Publications of Joseph D Rabinoff

%% Papers Published   
   Author = {Dupuy, T and Katz, E and Rabinoff, J and Zureick-Brown,
   Title = {Total p-differentials on schemes over Z/p2},
   Journal = {Journal of Algebra},
   Volume = {524},
   Pages = {110-123},
   Year = {2019},
   Month = {April},
   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}

   Author = {Foster, T and Rabinoff, J and Shokrieh, F and Soto,
   Title = {Non-Archimedean and tropical theta functions},
   Journal = {Mathematische Annalen},
   Volume = {372},
   Number = {3-4},
   Pages = {891-914},
   Year = {2018},
   Month = {December},
   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
   Doi = {10.1007/s00208-018-1646-3},
   Key = {fds345764}

   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},
   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}

   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},
   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
   Doi = {10.4153/CJM-2016-009-9},
   Key = {fds345766}

   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},
   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
   Doi = {10.1016/j.aim.2016.02.022},
   Key = {fds345767}

   Author = {Baker, M and Payne, S and Rabinoff, J},
   Title = {Nonarchimedean geometry, tropicalization, and metrics on
   Journal = {Algebraic Geometry},
   Volume = {3},
   Number = {1},
   Pages = {63-105},
   Year = {2016},
   Month = {January},
   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
   Doi = {10.14231/AG-2016-004},
   Key = {fds345768}

   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},
   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
   Doi = {10.1215/00127094-3673558},
   Key = {fds345769}

   Author = {Amini, O and Baker, M and Brugallé, E and Rabinoff,
   Title = {Lifting harmonic morphisms i: Metrized complexes and
             berkovich skeleta},
   Journal = {Research in Mathematical Sciences},
   Volume = {2},
   Number = {1},
   Year = {2015},
   Month = {December},
   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}

   Author = {Amini, O and Baker, M and Brugallé, E and Rabinoff,
   Title = {Lifting harmonic morphisms ii: Tropical curves and metrized
   Journal = {Algebra & Number Theory},
   Volume = {9},
   Number = {2},
   Pages = {267-315},
   Year = {2015},
   Month = {January},
   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}

   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
   Journal = {International Mathematics Research Notices},
   Volume = {2015},
   Number = {16},
   Pages = {7436-7472},
   Year = {2015},
   Month = {January},
   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
   Doi = {10.1093/imrn/rnu168},
   Key = {fds345771}

   Author = {Rabinoff, J},
   Title = {Tropical analytic geometry, Newton polygons, and tropical
   Journal = {Advances in Mathematics},
   Volume = {229},
   Number = {6},
   Pages = {3192-3255},
   Year = {2012},
   Month = {April},
   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
   Doi = {10.1016/j.aim.2012.02.003},
   Key = {fds345774}

   Author = {Rabinoff, J},
   Title = {Higher-level canonical subgroups for p-divisible
   Journal = {Journal of the Institute of Mathematics of
   Volume = {11},
   Number = {2},
   Pages = {363-419},
   Year = {2012},
   Month = {April},
   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}

   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},
   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}