Publications of Joseph D Rabinoff :chronological by type listing:
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@article{fds359812,
Author = {Katz, E and Rabinoff, J and Zureick-Brown, D},
Title = {Diophantine and tropical geometry, and uniformity of
rational points on curves},
Journal = {Proceedings of Symposia in Pure Mathematics},
Volume = {97},
Number = {2},
Pages = {231-279},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1090/pspum/097.2/01706},
Abstract = {We describe recent work connecting combinatorics and
tropical/ non-Archimedean geometry to Diophantine geometry,
particularly the uniformity conjectures for rational points
on curves and for torsion packets of curves. The method of
Chabauty–Coleman lies at the heart of this connection, and
we emphasize the clarification that tropical geometry
affords throughout the theory of p-adic integration,
especially to the comparison of analytic continuations of
p-adic integrals and to the analysis of zeros of integrals
on domains admitting monodromy.},
Doi = {10.1090/pspum/097.2/01706},
Key = {fds359812}
}
@article{fds345773,
Author = {Rabinoff, J},
Title = {Higher-level canonical subgroups for p-divisible
groups},
Journal = {Journal of the Institute of Mathematics of
Jussieu},
Volume = {11},
Number = {2},
Pages = {363-419},
Year = {2012},
Month = {April},
url = {http://dx.doi.org/10.1017/S1474748011000132},
Abstract = {Let R be a complete rank-1 valuation ring of mixed
characteristic (0, p), and let K be its field of fractions.
A g-dimensional truncated Barsotti-Tate group G of level n
over R is said to have a level-n canonical subgroup if there
is a K-subgroup of G ⊗ - R K with geometric structure (Z/p
nZ) g consisting of points 'closest to zero'. We give a
non-trivial condition on the Hasse invariant of G that
guarantees the existence of the canonical subgroup,
analogous to a result of Katz and Lubin for elliptic curves.
The bound is independent of the height and dimension of G.
© Cambridge University Press 2011.},
Doi = {10.1017/S1474748011000132},
Key = {fds345773}
}
@article{fds345775,
Author = {Rabinoff, J},
Title = {Hybrid grids and the Homing Robot},
Journal = {Discrete Applied Mathematics},
Volume = {140},
Number = {1-3},
Pages = {155-168},
Year = {2004},
Month = {May},
url = {http://dx.doi.org/10.1016/j.dam.2003.04.001},
Abstract = {In their paper (Inform. Process. Lett. 77 (2001) 261),
Wongngamnit and Angluin introduced a memory-efficient robot,
called the Homing Robot, which localizes in an occupancy
grid. We present a more general class of grids called hybrid
grids, and establish the least upper bound for the number of
moves the robot takes to localize. We also state analogous
results for a hexagonal tiling. © 2003 Elsevier B.V. All
rights reserved.},
Doi = {10.1016/j.dam.2003.04.001},
Key = {fds345775}
}
@article{fds345770,
Author = {Amini, O and Baker, M and Brugallé, E and Rabinoff,
J},
Title = {Lifting harmonic morphisms i: Metrized complexes and
berkovich skeleta},
Journal = {Research in Mathematical Sciences},
Volume = {2},
Number = {1},
Year = {2015},
Month = {December},
url = {http://dx.doi.org/10.1186/s40687-014-0019-0},
Abstract = {Let K be an algebraically closed, complete non-Archimedean
field. The purpose of this paper is to carefully study the
extent to which finite morphisms of algebraic K-curves are
controlled by certain combinatorial objects, called skeleta.
A skeleton is a metric graph embedded in the Berkovich
analytification of X. A skeleton has the natural structure
of a metrized complex of curves. We prove that a finite
morphism of K-curves gives rise to a finite harmonic
morphism of a suitable choice of skeleta. We use this to
give analytic proofs of stronger ‘skeletonized’ versions
of some foundational results of Liu-Lorenzini, Coleman, and
Liu on simultaneous semistable reduction of curves. We then
consider the inverse problem of lifting finite harmonic
morphisms of metrized complexes to morphisms of curves over
K. We prove that every tamely ramified finite harmonic
morphism of Ω-metrized complexes of k-curves lifts to a
finite morphism of K-curves. If in addition the ramification
points are marked, we obtain a complete classification of
all such lifts along with their automorphisms. This
generalizes and provides new analytic proofs of earlier
results of Saïdi and Wewers. As an application, we discuss
the relationship between harmonic morphisms of metric graphs
and induced maps between component groups of Néron models,
providing a negative answer to a question of Ribet motivated
by number theory. This article is the first in a series of
two. The second article contains several applications of our
lifting results to questions about lifting morphisms of
tropical curves.},
Doi = {10.1186/s40687-014-0019-0},
Key = {fds345770}
}
@article{fds345772,
Author = {Amini, O and Baker, M and Brugallé, E and Rabinoff,
J},
Title = {Lifting harmonic morphisms ii: Tropical curves and metrized
complexes},
Journal = {Algebra and Number Theory},
Volume = {9},
Number = {2},
Pages = {267-315},
Year = {2015},
Month = {January},
url = {http://dx.doi.org/10.2140/ant.2015.9.267},
Abstract = {We prove several lifting theorems for morphisms of tropical
curves. We interpret the obstruction to lifting a finite
harmonic morphism of augmented metric graphs to a morphism
of algebraic curves as the nonvanishing of certain Hurwitz
numbers, and we give various conditions under which this
obstruction does vanish. In particular, we show that any
finite harmonic morphism of (nonaugmented) metric graphs
lifts. We also give various applications of these results.
