L-series of an elliptic curve
where is the point at infinity. Also, let . We define the local part at of the L-series to be:
The L-series of the elliptic curve is defined to be:
where the product is over all primes.
Note: The product converges and gives an analytic function for all . This follows from the fact that . However, far more is true:
Theorem (Taylor, Wiles).
This result was known for elliptic curves having complex multiplication (Deuring, Weil) until the general result was finally proven.
- 1 James Milne, Elliptic Curves, http://www.jmilne.org/math/CourseNotes/math679.htmlonline course notes.
- 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
- 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
- 4 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.
|Title||L-series of an elliptic curve|
|Date of creation||2013-03-22 13:49:43|
|Last modified on||2013-03-22 13:49:43|
|Last modified by||alozano (2414)|
|Synonym||L-function of an elliptic curve|
|Defines||L-series of an elliptic curve|
|Defines||local part of the L-series|