Current Topics Seminar
| Speaker | Robert Bradshaw |
| Title | Elliptic Curves and the Birch and Swinnerton-Dyer conjecture |
| Date | April 24 4:00pm PDL C-36 |
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Elliptic curves are rich algebraic objects that often come up in the study
of number theory. For example, Wiles was able to finally prove Fermat's Last
Theorem by showing certain elliptic curves are all attached to modular
forms. In the early 1900's Mordell proved that the rational points on an
elliptic curve E form a finitely-generate abelian group, and though the
torsion part is well understood the rank is somewhat mysterious. On the
other hand, one can compute the number of points on E over every finite
field of prime order to construct the so-called L-function attached to E,
which is a complex analytic function that extends to the entire complex
plane. Numerical and heuristic evidence suggests that behavior of L at s=1
is intimately tied to the rank of the curve (namely it has order vanishing
equal to the rank) and there is even a conjectural formula for the
coefficient of the first non-vanishing term of the Taylor expansion about
s=1 in terms of algebraic invariants of the curve.
There are also p-adic analogues of some of the quantities involved in the BSD conjecture, and a p-adic BSD conjecture by Mazur, Tate, and Teitelbaum that is analogous to the classical one. Time permitting I will talk about the p-adic case as well.
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