Laura DeMarco, Holly Krieger, and Hexi Ye are the recipients of the 2020 Alexanderson Award for their paper, “Uniform Manin-Mumford for a family of genus 2 curves,” which was published this year in the Annals of Mathematics. The paper grew out of the AIM SQuaRE, Dynamical Andre-Oort Questions.

In this paper, they study the geometry of surfaces and how each sits inside a 4-dimensional torus, called its Jacobian. There is a large set of special points in this Jacobian, its “torsion” points; but in the early 1980s, Raynaud proved that the surface will intersect only finitely many of these points. This finiteness statement was called the Manin-Mumford conjecture. Soon after, Mazur posed an important question about these surfaces, asking if we can bound the number of the special points on the surface only in terms of its genus (the topology of the surface), or if there is additional geometric information we need to know to control the size of this intersection.

Although not obviously related to number theory, much progress on questions of this sort has been made over the years with tools from algebra and arithmetic geometry. But Mazur’s question about a uniform bound remained elusive.

In this paper, DeMarco, Krieger, and Ye also employ arithmetic tools, but their novel insight was to use ideas from dynamical systems to finally provide a uniform bound on the number of torsion points in a special setting. They study a family of surfaces in genus 2 that map to pairs of elliptic curves (surfaces of genus 1), and they give a positive answer to Mazur’s question for this family. The proofs employ a quantitative version of arithmetic equidistribution to relate the heights (i.e., the arithmetic complexity) of the surfaces to the number of torsion points.

Read the online version of the paper here.

Due to the pandemic there will not be a 2020 Alexanderson Lecture or ceremony, but we hope to have a celebration in the future.

Gerald Alexanderson

A member of the Santa Clara faculty since 1958, Jerry has served his institution and the broader mathematics community in various capacities. During this time, he was chair of the Mathematics department for 35 years and a member of the Faculty Senate Council. For thirty eight years he held the endowed Valeriote Professorship of Science chair. Known as an inspiring teacher and popular author, Alexanderson has cultivated a passion for problem solving and has promoted creative mathematical thinking as longstanding Associate Director of the prestigious William Lowell Putnam Competition. He is author of more than a dozen books, including textbooks in abstract algebra, and discrete and combinatorial mathematics. Alexanderson was the first recipient of Santa Clara University’s Bayma Award for Scholarship, and he received the Special Appreciation Award from the Dean of Arts and Sciences as well as the Special Recognition Award for Teaching, Research, and Service from the President of the university.

Alexanderson’s influence has extended to the national level, where he has played a leading and lasting role in the Mathematical Association of America (MAA). His contributions to the MAA have spanned more than 50 committees and 24 years on the Board of Governors, encompassing secretary, vice-president, and president of the Association and editor of Mathematics Magazine. Results of this work include the remodeling of the MAA Carriage House in Washington, D.C., into its Mathematical Sciences Conference Center. In this time, Jerry served on the Science Policy Committee of the American Mathematical Society (AMS) and was a consultant to the Editorial Board for the Bulletin of the AMS. In testament to his expansive record, Alexanderson received the MAA’s most prestigious award for distinguished service to Mathematics, the Yueh-Gin Gung and Dr. Charles Y. Hu award.