Professor Silverman studies number theory, elliptic curves, arithmetic and Diophantine geometry, number theoretic aspects of dynamical systems, and cryptography.

Research in Arithmetic Geometry
Professor Silverman's primary area of research is arithmetic geometry, especially the existence and distribution of integral and rational points on (elliptic) curves and on higher dimensional (abelian) varieties. A short list of some of his results includes:

• Lower bounds for canonical heights and application to uniform bounds for the number of integral points on elliptic curves in terms of the rank of the Mordell-Weil group. This joint work with Marc Hindry answers questions raised by Serge Lang.
  
• Limit formulas for the variation of the canonical height in families of abelian varieties, with applications to the injectivity of the specialization map.
  
• Application of the abc conjecture to Wieferich primes.
  
• A limit formula for the rank of an elliptic surface (joint work with Micheal Rosen) and application to the variation of the rank of specializations.
  
• Lower bounds for heights over abelian extensions and Lehmer's conjecture on abelian varieties (joint work with Matt Baker).
  
• Upper and lower bounds for divisibility sequences on tori, elliptic curves, and abelian varieties and the relation of such bounds to Vojta's conjecture.
  
• Independence of Heegner points defined over distinct quadratic imaginary fields (joint work with Micheal Rosen).
  

Professor Silverman has also written several books on arithmetic geometry at the advanced graduate and research level: The Arithmetic of Elliptic Curves (1986); Advanced Topics in the Arithmetic of Elliptic Curves (1994); and Diophantine Geometry: An Introduction (2000) (with Marc Hindry).

Research in Arithmetic Dynamics
Discrete dynamical systems over the real and complex numbers have a long and distinguished history. The more recent field of arithmetic dynamics attempts to understand the arithmetic properties of iteration of polynomial or rational functions applied to rational or algebraic points on a variety. Professor Silverman is one of the founders of this field and has many published research papers and graduate students in the area. Among Professor Silverman's contributions are construction of global and local dynamical height functions (with Greg Call), finiteness of integral points in orbits, construction of dynamical units (with Patrick Morton), finiteness of rational periodic points for Henon maps, construction of dynamical moduli spaces, and the Morton-Silverman Conjecture on uniform boundedness of algebraic periodic points. His current research program in arithmetic dynamics includes work on arithmetic and dynamical degree. He has written a book ( The Arithmetic of Dynamical Systems , Graduate Texts in Math. 241 , Springer-Verlag, N.Y., 2007) and a research monograph ( Moduli Spaces and Arithmetic Dynamics (CRM Monograph Series, Vol. 30) American Mathematical Society, 2012) in the area of arithmetic dynamics.

Research in Cryptography
Professor Silverman has done work on both elliptic curve cryptography (ECC) and on lattice based cryptography, especially helping with the development of the NTRU public key cryptosystem with his colleagues Jeffrey Hoffstein and Jill Pipher. His many papers on NTRU, some published and some technical notes posted on the web, have helped take NTRU from a theoretical construction to a viable commericial product. He continues to do research on both NTRU and ECC. He has written an undergraduate textbook ( An Introduction to Mathematical Cryptography , Undergraduate Texts in Mathematics, Springer-Verlag, 2008) with Hoffstein and Pipher.

Simons Collaboration Grant for Mathematicians, 2012-2017
NSF EAGER Grant, 2014-2016

NSF Focused Research Grant, 2009-2012

NSF Research Grant, 2006-2008

NSA Research Grant, 2003-2006, "Investigation of Elliptic Curves and Moduli of Dynamical Systems"

NSF Research Grant, 1999-2003, "Arithmetic of Elliptic Curves"

NSF Research Grants, 1986-1998

Moduli Spaces and Arithmetic Dynamics (CRM Monograph Series, Vol. 30) American Mathematical Society, 2012.

