
Research articles

Constructing Picard Curves
with Complex Multiplication using the Chinese Remainder Theorem,
with Sonny Arora.
Preprint, 2017.


Supersingular isogeny graphs and endomorphism rings: reductions
and solutions, with Sean
Hallgren, Kristin Lauter, Travis
Morrison, and Christophe Petit. To appear in Eurocrypt 2018.


Universally and
existentially definable subsets of global fields, with Travis
Morrison. Accepted for publication in Math Res. Lett., 2017.


As Easy as Q:
Hilbert's Tenth Problem for subrings of the rationals and number
fields, with Russell Miller, Jennifer Park, and Alexandra
Shlapentokh. Transactions of
the AMS, Volume 369, Number 11, pages 82918315, 2017.


Hilbert's Tenth Problem over function fields of
positive characteristic not containing the algebraic closure of a
finite field,
with Alexandra Shlapentokh. J. Eur. Math.
Soc. (JEMS) Volume 19, Issue 7, pages 21032138, 2017.


Constructing elliptic curves
and curves of genus 2 over finite fields, Contemporary Developments
in Finite Fields and Applications, World Scientific, pp. 4861, 2016.


Weak Instances of PLWE,
with Sean Hallgren, and Kristin Lauter, Selected Areas in
CryptographySAC 2014, Springer LNCS 8781, pp. 183194, 2014.


A quantum algorithm for computing the unit
group of an arbitrary degree number field,
with Sean Hallgren, Alexei Kitaev, and Fang Song, STOC 2014.


Computing the unit group, class group and compact
representations in algebraic function fields, with Sean
Hallgren. Tenth Algorithmic Number Theory
Symposium (ANTS X) Proceedings.
Open Book Series, volume 1, Mathematical Sciences Publishers,
Berkeley, 2013 (electronic).


Hilbert's
Tenth Problem for function fields of varieties over algebraically
closed fields of positive characteristic. Monatsh. Math. 168 (2012), no. 1, 116, 2012.


Hilbert's Tenth Problem and Mazur's Conjectures in
complementary subrings of number fields,
with Graham Everest and Alexandra Shlapentokh. Mathematical Research
Letters 18, no. 06, 11411162, 2011.


Pairings on Hyperelliptic Curves,
with Jennifer Balakrishnan, Juliana Belding, Sarah Chisholm,
Katherine Stange, and Edlyn Teske. WIN  Women in Numbers: Research Directions in
Number Theory, Fields Institute Communications, vol. 60, Amer.
Math. Soc., Providence, RI, pages 87120, 2011.


Quantum algorithms for ray class groups and some subfields of Hilbert class fields,
with Sean Hallgren. Symposium on
Discrete Algorithms (SODA) 2010 Proceedings, Society for Industrial and
Applied Mathematics (SIAM), pages 471483, 2010.


A CRT algorithm for constructing genus 2 curves over finite fields,
with Kristin Lauter. Arithmetic,
Geometry and Coding Theory (AGCT10), Séminaires
et Congrès 21 (2009), pages 161176. Société Mathématique de
France, Paris, 2009.


Undecidability in function fields of positive characteristic,
with Alexandra Shlapentokh. Int. Math. Res. Not., Vol. 2009: article ID rnp079, 36
pages, doi: 10.1093/imrn/rnp079, 2009.


Descent on elliptic curves and Hilbert's Tenth Problem,
with Graham Everest. Proc. Amer. Math. Soc., 137(6):19511959, 2009.


Hilbert's Tenth
Problem for function fields of
characteristic zero. Model Theory with
Applications to Algebra and Analysis, Volume 2, Cambridge
University Press, 2008, pages 237254.


On the
computation of the Cassels pairing for certain Kolyvagin
classes in the ShafarevichTate group, with Dimitar
Jetchev and Kristin Lauter. PairingBased Cryptography  Pairing 2008,
Springer LNCS volume 5209, 2008, pages 113125.


Hilbert's Tenth Problem for function fields of varieties over number fields
and padic fields, J. Algebra , 310(2007), pages 775792.


Integrality at a prime for global
fields and the perfect closure of global fields of characteristic p>2,
J. Number Theory, Volume 114, 2005, pages 170181.


Hilbert's Tenth Problem for function fields of
varieties over C, Int. Math. Res. Not., Issue 59, 2004, pages
31913205.


Improved Weil and Tate pairings for elliptic and hyperelliptic
curves, with Kristin Lauter and Peter L. Montgomery,
in ANTSVI proceedings, 2004, pages 169183 (Springer
LNCS link).


Hilbert's Tenth Problem and Arithmetic Geometry, Ph.D. thesis, May 2003.


Hilbert's Tenth Problem for
algebraic function fields of characteristic 2, Pacific J. Math., 210
(2), 2003, pages 261281.


Fast elliptic curve
arithmetic and improved Weil pairing evaluation, with Kristin Lauter and
Peter L. Montgomery,
Topics in Cryptology  CTRSA 2003, (Springer
LNCS link).

