Kirsten Eisenträger

Professor of Mathematics

Francis R. Pentz and Helen M. Pentz Professor of Science

Department of Mathematics

The Pennsylvania State University

109 McAllister Building

University Park, PA 16802

Phone: (814) 863-4127

e-mail: eisentra@math.psu.edu

Office: 330 McAllister Bldg.

I am a professor in the Department of Mathematics at Penn State, and the Francis R. Pentz and Helen M. Pentz Professor of Science. I am interested in number theory, arithmetic geometry, and applications to cryptography. I am a Fellow of the American Mathematical Society. My research is currently supported by a grant from the Department of the Army, and an NSF grant. In 2008 I was awarded a Sloan Research Fellowship, and in 2011 an NSF CAREER award. There are currently three Ph.D. students working with me.

Papers:

Research articles

	A topological approach to undefinability in algebraic extensions of Q, with Russell Miller, Caleb Springer, and Linda Westrick, 2020.
	Computing endomorphism rings of supersingular elliptic curves and connections to path-finding in isogeny graphs, with Sean Hallgren, Chris Leonardi, Travis Morrison, and Jennifer Park. to appear in Fourteenth Algorithmic Number Theory (ANTS XIV) Proceedings.
	Cycles in the supersingular l-isogeny graph and corresponding endomorphisms, with Efrat Bank, Catalina Camacho-Navarro, Travis Morrison, and Jennifer Park. Research Directions in Number Theory – Women in Numbers 4 (WIN 4) Proceedings, pages 41-66, Springer, 2019.
	Constructing Picard Curves with Complex Multiplication using the Chinese Remainder Theorem , with Sonny Arora. Thirteenth Algorithmic Number Theory Symposium (ANTS XIII) Proceedings}, pages 21-36, 2019.
	Supersingular isogeny graphs and endomorphism rings: reductions and solutions, with Sean Hallgren, Kristin Lauter, Travis Morrison, and Christophe Petit. Published in Eurocrypt 2018 Proceedings, pages 329-368.
	Universally and existentially definable subsets of global fields, with Travis Morrison. Math. Res. Lett., Volume 25, Number 4, pages 1173-1204, 2018.
	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 8291-8315, 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 2103-2138, 2017.
	Constructing elliptic curves and curves of genus 2 over finite fields, Contemporary Developments in Finite Fields and Applications, World Scientific, pp. 48-61, 2016.
	Weak Instances of PLWE, with Sean Hallgren, and Kristin Lauter, Selected Areas in Cryptography-SAC 2014, Springer LNCS 8781, pp. 183-194, 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, 1-16, 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, 1141-1162, 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 87-120, 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 471-483, 2010.

	A CRT algorithm for constructing genus 2 curves over finite fields, with Kristin Lauter. Arithmetic, Geometry and Coding Theory (AGCT-10), Séminaires et Congrès 21 (2009), pages 161-176. 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):1951-1959, 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 237-254.
	On the computation of the Cassels pairing for certain Kolyvagin classes in the Shafarevich-Tate group, with Dimitar Jetchev and Kristin Lauter. Pairing-Based Cryptography – Pairing 2008, Springer LNCS volume 5209, 2008, pages 113-125.
	Hilbert’s Tenth Problem for function fields of varieties over number fields and p-adic fields, J. Algebra , 310(2007), pages 775-792.

	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 170-181.
	Hilbert’s Tenth Problem for function fields of varieties over C, Int. Math. Res. Not., Issue 59, 2004, pages 3191-3205.
	Improved Weil and Tate pairings for elliptic and hyperelliptic curves, with Kristin Lauter and Peter L. Montgomery, in ANTS-VI proceedings, 2004, pages 169-183 (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 261-281.
	Fast elliptic curve arithmetic and improved Weil pairing evaluation, with Kristin Lauter and Peter L. Montgomery, Topics in Cryptology – CT-RSA 2003, (Springer LNCS link).

Other articles

The Theorem of Honda and Tate, presented in VIGRE Number Theory working group, December 2004.

Teaching:

All teaching materials are available on Canvas.

In the Fall of 2019 I am teaching Math 570, Elliptic Curves.

In the Spring of 2019 I taught Math 436, Linear Algebra.

In the Fall of 2018 I taught Math 536, Abstract Algebra.

In the Spring of 2018 I taught Math 435, Abstract Algebra, and Math 436, Linear Algebra.

In the Fall of 2017 I taught Math 536, Abstract Algebra.

In the Spring of 2017 I taught a topics course in Number Theory.

In the Fall of 2016 I taught two sections of Math 436, Linear Algebra.

In the Spring of 2016 I taught Math 536, Abstract Algebra.

In the Fall of 2015 I taught Math 567, Number Theory I, and Math 485, Graph Theory.

In the Spring of 2015 I taught Math 567, Number Theory I.

In the Fall of 2014 I taught two sections of Math 436, Linear Algebra.

In the Spring of 2013 I taught Math 536, Abstract Algebra.

In the Fall of 2012 I taught Math 567, Number Theory I.

In the Fall of 2011 I taught Math 311M, Honors Concepts of Discrete Mathematics.

In the Spring of 2011 I taught Math 597C, Topics in Number Theory.

In the Fall of 2010 I taught Math 567, Number Theory I.

In the Spring 2010 semester I taught Math 250, Section 3, Ordinary Differential Equations, and Math 536, Algebra II.

In the Spring 2009 semester I taught Math 568, Number Theory II.

In the Fall 2008 semester I taught Math 497A, Elliptic Curves and Applications to Cryptography.

In the Spring 2008 semester I taught Math 536, Algebra II.

In the Fall 2007 semester I taught Math 485, Graph Theory.