Starts 24 Jun 2016 15:30
Ends 24 Jun 2016 17:00
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Class numbers of imaginary quadratic fields
ICTP
Leonardo Building - Luigi Stasi Seminar Room
Abstract:
The class number problem of Gauss inquires about the class number (denoted h) of an imaginary quadratic field, which is one measure of the failure of unique factorization. He found there were 9 fields with class number 1 (the last being Q(√-163), and conjectured there were no more. This case of h=1 was resolved by very different methods by Heegner and Baker (who won the Fields Medal) and Stark in the 60s. Soon after, Goldfeld proposed a plan to effectively show that h→∞, his idea being dependent on the existence of an elliptic curve having analytic rank 3 (and meeting other minor technical conditions). The proof of this was then completed in the 80s by Gross and Zagier, involving a height formula for Heegner points. We discuss an improvement to Goldfeld's theorem, showing that an elliptic curve of analytic rank 2 suffices.
The class number problem of Gauss inquires about the class number (denoted h) of an imaginary quadratic field, which is one measure of the failure of unique factorization. He found there were 9 fields with class number 1 (the last being Q(√-163), and conjectured there were no more. This case of h=1 was resolved by very different methods by Heegner and Baker (who won the Fields Medal) and Stark in the 60s. Soon after, Goldfeld proposed a plan to effectively show that h→∞, his idea being dependent on the existence of an elliptic curve having analytic rank 3 (and meeting other minor technical conditions). The proof of this was then completed in the 80s by Gross and Zagier, involving a height formula for Heegner points. We discuss an improvement to Goldfeld's theorem, showing that an elliptic curve of analytic rank 2 suffices.