AAD2015 - Program

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Both talks take place on March 12th, 2015.

Location

Historical building of the Humboldt University
Unter den Linden 6,
Karl WeierstraƟ Lecture Hall (3038)

14:30-15:30 Benedict Gross
(Harvard University)


The arithmetic of hyperelliptic curves

Abstract: Abel began the study of hyperelliptic curves in his attempt to generalize Euler's theory of elliptic integrals. I will recall some of this theory, then turn to the study of rational points for hyperelliptic curves defined over Q.

When the genus is at least two, most hyperelliptic curves have no rational points, and a positive proportion have no rational points over any number field of odd degree. The latter result is joint work with Manjul Bhargava and Jerry Wang, and involves the study of the Jacobian and some of its principal homogeneous spaces.
Coffee
16:00-17:00 János Kollár
(Princeton University)


Quadratic families of elliptic curves and unirationality of degree 1 conic bundles

Abstract: We consider elliptic curves whose coefficients are degree 2 polynomials in a variable t. We prove that for infinitely many values of t the resulting elliptic curve has rank at least 1. All such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers. The main step of the proof is to show that such conic bundles are unirational. Joint work with M. Mella.