- Jumps in the height of the Ceresa cycle. (with Robin de Jong)
We study the jumps in the archimedean height of the Ceresa cycle, as introduced by R. Hain in his work on normal functions on moduli spaces of curves, and as further analyzed by P. Brosnan and G. Pearlstein in terms of asymptotic Hodge theory. Our work is based on a study of the asymptotic behavior of the Hain-Reed beta-invariant in degenerating families of curves. We show that the height jump of the Ceresa cycle at a given stable curve is equal to the so-called "slope" of the dual graph of the curve, and we characterize those stable curves for which the height jump vanishes. We also obtain an analytic formula for the height of the Ceresa cycle for a curve over a function field over the complex numbers, and characterize in analytic terms when the height of the Ceresa cycle vanishes.
@article {arXiv:2104.10060,
AUTHOR = {de Jong, Robin and Shokrieh, Farbod},
TITLE = {Jumps in the height of the {C}eresa cycle},
JOURNAL = {J. Differ. Geom.},
FJOURNAL = {Journal of Differential Geometry},
YEAR = {2022},
NOTE = {To appear. Preprint available at \href{https://arxiv.org/abs/2104.10060}{{\tt ar{X}iv:2104.10060}}},
URL = {https://arxiv.org/abs/2104.10060},
}
- Tautological cycles on tropical Jacobians. (with Andreas Gross)
The classical Poincaré formula relates the rational homology classes of tautological cycles on a Jacobian to powers of the class of Riemann theta divisor. We prove a tropical analogue of this formula. Along the way, we prove several foundational results about real tori with integral structures (and, therefore, tropical abelian varieties). For example, we prove a tropical version of the Appell-Humbert theorem. We also study various notions of equivalences between tropical cycles and their relation to one another.
@unpublished {arXiv:1910.07165,
AUTHOR = {Gross, Andreas and Shokrieh, Farbod},
TITLE = {Tautological cycles on tropical {J}acobians},
JOURNAL = {Algebra Number Theory},
FJOURNAL = {Algebra \& Number Theory},
YEAR = {2022},
NOTE = {To appear. Preprint available at \href{https://arxiv.org/abs/1910.07165}{{\tt ar{X}iv:1910.07165}}},
URL = {https://arxiv.org/abs/1910.07165},
}
- Faltings height and Néron-Tate height of a theta divisor. (with Robin de Jong)
@article {MR4371040,
AUTHOR = {de Jong, Robin and Shokrieh, Farbod},
TITLE = {Faltings height and {N}\'{e}ron-{T}ate height of a theta divisor},
JOURNAL = {Compos. Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {158},
YEAR = {2022},
NUMBER = {1},
PAGES = {1--32},
ISSN = {0010-437X},
MRCLASS = {11G10 (11G50 14G40 14K25)},
MRNUMBER = {4371040},
DOI = {10.1112/s0010437x21007661},
URL = {https://doi.org/10.1112/s0010437x21007661},
}
- Tropical moments of tropical Jacobians. (with Robin de Jong)
Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently computable formula for the tropical moment of a tropical Jacobian in terms of potential theory on the underlying metric graph. We show that there exists a universal linear relation between the tropical moment, a certain capacity called the tau invariant, and the total length of a metric graph. To put our formula in a broader context, we relate our work to the computation of heights attached to principally polarized abelian varieties.
@article {arXiv:1810.02639,
AUTHOR = {de Jong, Robin and Shokrieh, Farbod},
TITLE = {Tropical moments of tropical {J}acobians},
JOURNAL = {Canad. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics},
YEAR = {2022},
DOI = {10.4153/S0008414X22000220},
URL = {https://doi.org/10.4153/S0008414X22000220},
}
- Metric graphs, cross ratios, and Rayleigh's laws. (with Robin de Jong)
We systematically study the notion of cross ratios and energy pairings on metric graphs and electrical networks. We show that several foundational results on electrical networks and metric graphs immediately follow from the basic properties of cross ratios. For example, the projection matrices of Kirchhoff have natural (and efficiently computable) expressions in terms of cross ratios. We prove a generalized version of Rayleigh's law, relating energy pairings and cross ratios on metric graphs before and after contracting an edge segment. Quantitative versions of the Rayleigh's law for effective resistances, potential kernels, and cross ratios will follow as immediate corollaries.
