Research

I recently started working on rank growth/stability of (hyper)elliptic curves over towers of number fields. In the early 2000s, Poonen published a paper linking rank stability of elliptic curves over number filed extensions with Hilbert's Tenth Problem. Many people studied rank growth/stability of elliptic curves over number field extensions, including, most recently, Kundu, Lei, and Sprung. Finally, Alpöge, Bhargava, Ho, and Shnidman recently generalized this work to abelian varieties. 

My PhD research was on Brauer-Manin obstructions (BMOs) to Weak approximation on certain Kummer surfaces. In particular, products of complex multiplication elliptic curves.We discussed BMO for most CM cases, especially the obstruction coming from the transcendental part of the Brauer group, other than the case when we have CM by Z[i], which has been dealt with by Ieronymou and Skorobogatov. Before that, I worked on rational points on elliptic curves, particularly on the ranks of elliptic curves over the field of rationals. 

Publications

• M. Alaa Tawfik, R. Newton. Transcendental Brauer–Manin obstructions on singular K3 surfaces, Res. number theory 11, 16 (2025).
  https://doi.org/10.1007/s40993-024-00580-z 

• M. Alaa, M. Sadek. High rank quadratic twists of pairs of elliptic curves, Journal of Number Theory, (174), 2017, 436 - 444. https://www.sciencedirect.com/science/article/pii/S0022314X17300203 

• M. Alaa, M. Sadek. On geometric progressions on hyperelliptic curves, Journal of Integer Sequences, Volume 19 (2016), Article 16.6.3.  https://cs.uwaterloo.ca/journals/JIS/VOL19/Sadek/sadek3.pdf 

Master's Thesis

Elliptic Surfaces with positive Mordell-Weil rank and Quadratic twists of elliptic curves. Cairo University, 2017.