Alexander Smith (mathematician)

Alexander Smith (born March 4, 1993 in Massachusetts ) is an American mathematician who studies number theory.

In 2019 he received the first David Goss Prize in number theory.

He received the award for outstanding work on - class groups of imaginary-quadratic number fields (he showed their distribution according to the Cohen-Lenstra heuristic ) and - Selmer groups of quadratic twists of elliptic curves (he showed - in the case of full rational 2-torsion and none rational cyclic subgroups of order 4 - that they satisfy the Delaunay heuristic and Goldfeld's conjecture). Specifically, he showed: ${\displaystyle 2^{\infty }}$ ${\displaystyle 2^{\infty }}$

Let E be an elliptic curve over the rational numbers with full rational 2-torsion and without a rational cyclic subgroup of order 4. Then, assuming the conjecture of Birch and Swinnerton-Dyer, Dorian Goldfeld's conjecture : have half the quadratic twists of E analytical rank 0, the other half rank 1, while the portion with a higher analytical rank disappears asymptotically (with respect to the upper limit x for the twists). As a corollary he was able to show that the positive square-free integers are almost all non-congruent. ${\displaystyle n=1,2,3\mod 8}$

In addition, his work on congruent numbers as a student at Princeton University was highlighted (Undergraduate Senior Thesis with Shou-Wu Zhang ). He proved that at least 55.9 percent of all square-free positive integers are congruent numbers. At the same time, he also showed that Birch and Swinnerton-Dyer's conjecture applies to at least an equal proportion of special elliptic curves whose connection to the problem of congruent numbers was uncovered by Jerrold Tunnell . ${\displaystyle n=5,6,7\mod 8}$

He is (2019) a PhD student at Harvard University .

Fonts (selection)

• The congruent numbers have positive natural density, 2016, Arxiv
• ${\displaystyle 2^{\infty }}$- Selmer groups, - class groups and Goldfeld's conjecture, 2017, Arxiv ${\displaystyle 2^{\infty }}$

Individual evidence

1. ↑ Dorian Goldfeld, Simone Munao, Alexander Smith wins the first David Goss Prize in Number Theory, Notices of the AMS, Volume 66, 2019, No. 11, pp. 1875f