Chapter 24 Elliptic curves and their \(L\)-functions
Section 24.1 Elliptic curves and their \(L\)-functions
An elliptic curve over a field \(K\) is a nonsingular cubic plane curve with a distinguished \(K\)-rational point. (Warning: in general, the existence of such a point is a nontrivial condition.) If \(K\) has characteristic \(>2\text{,}\) any elliptic curve can be put in the form \(y^2 = P(x)\text{,}\) where \(P(x) = x^3 + ax^2 + bx + c\) is a polynomimal with no repeated roots and the distinguished point is the unique point at infinity on this curve in the projective plane. (This is a haphazard summary of the accurate definition, for which see [24].)
If \(E: y^2 = P(x)\) is an elliptic curve over \(\QQ\text{,}\) then for all but finitely many primes \(p\text{,}\) the reduction of \(E\) modulo \(p\) is a nonsingular cubic, and hence elliptic curve over \(\FF_p\text{.}\) (Warning: the finite set of bad primes depends on the choice of the equation \(y^2 = P(x)\text{,}\) not just on the isomorphism class of \(E\text{.}\) There is an optimal choice of the defining equation, but we won't use that here.) We define the \(L\)-function of \(E\) as the product
\begin{equation*}
L(E,s) = \prod_p (1 - a_p p^{-s} + p^{1-2s})^{-1}
\end{equation*}
over all of the nonexceptional primes, where \(p + 1 - a_p\) is the number of points on \(E\) modulo \(\FF_p\text{.}\) (Remember that \(E\) is being drawn in the projective plane, so you have to count the one point \([0:1:0]\) at infinity.)
It's obvious that there are at most \(2p+1\) points on \(E\text{,}\) two for each possible \(x\)-coordinate plus one point at infinity; that implies that \(L(E,s)\) converges absolutely for \(\Real(s) > 2\text{.}\) Actually one can do better.
In a few cases, the \(a_p\) exhibit predictable behavior. For instance, if \(E\) is the curve \(x^3 + y^3 = 1\text{,}\) then for \(p \equiv 2 \pmod{3}\text{,}\) we have \(a_p = 0\text{,}\) whereas for \(p \equiv 1 \pmod{3}\text{,}\) we can write \(a_p\) in terms of integers \(A,B\) for which \(A^2 + 3B^2 = p\) (this was first observed by Gauss). In most cases, however, no such easy formula exists.
By contrast, suppose \(E\) is a nonsingular conic curve passing through at least one \(\QQ\)-rational point; then \(\#E(\FF_p) = p+1\) always. (The points on the curve are in bijection with the lines through the given point.)
Theorem 24.2.
The function \(L(E,s)\) extends to a holomorphic function on \(\CC\text{,}\) satisfying a functional equation between \(s\) and \(2-s\text{.}\)This is a consequence of the modularity of elliptic curves. This theorem is the result of work of Wiles, Taylor–Wiles, Diamond, Fujiwara, Conrad–Diamond–Taylor, and Breuil–Conrad–Diamond–Taylor. (Whew!) When combined with a theorem of Ribet (part of Serre's conjecture), the modularity of elliptic curves (actually just the special case proved by Wiles) resolves the Fermat problem; see [2].
There is also an amazing conjecture relating the \(L\)-function to the \(\QQ\)-rational points of \(E\text{.}\) (The Mordell–Weil theorem states that \(E(\QQ)\) is a finitely generated abelian group.)
Conjecture 24.3. Birch–Swinnerton-Dyer.
The order of vanishing of \(L(E,s)\) at \(s=1\) equals the rank of the finitely generated abelian group \(E(\QQ)\text{.}\)This is known by work of Kolyvagin, Kato, Gross-Zagier, etc. in case the order of vanishing is 0 or 1.
