About

Joseph H. Silverman is a professor whose academic lineage and mentorship are documented through the Mathematics Genealogy Project. He earned his Ph.D. from Harvard University in 1982 with a dissertation titled "The Neron-Tate Height on Elliptic Curves." His research focus includes arithmetic dynamics, elliptic curves, and number theory, as evidenced by the topics of his numerous Ph.D. students' dissertations at Brown University. These topics range from dynamical Galois representations, algebraic dynamics, and moduli spaces to arithmetic dynamics of rational maps, height functions, and Galois theory. Silverman has advised a significant number of doctoral students, indicating a strong role in advancing research and education in these mathematical areas. His academic genealogy traces back through prominent mathematicians, including John Tate, Jr., Emil Artin, and Carl Friedrich Gauss, highlighting a distinguished scholarly heritage.

Research topics

• Computer Science
• Algorithm
• Mathematics
• Mathematical analysis
• Geometry

Selected publications

• Propagation of Zariski dense orbits

  Revista Matemática Iberoamericana · 2026-02-18

  articleOpen accessSenior author

  Let X/K be a smooth projective variety defined over a number field, and let f\colon X\to{X} be a morphism defined over K . We formulate a number of statements of varying strengths asserting, roughly, that if there is at least one point P_{0} \in X(K) whose f -orbit \mathcal{O}_{f}(P_{0}):=\{f^{n}(P):n\in\mathbb{N}\} is Zariski dense, then after replacing K by a finite extension, there are many f -orbits in X(K) . For example, a weak conclusion would be that X(K) is not the union of finitely many (grand) f -orbits, while a strong conclusion would be that any set of representatives for the Zariski dense grand f -orbits in X(K) is itself Zariski dense. We prove statements of this sort for various classes of varieties and maps, including projective spaces, abelian varieties, and surfaces.

  PublisherDOI

• Automorphism groups of commuting polynomial maps of the affine plane

  Rendiconti Lincei Matematica e Applicazioni · 2026-03-06

  articleOpen access1st authorCorresponding

  Let \mathcal{L} be a finite-dimensional semisimple Lie algebra of rank N over an algebraically closed field of characteristic 0 . Associated with \mathcal{L} is a family of polynomial folding maps \mathsf{F}_{n}:\mathbb{A}^{N}\to\mathbb{A}^{N}\quad\text{for }n\ge1 having the property that \mathsf{F}_{n} has topological degree n^{N} and \mathsf{F}_{m}\circ\mathsf{F}_{n}=\mathsf{F}_{n}\circ\mathsf{F}_{m}\quad\text{for all }m,n\ge1. We derive formulas for the leading terms of the folding maps on \mathbb{A}^{2} associated with the Lie algebras \mathcal{A}_{2} , \mathcal{B}_{2} , and \mathcal{G}_{2} , and we use these formulas to compute the affine automorphism group of each folding map.

  PublisherDOI

• An introduction to lattices, lattice reduction, and lattaice-based cryptography

  IAS/Park City mathematics series · 2025-01-01

  other1st authorCorresponding
  PublisherDOI

• A uniform quantitative Manin–Mumford theorem for curves over function fields

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2025-08-20

  articleOpen access

  Abstract We prove that any smooth projective geometrically connected non-isotrivial curve of genus <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> g\geq 2 over a one-dimensional function field of any characteristic has at most <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mn>16</m:mn> <m:mo>⁢</m:mo> <m:msup> <m:mi>g</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mn>32</m:mn> <m:mo>⁢</m:mo> <m:mi>g</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>124</m:mn> </m:mrow> </m:math> 16g^{2}+32g+124 torsion points for any Abel–Jacobi embedding of the curve into its Jacobian. The proof uses Zhang’s admissible pairing on curves, the arithmetic Hodge index theorem over function fields, and the metrized graph analogue of Elkies’ lower bound for the Green function. More generally, we prove an explicit Bogomolov-type result bounding the number of geometric points of small Néron–Tate height on the curve embedded into its Jacobian.