Height estimates for equidimensional dominant rational maps, J. Ramanujan Math. Soc. 26 (2011), 145-163.

with Katherine Stange, Amicable pairs and aliquot cycles for elliptic curves, Exper. Math., 20(3) (2011), 329-357.

with Patrick Ingram, Primitive divisors in arithmetic dynamics, Proc. Camb. Philos. Soc. (2009), 146, #2, 289-302.

with Liang-Chung Hsia, On a dynamical Brauer-Manin obstruction, Proceedings of the Journees Arithmetiques 2007, J. Th\'eor. Nombres Bordeaux 21 (2009), 235-250.

with Jose Felipe Voloch, A Local-Global Criterion for Dynamics on P^1, Acta Arithmetica 137.3 (2009), 285-294.

Lifting and elliptic curve discrete logarithms, Selected Areas of Cryptography (SAC 2008), Lecture Notes in Computer Science 5381, Springer-Verlag, Berlin, 2009, 82-102.

with Jill Pipher and Jeffrey Hoffstein, An Introduction to Mathematical Cryptography, Undergraduate Texts in Mathematics, Springer-Verlag, 2008.

An Introduction to Mathematical Cryptography, Undergraduate Texts in Mathematics, Springer-Verlag, 2008.

Variation of periods modulo p in arithmetic dynamics, New York Journal of Math. 14 (2008), 601-616 (electronic).

The Arithmetic of Dynamical Systems, Graduate Texts in Math. 241, Springer-Verlag, N.Y., 2007.

with M. Rosen, On the independence of Heegner points associated to distinct quadratic imaginary fields, Journal of Number Theory 127 (2007), 10-36.

with S. Kawaguchi, Dynamics of projective morphisms having identical canonical heights, Proc. London Math. Soc. 95 (2007), 519-544.

Height bounds and preperiodic points for families of jointly regular affine mapsQuart. J. Pure Appl. Math. 2 (2006), 135-145.

p-adic properties of division polynomials and elliptic divisibility sequences, Mathematische Annalen 332 (2005), 443-471.

Generalized greatest common divisors, divisibility sequences, and Vojta's conjecture on blowups Monatsch. Math. 145 (2005), 333-350.

The rank of an elliptic surface over abelian unramified base extensions, J. Reine Angew. Math. 577 (2004), 153-169.

with Matthew Baker, A lower bound for the canonical height on abelian varieties over abelian extensions, Mathematical Research Letters 11 (2004), 377-396.

with J. Hoffstein, N. Howgrave-Graham, J. Pipher, W. Whyte, NTRUSign: Digital Signatures Using the NTRU Lattice, Topics in Cryptology - CT-RSA 2003, San Francisco, February 2003, Lecture Notes in Computer Science 2612, Springer-Verlag, Berlin, 2003, 122-140.

with N. Howgrave-Graham, P. Nguyen, D. Pointcheval, J. Proos, A. Singer, W. Whyte, The impact of decryption failure on the security of NTRU encryption, Advances in Cryptology -Crypto 2003, Lecture Notes in Computer Science 2729, Springer-Verlag, 2003.

Cryptography and Lattices Conference (CaLC 2001), Lecture Notes in Computer Science 2461, Springer-Verlag, 2001, 1-215.

with A. May, Dimension reduction methods for convolution modular lattices, Cryptography and Lattices Conference (CaLC 2001), Lecture Notes in Computer Science 2461, Springer-Verlag, 2001, 110-125.

with M. Hindry, Diophantine Geometry: An Introduction, Graduate Texts in Math. 201, Springer-Verlag, New York, 2000, 1-558.

with M. Hindry, Sur le nombre de points de torsion rationnels sur une courbe elliptique, C.R. Acad. Sci. Paris 329 (1999), 97--100.

with Jeffrey Hoffstein, Jill Pipher, NTRU: A Ring Based Public Key Cryptosystem, Algorithmic Number Theory (ANTS III), Portland, OR, June 1998, Lecture Notes in Computer Science 1423, Springer-Verlag, Berlin, 1998, 267-288.

with J. Suzuki, Elliptic curve discrete logarithms and the index calculus, Advances in Cryptology-AsiaCrypt 98, Beijing, October 1998, Lecture Notes in Computer Science 1514, Springer-Verlag, Berlin, 1998, 110-125.

with M. Rosen, On the rank of an elliptic surface, Invent. Math. 133 (1998), 43-67.

