### About me

- I'm a Professor of Mathematics at the University of Connecticut.
- Information about CTNT - Connecticut Summer School in Number Theory.
- A video survey of my research area at CIRM (Luminy, France), and my VaNTAGe talk on ranks of elliptic curves.
- Books that I have published: AMS books, see also this page about "Number Theory and Geometry."
- My YouTube channel (mini-courses and lectures).
- My Blog: A Field Guide To Mathematics.
- Álvaro in the news.

### Contact Info

Phone: | (860) 486 3850 |
---|---|

E-mail: | alvaro.lozano-robledo@uconn.edu |

Address: | MONT 233 Department of Mathematics University of Connecticut 341 Mansfield Road U1009 Storrs, CT 06269-1009 USA |

Elliptic curve addition and doubling of rational points.

### Selected Publications

**Books **:

- Number Theory and Geometry: An Introduction to Arithmetic Geometry, AMS, Pure and Applied Undergraduate Texts, Book 35, 2019.
- Elliptic Curves, Modular Forms and Their L-functions, AMS, Student Mathematical Library, Vol. 58, 2011.

**Selected Articles ** (all preprints and offprints are available here):

*Galois representations attached to elliptic curves with complex multiplication*, in**Algebra and Number Theory**, 16:4 (2022). (arXiv preprint: 1809.02584.)*A probabilistic model for the distribution of ranks of elliptic curves over Q*, in**Journal of Number Theory**, Volume 221, April 2021, 270-338. (arXiv preprint: 1611.01999; see also my VaNTAGe talk on this topic.)*Uniform boundedness in terms of ramification*, in**Research in Number Theory,**2018, 4:6.*Torsion subgroups of rational elliptic curves over the compositum of all cubic fields*(with Harris Daniels, Filip Najman, and Andrew Sutherland), arXiv preprint: 1509.00528 (current version corrects an error in the proofs of Lemma 3.2 and Theorem 3.6), in**Mathematics of Computation**, 87 (2018), 425-458.*On the minimal degree of definition of p-primary torsion subgroups of elliptic curves*(with Enrique González-Jiménez), in**Mathematical Research Letters**., Vol 24 (2017), Num. 4, pp. 1067-1096. Magma scripts.(with Harris Daniels), in the*What is... an elliptic curve*?**Notices of the American Mathematical Society**., Vol 64, Issue 3, March 2017, pp. 241-243. (Published version)*Elliptic curves with abelian division fields*(with Enrique González-Jiménez), in**Mathematische Zeitschrift,**August 2016, Volume 283, Issue 3, pp. 835–859.*Ramification in the division fields of elliptic curves with potential supersingular reduction*, in**Research in Number Theory**2016**2**:8, pp. 1-25.*Division fields of elliptic curves with minimal ramification*, in the**Revista Matematica Iberoamericana,**Volume 31, Issue 4, 2015, pp. 1311–1332.*On the field of definition of p-torsion points on elliptic curves over the rationals*, in the**Mathematische Annalen**, Vol 357, Issue 1 (2013), pp. 279-305.*On products of quadratic twists and ranks of elliptic curves over large fields*(joint with Bo-Hae Im), in**Journal of the London Mathematical Society**, Vol 79, part 1, Feb. 2009, pp. 1-14.