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My research is in algorithmic number theory and arithmetic geometry. My work is supported by the Luce Foundation, the National Science Foundation, and the Simons Foundation.

I am part of the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation.

There are currently two Ph.D. students working with me.

**Teaching**

Fall 2021: MA 841 (Greatest hits in arithmetic geometry)

Spring 2021: MA 242 (Linear algebra)

Fall 2020: MA 841 (Greatest hits in number theory)

Spring 2020: MA 124 (Calculus II)

**Editorial/board positions**

Quarterly Journal of Mathematics

Journal de Théorie des Nombres de Bordeaux

Research in Number Theory (Editor-in-Chief)

Number Theory Foundation (Board of Directors)

ICERM (Scientific Advisory Board)

**Publications and preprints**

- Ogg's Torsion conjecture: Fifty years later (with B. Mazur and an appendix by N. Dogra), 2023.
- Quadratic Chabauty for modular curves: Algorithms and examples (with N. Dogra, J. S. Müller, J. Tuitman, J. Vonk),
*Compositio Mathematica*,**159**, no. 6, June 2023, 1111-1152. - Computational tools for quadratic Chabauty (with J. S. Müller; last updated April 2023).
- Variants of Lehmer's speculation for newforms (with W. Craig, K. Ono, W.-L. Tsai),
*Advances in Mathematics*, Volume 428, 1 September 2023, 109-141. (Corrigendum) - A tale of three curves,
*Snapshots of modern mathematics from Oberwolfach*, 2022. - Even values of Ramanujan's tau-function (with K. Ono and W.-L. Tsai),
*La Matematica*,**1**, 395-403 (2022). - Variations of Lehmer's Conjecture for Ramanujan's tau-function (with W. Craig, K. Ono),
*Journal of Number Theory*(Prime), 237 (2022), 3-14. - Two recent
*p*-adic approaches towards the (effective) Mordell conjecture (with A. J. Best, F. Bianchi, B. Lawrence, J. S. Müller, N. Triantafillou, J. Vonk),*Arithmetic L-functions and differential geometric methods*, 2021, 31-74. - Explicit quadratic Chabauty over number fields (with A. Besser, F. Bianchi, J. S. Müller),
*Israel Journal of Mathematics*, (2021), 1-48. - What is...a Coleman integral?,
*Notices of the AMS*,**66**, no. 5 (2019), 738-740. -
Chabauty-Coleman experiments for genus 3 hyperelliptic curves (with F. Bianchi, V. Cantoral-Farfán, M. Çiperiani, A. Etropolski),
*Research Directions in Number Theory*, Association for Women in Mathematics Series, Vol. 19, Springer, 2019, 67-90. - An effective Chabauty-Kim theorem (with N. Dogra),
*Compositio Mathematica*,**155**, no. 6 (2019), 1057-1075. - Explicit Chabauty-Kim for the split Cartan modular curve of level 13 (with N. Dogra, J. S. Müller, J. Tuitman, J. Vonk),
*Annals of Mathematics*,**189**, No. 3 (May 2019), 885-944. - Explicit Coleman integration for curves (with J. Tuitman),
*Mathematics of Computation*,**89**(2020), 2965-2984. - Quadratic Chabauty and rational points II: Generalised height functions on Selmer varieties (with N. Dogra),
*International Mathematics Research Notices*, (2020), rnz362. - Quadratic Chabauty and rational points I:
*p*-adic heights (with N. Dogra and an appendix by J. S. Müller),*Duke Mathematical Journal*,**167**, no. 11 (2018), 1981-2038. - Constructing genus 3 hyperelliptic Jacobians with CM (with S. Ionica, K. Lauter, C. Vincent),
*LMS Journal of Computation and Mathematics***19**Issue A (Algorithmic Number Theory Symposium XII), (2016), 283-300. - Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks (with W. Ho, N. Kaplan, S. Spicer, W. Stein, J. Weigandt),
*LMS Journal of Computation and Mathematics***19**Issue A (Algorithmic Number Theory Symposium XII), (2016), 351-370. - Computing integral points on hyperelliptic curves using quadratic Chabauty (with A. Besser, J. S. Müller),
*Mathematics of Computation***86**(2017), 1403-1434. - Shadow lines in the arithmetic of elliptic curves (with M. Çiperiani, J. Lang, B. Mirza, R. Newton),
*Directions in Number Theory: Proceedings of the WIN3 Workshop*, Springer, 2016, 33-55. - Explicit
*p*-adic methods for elliptic and hyperelliptic curves,*Advances on Superelliptic Curves and their Applications*, IOS Press, 2015, 260-285. - Quadratic Chabauty:
*p*-adic height pairings and integral points on hyperelliptic curves (with A. Besser, J. S. Müller),*Journal für die reine und angewandte Mathematik*,**720**(2016), 51-79. - A
*p*-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties (with J. S. Müller, W. Stein),*Mathematics of Computation*,**85**(2016), 983-1016. - A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves
(with I. Dan-Cohen, M. Kim, S. Wewers),
*Mathematische Annalen*,**372**(2018), no. 1-2, 369-428. (Errata) - On 3-adic heights on elliptic curves,
*Journal of Number Theory*(Special issue - in honor of Wen-Ching Winnie Li),**161**(2016), 119-134. - Coleman integration for even degree models of hyperelliptic curves,
*LMS Journal of Computation and Mathematics*,**18**(2015), 258-265. *p*-adic heights of Heegner points and Lambda-adic regulators (with M. Çiperiani, W. Stein),*Mathematics of Computation*,**84**(2015), 923-954.- Comparing arithmetic intersection formulas for
denominators of Igusa class polynomials (with J. Anderson,
K. Lauter, J. Park, B. Viray),
*Women in Numbers 2: Research Directions in Number Theory*, Contemporary Mathematics,**606**, AMS, 2013, 65-82. - Coleman-Gross height pairings
and the
*p*-adic sigma function (with A. Besser),*Journal für die reine und angewandte Mathematik*,**698**(2015), 89-104. - Iterated Coleman integration for hyperelliptic curves,
*ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium*, Open Book Series**1**, MSP, 2013, 41-61. - Computing
local
*p*-adic height pairings on hyperelliptic curves (with A. Besser),*International Mathematics Research Notices*(2012)**2012**(11): 2405-2444. (Errata) - Appendix and erratum to Minhyong Kim, "Massey products for elliptic curves of rank 1" (with K. Kedlaya, M. Kim),
*Journal of the American Mathematical Society***24**(2011), 281-291. - Pairings on hyperelliptic curves (with J. Belding, S. Chisholm, K. Eisentraeger, K. Stange, E. Teske),
*WIN - Women in Numbers*, Fields Institute Communications,**60**, AMS, 2011, 87-120. - Explicit Coleman integration for hyperelliptic curves (with R. Bradshaw, K. Kedlaya),
*Algorithmic Number Theory (ANTS-IX)*, LNCS**6197**, Springer, 2010, 16-31.