@article {arXiv:1810.02638,
AUTHOR = {de Jong, Robin and Shokrieh, Farbod},
TITLE = {Metric graphs, cross ratios, and {R}ayleigh's laws},
JOURNAL = {Rocky Mt. J. Math.},
FJOURNAL = {Rocky Mountain Journal of Mathematics},
YEAR = {2022},
NOTE = {To appear. Preprint available at \href{https://arxiv.org/abs/1810.02638}{{\tt ar{X}iv:1810.02638}}},
URL = {https://arxiv.org/abs/1810.02638},
}
- Effective divisor classes on metric graphs. (with Andreas Gross and Lilla Tóthmérész)
We introduce the notion of semibreak divisors on metric graphs and prove that every effective divisor class (of degree at most the genus) has a semibreak divisor representative. This appropriately generalizes the notion of break divisors (in degree equal to genus). We provide an algorithm to efficiently compute such semibreak representatives.
Semibreak divisors provide the tool to establish some basic properties of effective loci inside Picard groups of metric graphs. We prove that effective loci are pure-dimensional polyhedral sets. We also prove that a 'generic' divisor class (in degree at most the genus) has rank zero, and that the Abel-Jacobi map is 'birational' onto its image. These are analogues of classical results for Riemann surfaces.
@article {arXiv:1807.00843,
AUTHOR = {Gross, Andreas and Shokrieh, Farbod and T{\'o}thm{\'e}r{\'e}sz, Lilla},
TITLE = {Effective divisor classes on metric graphs},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
YEAR = {2022},
DOI = {10.1007/s00209-022-03056-x},
URL = {https://doi.org/10.1007/s00209-022-03056-x},
}
- Cycles, cocycles, and duality on tropical manifolds. (with Andreas Gross)
We prove a Poincaré duality for the Chow rings of smooth fans whose support are tropical linear spaces. As a consequence, we show that cycles and cocycles on tropical manifolds are Poincaré dual to each other. This allows us to define pull-backs of tropical cycles along arbitrary morphisms with smooth target.
@article {MR4246795,
AUTHOR = {Gross, Andreas and Shokrieh, Farbod},
TITLE = {Cycles, cocycles, and duality on tropical manifolds},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {149},
YEAR = {2021},
NUMBER = {6},
PAGES = {2429--2444},
ISSN = {0002-9939},
MRCLASS = {14T10 (05B35 14C17)},
MRNUMBER = {4246795},
DOI = {10.1090/proc/15468},
URL = {https://doi.org/10.1090/proc/15468},
}
- Limits of canonical forms on towers of Riemann surfaces. (with Hyungryul Baik and Chenxi Wu)
- Journal für die reine und angewandte Mathematik (Crelle's Journal), Volume 764, Pages 287-304, 2020.
- abstract | DOI | arXiv |
bibtex
@article {MR4116639,
AUTHOR = {Baik, Hyungryul and Shokrieh, Farbod and Wu, Chenxi},
TITLE = {Limits of canonical forms on towers of {R}iemann surfaces},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's Journal]},
VOLUME = {764},
YEAR = {2020},
PAGES = {287--304},
ISSN = {0075-4102},
MRCLASS = {30F10 (14H30 57K20)},
MRNUMBER = {4116639},
DOI = {10.1515/crelle-2019-0007},
URL = {https://doi.org/10.1515/crelle-2019-0007},
}
- Canonical measures on metric graphs and a Kazhdan's theorem. (with Chenxi Wu)
We extend the notion of canonical measures to all (possibly non-compact) metric graphs. This will allow us to introduce a notion of hyperbolic measures
on universal covers of metric graphs. Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. We will prove a generalized version of this theorem for metric graphs, allowing any infinite Galois cover to replace the universal cover. We will show all such limiting measures satisfy a version of Gauss-Bonnet formula which, using the theory of von Neumann dimensions, can be interpreted as a trace formula
. In the special case where the infinite cover is the universal cover, we will provide explicit methods to compute the corresponding limiting (hyperbolic) measure. Our ideas are motivated by non-Archimedean analytic and tropical geometry.
