Current Topics Seminar
| Speaker | Robert Miller |
| Title | The PHS Method for Elliptic Curves |
| Date | May 7 4:00pm PDL C-36 |
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An elliptic curve (over Q) is a cubic algebraic curve with a group
law. One way of studying the arithmetic of an elliptic curve E is by
studying its Weil-Châtelet group. This is the set of principal
homogeneous spaces (PHS) C/E, which are curves which act on the
elliptic curve in a simply transitive, algebraic way. Of particular
interest are the elements of the Weil-Châtelet group which possess
points defined over every localization Qp and R, and yet are
insoluble over Q (even though the converse is always true, this can
and does happen). These form the Shafarevich-Tate group, a group whose
finiteness is not known in general, but whose order is conjecturally
given in all cases by Birch and Swinnterton-Dyer. PHS computations can
be used to verify this conjecture in many specific cases.
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