Professor Benedict Gross

Harvard University

**Tuesday, April 7, 2015**

3:30pm, Room 102 Miller Learning Center

*The ranks of elliptic curves*

The study of integral and rational points on elliptic curves, typically given as cubic equations in two variables, goes back to antiquity. I will discuss some of Fermat's work on the subject and will state the theorem of Mordell and Weil on the group of rational points. I will then present a heuristic for the conjecture of Birch and Swinnerton-Dyer on the rank of this group, and discuss the progress we have made, both in constructing rational points and in bounding the rank from above.

**Wednesday, April 8, 2015**

3:30pm, Room 328 Boyd Graduate Studies Building

*The arithmetic of hyperelliptic curves*

Abel began the study of hyperelliptic curves, with his results generalizing those of Euler on elliptic integrals. After Abel, Legendre, Jacobi, and Riemann made decisive progress on the study of the periods of these curves over the real and complex numbers. In this talk, I will consider the arithmetic of hyperelliptic curves defined over the rational numbers. An important theorem of Faltings states that there are only finitely many rational solutions when the genus is at least 2. We will consider the average number of rational solutions, as one varies over families of curves of a fixed genus.

**Thursday, April 9, 2015**

4:00pm, Room 304 Boyd Graduate Studies Building

*Pencils of quadrics*

Quadric hypersurfaces, defined by homogeneous equations of degree 2, are the simplest projective varieties other than linear subspaces. In this talk I will review the theory of quadratic forms over a general field, and discuss the smooth intersection of two quadric hypersurfaces in projective space. The Fano scheme of maximal linear subspaces contained in this intersection is either finite or is a principal homogeneous space for the Jacobian of a hyperelliptic curve. This gives an important tool for the arithmetic study of these curves.

*Refreshments will be served preceding each lecture.*

*There will be a banquet honoring Prof. Gross, the evening of April 7, 2015 details to follow. *

Please contact Julie McEver, julie@math.uga.edu.

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