448 research outputs found
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
On nondegeneracy of curves
A curve is called nondegenerate if it can be modeled by a Laurent polynomial
that is nondegenerate with respect to its Newton polytope. We show that up to
genus 4, every curve is nondegenerate. We also prove that the locus of
nondegenerate curves inside the moduli space of curves of fixed genus g > 1 is
min(2g+1,3g-3)-dimensional, except in case g=7 where it is 16-dimensional
Nondegenerate curves of low genus over small finite fields
In a previous paper, we proved that over a finite field of sufficiently
large cardinality, all curves of genus at most 3 over k can be modeled by a
bivariate Laurent polynomial that is nondegenerate with respect to its Newton
polytope. In this paper, we prove that there are exactly two curves of genus at
most 3 over a finite field that are not nondegenerate, one over F_2 and one
over F_3. Both of these curves have remarkable extremal properties concerning
the number of rational points over various extension fields.Comment: 8 pages; uses pstrick
On explicit descent of marked curves and maps
We revisit a statement of Birch that the field of moduli for a marked
three-point ramified cover is a field of definition. Classical criteria due to
D\`ebes and Emsalem can be used to prove this statement in the presence of a
smooth point, and in fact these results imply more generally that a marked
curve descends to its field of moduli. We give a constructive version of their
results, based on an algebraic version of the notion of branches of a morphism
and allowing us to extend the aforementioned results to the wildly ramified
case. Moreover, we give explicit counterexamples for singular curves.Comment: 35 page
Explicit methods for Hilbert modular forms
We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of
Hilbert modular forms over a totally real field. We provide many explicit
examples as well as applications to modularity and Galois representations.Comment: 52 pages, 10 figures, many table
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