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I was told of a relation between the number of fixed points of the Atkin-Lehner involution and the class number of certain number fields.

Can someone point me to a reference where I could learn about it? (I am not very familiar with CM theory so I don't want a one-line proof without further explanation).

Thanks.

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This might be not very elegant on my side, but you can take a look to section 2 of: www1.iwr.uni-heidelberg.de/groups/arith-geom/centeleghe/up.pdf The idea is to interpret the Atkin-Lehner involution on $X_0(p)$ in terms of moduli of elliptic curves, and then use the relationship between fractional ideals of $Q(\sqrt{-p})$ and certain CM elliptic curves. –  Tommaso Centeleghe Jun 29 '11 at 20:43
    
Hi Tommaso, this might be a silly question but why is \sum = h(-4p) ? Or why is \sum = h(-p) + h(-4p) (in the other case)? –  expmat Jun 30 '11 at 15:02
    
Dear expmat, This is exactly what I'm referring to in my answer. For $X_0(p)$, we must have $D = 1$, $N=p$, $m=p$ and so if $p\equiv 1\bmod 4$, $\mathbf{Z}[\sqrt{-p}]$ is maximal and the number of fixed points is $h(-4p)$. If $p\equiv 3\bmod 4$, an embedding of $\mathbf{Z}[\sqrt{-p}]$ must be optimal either for $\mathbf{Z}[\sqrt{-p}]$ or for $\mathbf{Z}[\dfrac{1 + \sqrt{-p}}{2}]$ so you have to count optimal embeddings of both. Hence the number of fixed points is $h(-p) + h(-4p)$. –  stankewicz Jun 30 '11 at 23:30
    
Regarding the exposition on definite quaternion algebras, you could take a look at Pizer's articles, and I suggest to look at Gross article (Heights and special values of L-series) where he first talks about the optimal embeddings. The idea is that the number of optimal embeddings is roughly speaking the number of bilateral ideals times the class number of the order you are embedding (in case there is such an ideal). So the formula written below gives you $2^t$, where $t$ is the number of prime divisors of ND (the level) which is exactly the number of bilateral ideals (if ND is square free). –  A. Pacetti Jul 1 '11 at 12:02
    
@expmat. If you are interested in fixed points of AL involution on $X_0(p)$ then you should classify elliptic curves over C, up to isomorphism, that admit an endomorphism whose square is -p. If you think about the correspondence between elliptic curves up to isom. and lattices up to homothety then what you are doing is classifying lattices inside C that are stable under multiplication by $\sqrt{-p}$. This leads to the class number interpretation. You will have to consider the class number of all quadratic imaginary orders containing $\sqrt{-p}$. If p\equiv 3 mod 4 then there are two of them. –  Tommaso Centeleghe Jul 2 '11 at 14:28

3 Answers 3

Given that I don't know exactly which relation you're talking about, I'll give you something old and something new:

A priori, asking for a formula for the number of fixed points of Atkin-Lehner is asking for the trace of the matrix representing the Atkin-Lehner involution. Hence you're asking for a trace formula, in particular the Eichler-Selberg trace formula. The original reference for that, featuring many relations between class numbers is

Eichler, M. Modular correspondences and their representations. J. Indian Math. Soc. (N.S.) {\bf 20} (1956), 163-206.

On the other hand a more modern view of fixed points of an Atkin-Lehner involution $w_m$ is that they're in bijection with conjugacy classes of embeddings $\mathbf{Z}[\sqrt{-m}] \hookrightarrow \mathcal{O}_0(N)$, the order used to define the Shimura curve $X^D_0(N)$. You said you wanted me to sweep the CM theory under the rug, so I won't elaborate on Shimura curves.

Anyways, this can by done by counting conjugacy classes of optimal embeddings of either $R = \mathbf{Z}[\sqrt{-m}]$ or $\mathbf{Z}[\dfrac{1 + \sqrt{-m}}{2}]$ into your quaternion order.

For counting these things, probably the book of Vigneras is best, but I like the exposition of Santiago Molina here http://www.crm.es/Publications/10/Pr928.pdf or here http://arxiv.org/PS_cache/arxiv/pdf/0912/0912.5217v4.pdf (same paper in section 2)

In particular let's simplify things and say both that $N$ is squarefree and $\mathbf{Z}[\sqrt{-m}]$ is a maximal order so every embedding is optimal. In this case the number of fixed points of $w_m$ is

$$h(-4m)\prod_{p|D}\left(1 -\left(\dfrac{-4m}{p}\right)\right)\prod_{q|N}\left(1 +\left(\dfrac{-4m}{q}\right)\right)$$

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This is an interesting answer, which I have learned from, but I worry that it is addressing a more difficult question than the intended one. Let me leave a comment that may help: This answer (at least the second part) is about the Atkin-Lehner involution on Shimura curves. My answer is about the Atkin-Lehner involution on modular curves, which are older objects, and are the ones which are most immediately related to modular forms. –  David Speyer Jul 2 '11 at 14:23

I learned from Harald Helfgott's thesis (on the arxiv here) that this connection between class numbers and the Fricke involution goes back to Fricke. See his Appendix A.4.

