L-series of CM Elliptic Curves


This will be the last post in the CM elliptic curve series. Last time we covered the main theorem of complex multiplication. Today we’ll very, very briefly sketch one amazing use of the main theorem. We’ll first talk about how to associate a Grössencharacter to a CM elliptic curve and then use this to better describe the {L}-series of an elliptic curve.

Here’s why this will be amazing. Awhile ago we talked about {L}-series of varieties and various modularity conjectures. One of the huge, major theorems of modern number theory (which wasn’t proved in full until 2003, and built on all of Wiles and Taylor’s results) is the so-called Modularity Theorem. It says that an elliptic curve {E/\mathbb{Q}} is modular and hence its {L}-series has an analytic continuation to all of {\mathbb{C}}.

This is still open (as far as I know) for elliptic curves over an arbitrary number field, but today we’ll see that we can use the theory we’ve built to show that any CM elliptic curve over any number field has an {L}-series that analytically continues to the whole plane.

Fix {E/L} an elliptic curve over a number field with CM by {R_K}, the ring of integers in a quadratic imaginary field {K}. We use the main theorem of CM to do the following. Fix an idele {x\in \mathcal{J}_L} and let {s=N_{L/K}(x)\in\mathcal{J}_L}. There is a unque {\alpha\in K^*} such that {\alpha R_K=(s)} and for any fractional {\frak{a}} in {K} and any analytic iso {f: \mathbb{C}/\frak{a}\stackrel{\sim}{\rightarrow} E(\mathbb{C})} we get a commutative diagram:

\displaystyle \begin{matrix} K/\frak{a} & \rightarrow & K/\frak{a} \\ \downarrow & & \downarrow \\ E(L^{ab}) & \rightarrow & E(L^{ab}) \end{matrix}

What this gives us is a map {\alpha_{E/L}: \mathcal{J}_L\rightarrow K^*\subset\mathbb{C}^*}. Recall that a Grössencharacter of a number field {L} is such a map that is trivial on {L^*}. We can alter this to the map {\Psi_{E/L}:\mathcal{J}_L\rightarrow \mathbb{C}^*} by {\Psi_{E/L}(x)=\alpha_{E/L}(x)(N_{L/K}(x^{-1}))_\infty}. It turns out this is our desired Grössencharacter.

Recall how we formed the L-series of a variety over {\mathbb{Q}}. There is nothing special about {\mathbb{Q}} going on in that constuction and so the same thing can be done for any elliptic curve {E/L} where {L} is a number field. Basically you piece it together as a product over primes of some expression involving the characteristic polynomial of the Frobenius elements acting on the cohomology of the reductions of {E} mod these primes.

Given a Grössencharacter {\Psi: \mathcal{J}_L\rightarrow \mathbb{C}^*} we can define the Hecke {L}-series to be {\displaystyle L(s,\Psi)=\prod_{\frak{p}}(1-\Psi(\frak{p})q_\frak{p}^{-s})^{-1}}, where {q_\frak{p}} is the size of the residue field at {\frak{p}}. Hecke proved that this {L}-series has an analytic continuation to the complex plane.

Duering proved that if {E/L} is an elliptic curve with CM by {R_K}, then {L(E/L, s)=L(s, \Psi_{E/L})L(s, \overline{\Psi_{E/L}})}. In fact, even better is that if {K} is not contained in {L}, then the {L}-series of the elliptic curve over {L} is precisely the Hecke {L}-series of the Grössencharacter attached to the base-changed elliptic curve {E/KL}. In either case, we see that it is much easier to prove that the {L}-series of an elliptic curve with CM by the ring of integers in a quadratic imaginary field has an analytic continuation.

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