311 citations · 1.2k across the 55 of their papers we have counts for
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Shimura curves for level-3 subgroups of the (2,3,7) triangle group, and some other examples
Noam D. Elkies
We determine the Shimura modular curve X_0(3) and the Jacobian of the Shimura modular curve X_1(3) associated with the congruence subgroups Gamma_0(3), Gamma_1(3) of the (2,3,7) tr…
Elliptic Curves x^3 + y^3 = k of High Rank
Noam D. Elkies, Nicholas F. Rogers
We use rational parametrizations of certain cubic surfaces and an explicit formula for descent via 3-isogeny to construct the first examples of elliptic curves E_k: x^3 + y^3 = k o…
Elliptic curves of large rank and small conductor
Noam D. Elkies, Mark Watkins
For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to ov…
Sylvester-Gallai Theorems for Complex Numbers and Quaternions
Noam Elkies, Lou M. Pretorius, Konrad J. Swanepoel
A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai Theo…