For example, we show that linear equivalence of divisors on
a tropical curve C coincides with the equivalence relation
generated by declaring that the fibers of every finite
harmonic morphism from C to the tropical projective line are
equivalent. We study liftability of metrized complexes
equipped with a finite group action, and use this to
classify all augmented metric graphs arising as the
tropicalization of a hyperelliptic curve. We prove that
there exists a d-gonal tropical curve that does not lift to
a d-gonal algebraic curve. This article is the second in a
series of two. Throughout this paper, unless explicitly
stated otherwise, K denotes a complete algebraically closed
nonarchimedean field with nontrivial valuation val V K →
RU{∞}. Its valuation ring is denoted R, its maximal ideal
ismR, and the residue field is k = R/mR. We denote the value
group of K by ˄ = val.(KX) С R.},
Doi = {10.2140/ant.2015.9.267},
Key = {fds345772}
}
@article{fds345764,
Author = {Foster, T and Rabinoff, J and Shokrieh, F and Soto,
A},
Title = {Non-Archimedean and tropical theta functions},
Journal = {Mathematische Annalen},
Volume = {372},
Number = {3-4},
Pages = {891-914},
Year = {2018},
Month = {December},
url = {http://dx.doi.org/10.1007/s00208-018-1646-3},
Abstract = {We define a tropicalization procedure for theta functions on
abelian varieties over a non-Archimedean field. We show that
the tropicalization of a non-Archimedean theta function is a
tropical theta function, and that the tropicalization of a
non-Archimedean Riemann theta function is a tropical Riemann
theta function, up to scaling and an additive constant. We
apply these results to the construction of rational
functions with prescribed behavior on the skeleton of a
principally polarized abelian variety. We work with the
Raynaud–Bosch–Lütkebohmert theory of non-Archimedean
theta functions for abelian varieties with semi-abelian
reduction.},
Doi = {10.1007/s00208-018-1646-3},
Key = {fds345764}
}
@article{fds345768,
Author = {Baker, M and Payne, S and Rabinoff, J},
Title = {Nonarchimedean geometry, tropicalization, and metrics on
curves},
Journal = {Algebraic Geometry},
Volume = {3},
Number = {1},
Pages = {63-105},
Year = {2016},
Month = {January},
url = {http://dx.doi.org/10.14231/AG-2016-004},
Abstract = {We develop a number of general techniques for comparing
analytifications and tropicalizations of algebraic
varieties. Our basic results include a projection formula
for tropical multiplicities and a generalization of the
Sturmfels-Tevelev multiplicity formula in tropical
elimination theory to the case of a nontrivial valuation.
For curves, we explore in detail the relationship between
skeletal metrics and lattice lengths on tropicalizations and
show that the maps from the analytification of a curve to
the tropicalizations of its toric embeddings stabilize to
isometries on finite subgraphs. Other applications include
generalizations of Speyer's well-spacedness condition and
the Katz- Markwig-Markwig results on tropical
j-invariants.},
Doi = {10.14231/AG-2016-004},
Key = {fds345768}
}
@article{fds345766,
Author = {Papikian, M and Rabinoff, J},
Title = {Optimal quotients of Jacobians with toric reduction and
component groups},
Journal = {Canadian Journal of Mathematics},
Volume = {68},
Number = {6},
Pages = {1362-1381},
Year = {2016},
Month = {December},
url = {http://dx.doi.org/10.4153/CJM-2016-009-9},
Abstract = {Let J be a Jacobian variety with toric reduction over a
local field K. Let J → E be an optimal quotient defined
over K, where E is an elliptic curve. We give examples in
which the functorially induced map φJ → φE on component
groups of the Neron models is not surjective. This answers a
question of Ribet and Takahashi. We also give various
criteria under which φJ → φE E is surjective and discuss
when these criteria hold for the Jacobians of modular
curves.},
Doi = {10.4153/CJM-2016-009-9},
Key = {fds345766}
}
@article{fds345767,
Author = {Gubler, W and Rabinoff, J and Werner, A},
Title = {Skeletons and tropicalizations},
Journal = {Advances in Mathematics},
Volume = {294},
Pages = {150-215},
Year = {2016},
Month = {May},
url = {http://dx.doi.org/10.1016/j.aim.2016.02.022},
Abstract = {Let K be a complete, algebraically closed non-archimedean
field with ring of integers K∙ and let X be a K-variety.