@article {MR3935033,
AUTHOR = {Shokrieh, Farbod and Wu, Chenxi},
TITLE = {Canonical measures on metric graphs and a {K}azhdan's theorem},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {215},
YEAR = {2019},
NUMBER = {3},
PAGES = {819--862},
ISSN = {0020-9910},
MRCLASS = {14T05 (05C12 05C63 14H30 20F65 57M60)},
MRNUMBER = {3935033},
MRREVIEWER = {Hannah Markwig},
DOI = {10.1007/s00222-018-0838-5},
URL = {https://doi.org/10.1007/s00222-018-0838-5},
}
- Non-Archimedean and tropical theta functions. (with Tyler Foster, Joseph Rabinoff, and Alejandro Soto)
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.
@article {MR3880286,
AUTHOR = {Foster, Tyler and Rabinoff, Joseph and Shokrieh, Farbod and Soto, Alejandro},
TITLE = {Non-{A}rchimedean and tropical theta functions},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {372},
YEAR = {2018},
NUMBER = {3-4},
PAGES = {891--914},
ISSN = {0025-5831},
MRCLASS = {14T05 (14K25)},
MRNUMBER = {3880286},
MRREVIEWER = {Shu Kawaguchi},
DOI = {10.1007/s00208-018-1646-3},
URL = {https://doi.org/10.1007/s00208-018-1646-3},
}
- Divisors on graphs, binomial and monomial ideals, and
cellular resolutions. (with Fatemeh Mohammadi)
We study various binomial and monomial ideals arising in
the theory of divisors, orientations, and matroids on
graphs. We use ideas from potential theory on graphs and from
the theory of Delaunay decompositions for lattices to describe
their minimal polyhedral cellular free resolutions. We show
that the resolutions of all these ideals are closely related
and that their -graded Betti tables coincide. As corollaries, we
give conceptual proofs of conjectures and questions posed by
Postnikov and Shapiro, by Manjunath and Sturmfels, and by
Perkinson, Perlman, and Wilmes. Various other results related
to the theory of chip-firing games on graphs also follow from
our general techniques and results.
@article {MR3489059,
AUTHOR = {Mohammadi, Fatemeh and Shokrieh, Farbod},
TITLE = {Divisors on graphs, binomial and monomial ideals, and cellular resolutions},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {283},
YEAR = {2016},
NUMBER = {1-2},
PAGES = {59--102},
ISSN = {0025-5874},
MRCLASS = {13D02 (05C25 52C07)},
MRNUMBER = {3489059},
MRREVIEWER = {Sara Saeedi Madani},
DOI = {10.1007/s00209-015-1589-2},
URL = {https://doi.org/10.1007/s00209-015-1589-2},
}
- Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem. (with Yang An, Matthew Baker, and Greg Kuperberg)
@article {MR3264262,
AUTHOR = {An, Yang and Baker, Matthew and Kuperberg, Greg and Shokrieh, Farbod},
TITLE = {Canonical representatives for divisor classes on tropical curves and the matrix-tree theorem},
JOURNAL = {Forum Math. Sigma},
FJOURNAL = {Forum of Mathematics. Sigma},
VOLUME = {2},
YEAR = {2014},
PAGES = {Paper No. e24, 25},
MRCLASS = {14T05 (05A19 05C25 05E45 14C20)},
MRNUMBER = {3264262},
MRREVIEWER = {Luis Felipe Tabera},
DOI = {10.1017/fms.2014.25},
URL = {https://doi.org/10.1017/fms.2014.25},
}
- Divisors on graphs, connected flags, and syzygies. (with Fatemeh Mohammadi)
We study the binomial and monomial ideals arising from
linear equivalence of divisors on graphs from the point of
view of Gröbner theory. We give an explicit description
of a minimal Gröbner basis for each higher syzygy
module. In each case the given minimal Gröbner basis is
also a minimal generating set. The Betti numbers of the
binomial ideal and its natural initial ideal coincide and they
correspond to the number of connected flags
in the
graph. In particular the Betti numbers are independent of the
characteristic of the base field. For complete graphs the
problem was previously studied by Postnikov and Shapiro and
by Manjunath and Sturmfel. The case of a general graph was stated as an open problem.
@article {MR3291642,
AUTHOR = {Mohammadi, Fatemeh and Shokrieh, Farbod},
TITLE = {Divisors on graphs, connected flags, and syzygies},
JOURNAL = {Int. Math. Res. Not. IMRN},
FJOURNAL = {International Mathematics Research Notices. IMRN},
YEAR = {2014},
NUMBER = {24},
PAGES = {6839--6905},
ISSN = {1073-7928},
MRCLASS = {13F20 (05C25 13D02 13P10)},
MRNUMBER = {3291642},
MRREVIEWER = {Yuehui Zhang},
DOI = {10.1093/imrn/rnt186},
URL = {https://doi.org/10.1093/imrn/rnt186},
}
- Chip-firing games, potential theory on graphs, and
spanning trees. (with Matthew Baker)
- Journal of Combinatorial Theory,
Series A, Volume 120, Issue 1, Pages 164-182, 2013.