  PublisherOA PDFDOI

• A heuristic subexponential algorithm to find paths in Markoff graphs over finite fields

  Research in Number Theory · 2025-02-06 · 1 citations

  article1st authorCorresponding
  PublisherDOI

• Corrigendum to: “A Lehmer-type lower bound for the canonical height on elliptic curves over function fields” [J. Number Theory 262 (2024) 506–538]

  Journal of Number Theory · 2025-07-25

  articleOpen access1st authorCorresponding

  We correct a constant appearing in an inequality, and explain how the change propagates through the paper to change various other constants. The revised result is the lower bound h ˆ E ( P ) ≥ 1 18000 ⋅ h F ( j E ) 2 ⋅ [ K : F ] 2 , in which the fraction 1 18000 replaces the constant 1 10500 appearing in the original publication, and with the added requirement that [ K : F ] ≥ 6 .

  PublisherDOI

• A Lehmer-Type Lower Bound for the Canonical Height on Elliptic Curves Over Function Fields

  arXiv (Cornell University) · 2024-02-22

  preprintOpen access1st authorCorresponding

  Let $\mathbb{F}$ be the function field of a curve over an algebraically closed field with $\operatorname{char}(\mathbb{F})\ne2,3$, and let $E/\mathbb{F}$ be an elliptic curve. Then for all finite extensions $\mathbb{K}/\mathbb{F}$ and all non-torsion points $P\in{E(\mathbb{K})}$, the $\mathbb{F}$-normalized canonical height of $P$ is bounded below by \[ \hat{h}_E(P) \ge \frac{1}{10500\cdot h_{\mathbb{F}}(j_E)^{2}\cdot [\mathbb{K}:\mathbb{F}]^{2}}. \]

  PublisherOA PDFDOI

• Applications of Nanocomposites and Polymeric Materials in the Aerospace Industry

  2024-01-01 · 3 citations

  article1st authorCorresponding

  The idea of polymer nanocomposites began in the early-1900s when the discovery of chain-like structures were being formed by chemical reactions, which will be known as polymers. The first reports of nanocomposites were posted during the mid-1900s, and ever since, scientists have been trying to find new ways to implement them into current designs to take advantage of their advanced characteristics. Nanoscience phenomena have gained public attention for a few decades, but several aerospace industry challenges prevent these new, improved materials from being widely used and implemented. Since nanoscience is a relatively new field, it takes time to standardize and integrate nanomaterials and polymers into current designs and manufacturing processes. Also, providing their standardization and compliance with the safety requirements is crucial, which also demands time and more research. Given that Nanocomposites are still in the experimental phase, numerous companies are still focusing on older materials and technologies more known for their reliability. The adaptation of nanocomposite materials in both aircraft and spacecraft construction presents the opportunity to significantly reduce weight, which has been a crucial goal in the aerospace sector. Reduced weight will facilitate improved fuel efficiency, which in turn will reduce overall carbon emissions. Combined with nanocomposites’ increased durability to corrosion and extreme temperatures, this ensures aircraft systems’ longevity while demonstrating environmentally sustainable air travel. This paper will also include numerous lab results, consisting of the viscosity measurements of various nanocomposites and polymeric materials and other features, such as flame retardancy. Using the information collected, we can determine what additives best suit the aviation and aerospace industry to increase cost efficiency, structural integrity, and reduce manufacturing time. We will also provide physical examples of various polymeric materials, each with different physical properties and chemical additives.

  PublisherDOI

• A Lehmer-type lower bound for the canonical height on elliptic curves over function fields

  Journal of Number Theory · 2024-05-16 · 1 citations

  article1st authorCorresponding
  PublisherDOI

• Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions

  Simons symposia · 2024-08-02

  preprintOpen access1st authorCorresponding
  PublisherOA PDFDOI

Recent grants

• FRG: Collaborative Research: Algebraic Dynamics

  NSF · $142k · 2009–2013

• Investigations in Arithmetic Geometry and Arithmetic Dynamics

  NSF · $195k · 2007–2011

Frequent coauthors

• Samuel G. Armistead

  Brocade (United States)

  95 shared

• Jeffrey Hoffstein

  Providence College

  71 shared

• Jill Pipher

  57 shared

• Mohamad Al‐Sheikhly

  University of Maryland, College Park

  42 shared

• Marc Hindry

  39 shared

• P. Neta

  Material Measurement Laboratory

  30 shared

• Shu Kawaguchi

  30 shared

• John Tate

  Fred Hutch Cancer Center

  27 shared

Labs

• Joseph Silverman Research LabPI

Education

• Ph.D., Number Theory

  Brown University

• M.S.

  Brown University

• B.A.

  Brown University