We associate to the data of a strictly semistable K∙-model
X of X plus a suitable horizontal divisor H a skeleton
S(X,H) in the analytification of X. This generalizes
Berkovich's original construction by admitting unbounded
faces in the directions of the components of H. It also
generalizes constructions by Tyomkin and
Baker-Payne-Rabinoff from curves to higher dimensions. Every
such skeleton has an integral polyhedral structure. We show
that the valuation of a non-zero rational function is
piecewise linear on S(X,H). For such functions we define
slopes along codimension one faces and prove a slope formula
expressing a balancing condition on the skeleton. Moreover,
we obtain a multiplicity formula for skeletons and
tropicalizations in the spirit of a well-known result by
Sturmfels-Tevelev. We show a faithful tropicalization result
saying roughly that every skeleton can be seen in a suitable
tropicalization. We also prove a general result about
existence and uniqueness of a continuous section to the
tropicalization map on the locus of tropical multiplicity
one.},
Doi = {10.1016/j.aim.2016.02.022},
Key = {fds345767}
}
@article{fds345771,
Author = {Baker, M and Rabinoff, J},
Title = {The skeleton of the jacobian, the jacobian of the skeleton,
and lifting meromorphic functions from tropical to algebraic
curves},
Journal = {International Mathematics Research Notices},
Volume = {2015},
Number = {16},
Pages = {7436-7472},
Year = {2015},
Month = {January},
url = {http://dx.doi.org/10.1093/imrn/rnu168},
Abstract = {Let K be an algebraically closed field which is complete
with respect to a nontrivial, non-Archimedean valuation and
let be its value group. Given a smooth, proper, connected
K-curve X and a skeleton of the Berkovich analytification
Xan, there are two natural real tori which one can consider:
the tropical Jacobian Jac() and the skeleton of the
Berkovich analytification Jac(X)an. We show that the
skeleton of the Jacobian is canonically isomorphic to the
Jacobian of the skeleton as principally polarized tropical
abelian varieties. In addition, we show that the
tropicalization of a classical Abel-Jacobi map is a tropical
Abel-Jacobi map. As a consequence of these results, we
deduce that-rational principal divisors on, in the sense of
tropical geometry, are exactly the retractions of principal
divisors on X. We actually prove a more precise result which
says that, although zeros and poles of divisors can cancel
under the retraction map, in order to lift a-rational
principal divisor on to a principal divisor on X it is never
necessary to add more than g extra zeros and g extra poles.
Our results imply that a continuous function F: R is the
restriction to of.log | f | for some nonzero meromorphic
function f on X if and only if F is a-rational tropical
meromorphic function, and we use this fact to prove that
there is a rational map f: X → P3 whose tropicalization,
when restricted to, is an isometry onto its
image.},
Doi = {10.1093/imrn/rnu168},
Key = {fds345771}
}
@article{fds345763,
Author = {Dupuy, T and Katz, E and Rabinoff, J and Zureick-Brown,
D},
Title = {Total p-differentials on schemes over Z/p2},
Journal = {Journal of Algebra},
Volume = {524},
Pages = {110-123},
Year = {2019},
Month = {April},
url = {http://dx.doi.org/10.1016/j.jalgebra.2019.01.003},
Abstract = {For a scheme X defined over the length 2 p-typical Witt
vectors W2(k) of a characteristic p field, we introduce
total p-differentials which interpolate between
Frobenius-twisted differentials and Buium's p-differentials.
