592 research outputs found
Algorithms for Mumford curves
Mumford showed that Schottky subgroups of give rise to certain
curves, now called Mumford curves, over a non-Archimedean field K. Such curves
are foundational to subjects dealing with non-Archimedean varieties, including
Berkovich theory and tropical geometry. We develop and implement numerical
algorithms for Mumford curves over the field of p-adic numbers. A crucial and
difficult step is finding a good set of generators for a Schottky group, a
problem solved in this paper. This result allows us to design and implement
algorithms for tasks such as: approximating the period matrices of the
Jacobians of Mumford curves; computing the Berkovich skeleta of their
analytifications; and approximating points in canonical embeddings. We also
discuss specific methods and future work for hyperelliptic Mumford curves.Comment: 32 pages, 4 figure
Moduli of Tropical Plane Curves
We study the moduli space of metric graphs that arise from tropical plane
curves. There are far fewer such graphs than tropicalizations of classical
plane curves. For fixed genus , our moduli space is a stacky fan whose cones
are indexed by regular unimodular triangulations of Newton polygons with
interior lattice points. It has dimension unless or .
We compute these spaces explicitly for .Comment: 31 pages, 25 figure
An elliptic curve test of the L-Functions Ratios Conjecture
We compare the L-Function Ratios Conjecture's prediction with number theory
for the family of quadratic twists of a fixed elliptic curve with prime
conductor, and show agreement in the 1-level density up to an error term of
size X^{-(1-sigma)/2} for test functions supported in (-sigma, sigma); this
gives us a power-savings for \sigma<1. This test of the Ratios Conjecture
introduces complications not seen in previous cases (due to the level of the
elliptic curve). Further, the results here are one of the key ingredients in
the companion paper [DHKMS2], where they are used to determine the effective
matrix size for modeling zeros near the central point for this family. The
resulting model beautifully describes the behavior of these low lying zeros for
finite conductors, explaining the data observed by Miller in [Mil3].
A key ingredient in our analysis is a generalization of Jutila's bound for
sums of quadratic characters with the additional restriction that the
fundamental discriminant be congruent to a non-zero square modulo a square-free
integer M. This bound is needed for two purposes. The first is to analyze the
terms in the explicit formula corresponding to characters raised to an odd
power. The second is to determine the main term in the 1-level density of
quadratic twists of a fixed form on GL_n. Such an analysis was performed by
Rubinstein [Rub], who implicitly assumed that Jutila's bound held with the
additional restriction on the fundamental discriminants; in this paper we show
that assumption is justified.Comment: 35 pages, version 1.2. To appear in the Journal of Number Theor
Bitangents of tropical plane quartic curves
We study smooth tropical plane quartic curves and show that they satisfy
certain properties analogous to (but also different from) smooth plane quartics
in algebraic geometry. For example, we show that every such curve admits either
infinitely many or exactly 7 bitangent lines. We also prove that a smooth
tropical plane quartic curve cannot be hyperelliptic.Comment: 13 pages, 9 figures. Minor revisions; accepted for publication in
Mathematische Zeitschrif
Graphs of gonality three
In 2013, Chan classified all metric hyperelliptic graphs, proving that
divisorial gonality and geometric gonality are equivalent in the hyperelliptic
case. We show that such a classification extends to combinatorial graphs of
divisorial gonality three, under certain edge- and vertex-connectivity
assumptions. We also give a construction for graphs of divisorial gonality
three, and provide conditions for determining when a graph is not of divisorial
gonality three.Comment: 19 pages, 13 figures; corrected statements of Theorems 1.2 and 4.1,
as well as material in Section
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