The space of rational maps on P1, Duke Math. J. 94 (1998), 41-77.

The average rank of an algebraic family of elliptic curves, J. Reine Angew. Math. 504 (1998), 227-236.

A Friendly Introduction to Number Theory, Prentice-Hall, N.J., 1997, 1-288; 2nd ed. 2001, 1-390; 3rd ed. 2005, 1-434.

On the field of definition for dynamical systems on P1, Compos. Math. 98 (1995), 269-304.

co-editor with G. Cornell and G. Stevens, Modular Forms and Fermat's Last Theorem, a conference held at Boston University, 1995, Springer-Verlag, N.Y., 1997, 1-569

with P. Morton, Periodic points, multiplicities, and dynamical units, J. Reine Angew. Math. 461 (1995), 81-122.

Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Math. 151, Springer-Verlag, N.Y., 1994, 1-525.

Integer points, Diophantine approximation, and iteration of rational maps, Duke Math. J. 71 (1993), 793-829.

with G. Call, Canonical heights on varieties with morphisms, Compos. Math. 89 (1993), 163-205.

Variation of the canonical height on elliptic surfaces I: Three examples, J. Reine Angew. Math. 426 (1992), 151-178.

with J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Math., Springer-Verlag, N.Y., 1992, 1-281.

with J.F. Voloch, Multiple Weierstrass points, Compos. Math. 79 (1991), 123-134.

Rational points on symmetric products of a curve, Am. J. Math. 113 (1991), 471-508.

Rational points on K3 surfaces: A new canonical height, Invent. Math. 105 (1991), 347-373.

Hecke points on modular curves, Duke Math. J. 60 (1990), 401-423.

with M. Hindry, The canonical height and integral points on elliptic curves, Invent. Math. 93 (1988), 419-450.

A quantitative version of Siegel's theorem: Integral points on elliptic curves and Catalan curves, J. Reine Angew. Math. 378 (1987), 60-100.

Integral points on abelian varieties are widely spaced, Compos. Math. 61 (1987), 253-266.

Arithmetic distance functions and height functions in Diophantine geometry Math. Ann., 279 (1987), 193-216.

The Arithmetic of Elliptic Curves, Graduate Texts in Math. 106 , Springer-Verlag, N.Y., 1986, 1-400

Integral points on abelian varieties, Invent. Math. 81 (1985), 341-346.

Divisibility of the specialization map for families of elliptic curves, Amer. J. Math. 107 (1985), 555-565.

Lower bounds for height functions, Duke Math. J. 51 (1984), 395-403.

co-editor with G. Cornell, Arithmetic Geometry, a conference held at Storrs, Connecticut, 1984, Springer-Verlag, N.Y., 1986, 1-353

Representations of integers by binary forms and the rank of the Mordell-Weil group, Invent. Math. 74 (1983), 281-292.

Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197-211.

Integer points and the rank of Thue elliptic curves, Invent. Math. 66 (1982), 395-404.

Lower bound for the canonical height on elliptic curves, Duke Math. J. 48 (1981), 633-648.

Fellow of the American Mathematical Society, elected 2012

Mathematical Association of America Award for Distinguished Teaching, 2011

Elliptic Curve Cryptography Visionary Award, 2011

Guggenheim Foundation Fellowship, 1998-1999

American Mathematical Society Steele Prize for Mathematical Exposition, 1998

Brown University Award for Excellence in Teaching, 1996

Mathematical Association of America Lester Ford Award, 1994

Sloan Foundation Fellowship, 1987-1991

American Mathematical Society

Association of Women in Mathematics

Dr. Silverman teaches undergraduate mathematics courses in all areas, with an emphasis on algebraic subjects, and graduate courses in algebra, algebraic geometry, number theory, and arithmetic dynamics.