%% Papers Published @article{fds345763, Author = {Dupuy, T and Katz, E and Rabinoff, J and Zureick-Brown, D}, Title = {Total p-differentials on schemes over Z/p^{2}}, 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} } @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}, 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{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}, 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{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}, 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}, 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{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}, 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{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}, 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} } @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}, 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 & 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} } @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}, 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{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}, 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{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}, 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}, 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} }