other results supercede this in one way: They produce families of 3-branch point covers of arbitrary genus g>0 whose map to the moduli of curves of genus g is dominant. Yet, present applications could use the explicitness of this case in those higher genus results.
Improvements on this result:
A better result would should show exactly
(*) What is the dimension in terms of r
and n for the image in the
moduli space of the natural map.
A special case of such a result would be
(**) For what r and n do you have full moduli dimension.
One application of Alternating Groups and Moduli space
Lifting Invariants (§ 5.1) is a weaker version of (**):
The production of infinitely many values of (r,n)
for each value of g for which
the map has full moduli dimension. With such a result we can prove the
existence of "automorphic functions"
on the spaces labeled H+(An,
C3n)abs,rd (the other
component H-(An,
C3n) does
not support such a function) in a meaningful way. Functions
canonically defined by the moduli problem, that are automorphic for
certain subgroups of the Symplectic group defined by subspaces of the
Siegel Upper-half plane.
Those words, however, are not the full value of the functions. Their
existence and intrinsic properties are the serious point. This
application was based on M. Artibani and P. Pirola, Algebraic functions
with even monodromy, PAMS and J.-P. Serre,
Revêtements a ramification impaire et
thêta-caractéristiques, C. R. Acad. Sci. Paris 311 (1990), 547–552.
The results, however, for g
> 1, are far from the best possible, so the proof in the case g=1 still has value, and
application to moduli spaces of genus 1 curves that are upper
half-plane quotients, yet aren't modular curves. The case r=4 and g=1 (the space labeled H+(A4,
C34)abs,rd)
is explained in great detail in §6.3 of the paper attached to lum-fried0611594pap.html.