# L-series of an elliptic curve

Let $E$ be an elliptic curve^{} over $\mathbb{Q}$ with Weierstrass
equation:

$${y}^{2}+{a}_{1}xy+{a}_{3}y={x}^{3}+{a}_{2}{x}^{2}+{a}_{4}x+{a}_{6}$$ |

with coefficients ${a}_{i}\in \mathbb{Z}$. For $p$ a prime in $\mathbb{Z}$, define ${N}_{p}$ as the number of points in the reduction of the curve modulo $p$, this is, the number of points in:

$$\{O\}\cup \{(x,y)\in \mathbb{F}_{p}{}^{2}:{y}^{2}+{a}_{1}xy+{a}_{3}y-{x}^{3}-{a}_{2}{x}^{2}-{a}_{4}x-{a}_{6}\equiv 0modp\}$$ |

where $O$ is the point at infinity. Also, let ${a}_{p}=p+1-{N}_{p}$. We define the *local part at $p$ of
the L-series* to be:

$${L}_{p}(T)=\{\begin{array}{cc}1-{a}_{p}T+p{T}^{2}\text{, if}E\text{has good reduction at}p,\hfill & \\ 1-T\text{, if}E\text{has split multiplicative reduction at}p,\hfill & \\ 1+T\text{, if}E\text{has non-split multiplicative reduction at}p,\hfill & \\ 1\text{, if}E\text{has additive reduction at}p.\hfill & \end{array}$$ |

###### Definition.

The L-series of the elliptic curve $E$ is defined to be:

$$L(E,s)=\prod _{p}\frac{1}{{L}_{p}({p}^{-s})}$$ |

where the product is over all primes.

Note: The product converges and gives an analytic function^{} for all
$Re(s)>3/2$. This follows from the fact that $\mid {a}_{p}\mid \le 2\sqrt{p}$. However, far more is true:

###### Theorem (Taylor, Wiles).

The L-series $L\mathit{}\mathrm{(}E\mathrm{,}s\mathrm{)}$ has an analytic continuation to the entire
complex plane^{}, and it satisfies the following functional equation.
Define

$$\mathrm{\Lambda}(E,s)={({N}_{E/\mathbb{Q}})}^{s/2}{(2\pi )}^{-s}\mathrm{\Gamma}(s)L(E,s)$$ |

where ${N}_{E}\mathrm{/}\mathrm{Q}$ is the conductor^{} of $E$ and $\mathrm{\Gamma}$ is
the Gamma function^{}. Then:

$$\mathrm{\Lambda}(E,s)=w\mathrm{\Lambda}(E,2-s)\mathit{\hspace{1em}}withw=\pm 1$$ |

The number $w$ above is usually called the *root number* of
$E$, and it has an important conjectural meaning (see Birch and
Swinnerton-Dyer conjecture).

This result was known for elliptic curves having complex
multiplication^{} (Deuring, Weil) until the general result was
finally proven.

## References

- 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 |

Canonical name | LseriesOfAnEllipticCurve |

Date of creation | 2013-03-22 13:49:43 |

Last modified on | 2013-03-22 13:49:43 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 8 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 14H52 |

Synonym | L-function of an elliptic curve |

Related topic | EllipticCurve |

Related topic | DirichletLSeries |

Related topic | ConductorOfAnEllipticCurve |

Related topic | HassesBoundForEllipticCurvesOverFiniteFields |

Related topic | ArithmeticOfEllipticCurves |

Defines | L-series of an elliptic curve |

Defines | local part of the L-series |

Defines | root number |