They form a sheaf over the reduction X0, and behave as if
they were the sheaf of differentials of X over a deeper base
below W2(k). This allows us to construct the analogues of
Gauss–Manin connections and Kodaira–Spencer classes as
in the Katz–Oda formalism. We make connections to
Frobenius lifts, Borger–Weiland's biring formalism, and
Deligne–Illusie classes.},
Doi = {10.1016/j.jalgebra.2019.01.003},
Key = {fds345763}
}
@article{fds345774,
Author = {Rabinoff, J},
Title = {Tropical analytic geometry, Newton polygons, and tropical
intersections},
Journal = {Advances in Mathematics},
Volume = {229},
Number = {6},
Pages = {3192-3255},
Year = {2012},
Month = {April},
url = {http://dx.doi.org/10.1016/j.aim.2012.02.003},
Abstract = {In this paper we use the connections between tropical
algebraic geometry and rigid-analytic geometry in order to
prove two main results. We use tropical methods to prove a
theorem about the Newton polygon for convergent power series
in several variables: if f 1, ..., f n are n convergent
power series in n variables with coefficients in a
non-Archimedean field K, we give a formula for the
valuations and multiplicities of the common zeros of f 1,
..., f n. We use rigid-analytic methods to show that stable
complete intersections of tropical hypersurfaces compute
algebraic multiplicities even when the intersection is not
tropically proper. These results are naturally formulated
and proved using the theory of tropicalizations of
rigid-analytic spaces, as introduced by Einsiedler,
Kapranov, and Lind (2006) [14] and Gubler (2007) [20]. We
have written this paper to be as readable as possible both
to tropical and arithmetic geometers. © 2012 Elsevier
Inc..},
Doi = {10.1016/j.aim.2012.02.003},
Key = {fds345774}
}
@article{fds345765,
Author = {Gubler, W and Rabinoff, J and Werner, A},
Title = {Tropical skeletons},
Journal = {Annales de l'Institut Fourier},
Volume = {67},
Number = {5},
Pages = {1905-1961},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.5802/aif.3125},
Abstract = {In this paper, we study the interplay between tropical and
analytic geometry for closed subschemes of toric varieties.
Let K be a complete non-Archimedean field, and let X be a
closed subscheme of a toric variety over K. We define the
tropical skeleton of X as the subset of the associated
Berkovich space Xan which collects all Shilov boundary
points in the fibers of the Kajiwara-Payne tropicalization
map. We develop polyhedral criteria for limit points to
belong to the tropical skeleton, and for the tropical
skeleton to be closed. We apply the limit point criteria to
the question of continuity of the canonical section of the
tropicalization map on the multiplicity-one locus. This map
is known to be continuous on all torus orbits; we prove
criteria for continuity when crossing torus orbits. When X
is schön and defined over a discretely valued field, we
show that the tropical skeleton coincides with a skeleton of
a strictly semistable pair, and is naturally isomorphic to
the parameterizing complex of Helm-Katz.},
Doi = {10.5802/aif.3125},
Key = {fds345765}
}
@article{fds345769,
Author = {Katz, E and Rabinoff, J and Zureick-Brown, D},
Title = {Uniform bounds for the number of rational points on curves
of small mordell-weil rank},
Journal = {Duke Mathematical Journal},
Volume = {165},
Number = {16},
Pages = {3189-3240},
Year = {2016},
Month = {January},
url = {http://dx.doi.org/10.1215/00127094-3673558},
Abstract = {Let X be a curve of genus g ≥ 2 over a number field F of
degree d = [F : Q]. The conjectural existence of a uniform
bound N (g, d) on the number #X(F) of F-rational points of X
is an outstanding open problem in arithmetic geometry, known
by the work of Caporaso, Harris, and Mazur to follow from
the Bombieri-Lang conjecture. A related conjecture posits
the existence of a uniform bound Ntors,†(g, d) on the
number of geometric torsion points of the Jacobian J of X
which lie on the image of X under an Abel-Jacobi map. For
fixed X, the finiteness of this quantity is the
Manin-Mumford conjecture, which was proved by Raynaud. We
give an explicit uniform bound on #X(F) when X has
Mordell-Weil rank r ≤ g 3. This generalizes recent work of
Stoll on uniform bounds for hyperelliptic curves of small
rank to arbitrary curves. Using the same techniques, we give
an explicit, unconditional uniform bound on the number of
F-rational torsion points of J lying on the image of X under
an Abel-Jacobi map. We also give an explicit uniform bound
on the number of geometric torsion points of J lying on X
when the reduction type of X is highly degenerate. Our
methods combine Chabauty-Coleman's p-adic integration,
non-Archimedean potential theory on Berkovich curves, and
the theory of linear systems and divisors on metric
graphs.},
Doi = {10.1215/00127094-3673558},
Key = {fds345769}
}
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Mathematics Department