**Recent and Upcoming Events**

ANTS XVI, July 15 - 19, 2024 at MIT, co-organized with Kiran Kedlaya, Andrew Sutherland, and John Voight

The Mordell conjecture 100 years later, July 8 - 12, 2024 at MIT, co-organized with Philipp Habegger, Bjorn Poonen, Andrew Sutherland, and Wei Zhang

GeMsGetMath at BU (for high school students), August 13 - 18, 2023 at Boston University, co-organized with Margaret Beck

Diophantine Geometry, Spring 2023 at MSRI, co-organized with Mirela Çiperiani, Philipp Habegger, Wei Ho, Hector Pasten, Yunqing Tang, Shou-Wu Zhang

Park City Mathematics Institute 2022: Number Theory Informed by Computation, July 17 - August 6, 2022, Park City, co-organized with Bjorn Poonen and Akshay Venkatesh

Connecticut Summer School and Conference in Number Theory 2022, June 6 - 12, 2022, University of Connecticut, co-organized with Keith Conrad, Álvaro Lozano-Robledo, and Christelle Vincent

(Virtual) Park City Mathematics Institute 2021 Graduate Summer School (Week 3: Number Theory Informed by Computation), July 26 - 30, 2021, co-organized with Bjorn Poonen and Akshay Venkatesh

Rational Points and Galois Representations, May 10 - 12, 2021, co-organized with Netan Dogra and Carl Wang-Erickson, with JSB and Brian Lawrence as scientific organizers

Connecticut Summer School and Conference in Number Theory, June 8 - 14, 2020, co-organized with Keith Conrad, Álvaro Lozano-Robledo, and Christelle Vincent

Boston University-Keio University workshop 2019, June 24 - 28, 2019 at Boston University, co-organized with Ali Altug, Steve Rosenberg, and Kenichi Bannai

CMI-HIMR Summer School in Computational Number Theory, June 17 - 28, 2019 at the University of Bristol, co-organized with Tim Dokchitser

Arithmetic of Low-Dimensional Abelian Varieties, June 3 - 7, 2019 at ICERM, co-organized with Noam Elkies, Brendan Hassett, Bjorn Poonen, Andrew Sutherland, and John Voight

Arithmetic Geometry, Number Theory, and Computation, August 20 - 24, 2018 at MIT, co-organized with Noam Elkies, Brendan Hassett, Bjorn Poonen, Andrew Sutherland, and John Voight

WIN4: Women in Numbers 4, August 13 - 18, 2017 at the Banff International Research Station, co-organized with Chantal David, Michelle Manes, and Bianca Viray

Sage, free, open-source mathematical software