- abstract | DOI | arXiv |
bibtex
We study the interplay between chip-firing games and
potential theory on graphs, characterizing reduced divisors
(-parking functions) on graphs as the solution to an energy (or potential) minimization problem and providing an algorithm
to efficiently compute reduced divisors. Applications include
an efficient bijective
proof of Kirchhoff's matrix-tree
theorem and a new algorithm for finding random spanning
trees. The running times of our algorithms are analyzed using
potential theory, and we show that the bounds thus obtained
generalize and improve upon several previous results in the
literature.
@article {MR2971705,
AUTHOR = {Baker, Matthew and Shokrieh, Farbod},
TITLE = {Chip-firing games, potential theory on graphs, and spanning trees},
JOURNAL = {J. Combin. Theory Ser. A},
FJOURNAL = {Journal of Combinatorial Theory. Series A},
VOLUME = {120},
YEAR = {2013},
NUMBER = {1},
PAGES = {164--182},
ISSN = {0097-3165},
MRCLASS = {05C57 (05C05 05C50 31C20 65R10)},
MRNUMBER = {2971705},
MRREVIEWER = {Nick Gravin},
DOI = {10.1016/j.jcta.2012.07.011},
URL = {https://doi.org/10.1016/j.jcta.2012.07.011},
}
- Improved simulation of nondeterministic Turing machines. (with S. Kalyanasundaram, R. J. Lipton, K. W. Regan)
The standard simulation of a nondeterministic Turing
machine (NTM) by a deterministic one essentially searches a
large bounded-degree graph whose size is exponential in the
running time of the NTM. The graph is the natural one defined
by the configurations of the NTM. All methods in the
literature have required time linear in the size of
this graph. This paper presents a new simulation method that
runs in time . The search savings exploit the one-dimensional nature of Turing machine tapes. In addition, we remove most of the time dependence on nondeterministic choices of states and tape head movements.
@article {MR2885890,
AUTHOR = {Kalyanasundaram, Subrahmanyam and Lipton, Richard J. and Regan, Kenneth W. and Shokrieh, Farbod},
TITLE = {Improved simulation of nondeterministic {T}uring machines},
JOURNAL = {Theoret. Comput. Sci.},
FJOURNAL = {Theoretical Computer Science},
VOLUME = {417},
YEAR = {2012},
PAGES = {66--73},
ISSN = {0304-3975},
MRCLASS = {68Q05},
MRNUMBER = {2885890},
MRREVIEWER = {B. H. Mayoh},
DOI = {10.1016/j.tcs.2011.05.018},
URL = {https://doi.org/j.tcs.2011.05.018},
}
- The monodromy pairing and discrete logarithm on the
Jacobian of finite graphs.
Every graph has a canonical finite abelian group attached to
it. This group has appeared in the literature under a variety
of names including the sandpile group, critical group,
Jacobian group, and Picard group. The construction of this
group closely mirrors the construction of the Jacobian variety
of an algebraic curve. Motivated by this analogy, it was
recently suggested by Norman Biggs that the critical group of
a finite graph is a good candidate for doing discrete
logarithm based cryptography. In this paper, we study a
bilinear pairing on this group and show how to compute
it. Then we use this pairing to find the discrete logarithm
efficiently, thus showing that the associated cryptographic
schemes are not secure. Our approach resembles the MOV attack
on elliptic curves.
@article {MR2660333,
AUTHOR = {Shokrieh, Farbod},
TITLE = {The monodromy pairing and discrete logarithm on the {J}acobian of finite graphs},
JOURNAL = {J. Math. Cryptol.},
FJOURNAL = {Journal of Mathematical Cryptology},
VOLUME = {4},
YEAR = {2010},
NUMBER = {1},
PAGES = {43--56},
ISSN = {1862-2976},
MRCLASS = {05C50 (14H40 94A60)},
MRNUMBER = {2660333},
DOI = {10.1515/JMC.2010.002},
URL = {https://doi.org/10.1515/JMC.2010.002},
}