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Let me try to explain the CM connection, as I think it really is the most intuitive way to understand this. Fear not, this is not a one line answer!

I'll be working with $\mathbb{C}$ throughout. Every elliptic curve over $\mathbb{C}$ is of the form $\mathbb{C}/\Lambda$, where $\Lambda$ is a discrete rank two sublattice of $\mathbb{C}$. This description is not unique: If $\alpha$ is any nonzero complex number, then $\mathbb{C}/\Lambda$ and $\mathbb{C}/\alpha \Lambda$ define the same elliptic curve. If we have a two elliptic curves, $\mathbb{C}/\Lambda_1$ and $\mathbb{C}/\Lambda_2$, and a map $\phi$ between them, then there is a unique complex number $\beta$ such that $\beta \Lambda_1 \subseteq \Lambda_2$ and $\phi$ arises as the map which takes the coset $z+\Lambda_1$ to the coset $\beta z + \Lambda_2$.


I'll first explain the basic idea of complex multiplication, and then talk about the Atkin-Lehner case. Complex multiplication is all about describing the possible self maps of an elliptic curve. Consider a complex number $\beta$ and a lattice $\Lambda$. When does multiplication by $\beta$, as a map from $\mathbb{C}$ to $\mathbb{C}$, descend to a map from $\mathbb{C}/\Lambda$ to $\mathbb{C}/\Lambda$? This happens, if and only if $\beta \Lambda \subseteq \Lambda$.

Now, let's fix $\beta$ and consider which $\Lambda$ have this property. Notice that, if $\lambda$ is a nonzero element of $\Lambda$, and $\theta$ is any element in the ring $\mathbb{Z}[\beta]$, then $\theta \lambda$ is in $\lambda$. This means that $\mathbb{Z}[\beta] \cdot \lambda$ must form a discrete sublattice of $\Lambda$, so the ring $\mathbb{Z}[\beta]$ must be a discrete sublattice of $\mathbb{C}$.

Case 1: $\mathbb{Z}[\beta]$ is a rank $1$ sublattice of $\mathbb{C}$. In this case, $\beta$ is in $\mathbb{Z}$ and $\beta \Lambda \subseteq \Lambda$ for every $\Lambda$.

Case 2: $\mathbb{Z}[\beta]$ is a rank $2$ sublattice of $\mathbb{C}$. In this case (this is not obvious) $\mathbb{Z}[\beta]$ must either be of the form $\mathbb{Z}[\sqrt{-d}]$ or $\mathbb{Z}[(1+\sqrt{-d})/2]$, where $d>0$ and, in the latter case, $d$ must be $3 \mod 4$. In this case, there are finitely many lattices $\Lambda$ such that $\beta \Lambda \subseteq \Lambda$ (up to treating $\Lambda$ and $\alpha \Lambda$ as equivalent, as mentioned in the second paragraph.). The number of these lattices is more or less the class number of $\mathbb{Q}[\sqrt{-d}]$. (It is exactly this if $d$ is square free and $1$ or $2$ mod $4$; otherwise there are some details to fix up.)

Case 3: $\mathbb{Z}[\beta]$ is not a discrete sublattice of $\mathbb{C}$. As discussed above, in this case there are no $\Lambda$'s for which $\beta \Lambda \subseteq \Lambda$.


Now, for the Atkin-Lehner connection. The modular curve $Y_0(p)$ (the one without the cusps) parameterizes ordered pairs $(\Lambda_1, \Lambda_2)$, where $\Lambda_2$ is an index $p$ sublattice of $\Lambda_1$, and where $(\Lambda_1, \Lambda_2)$ is identified with $(\alpha \Lambda_1, \alpha \Lambda_2)$ for any nonzero complex number $\alpha$.

The Atkin-Lehner involution sends $(\Lambda_1, \Lambda_2)$ to $(\Lambda_2, p \Lambda_1)$. So a fixed point of Atkin-Lehner must correspond to a pair $(\Lambda_1, \Lambda_2)$ such that $(\Lambda_2, p \Lambda_1) = (\alpha \Lambda_1, \alpha \Lambda_2)$ for some $\alpha$. In particular, $$p \Lambda_1 = \alpha (\alpha \Lambda_1) = \alpha^2 \Lambda_1.$$

Set $\gamma = \alpha^2/p$. Then both $\gamma$ and $\gamma^{-1}$ take $\Lambda_1$ to itself, so both $\mathbb{Z}[\gamma]$ and $\mathbb{Z}[\gamma^{-1}]$ are discrete lattices. Looking at the case by case analysis above, one works out that $\gamma$ is one of $1$, $-1$, $\pm i$, $\pm e^{2 \pi i/6}$ and $\pm e^{4 \pi i/6}$. Now, $\alpha = \sqrt{p \gamma}$ and we have $\alpha \Lambda_1 = \Lambda_2 \subset \Lambda_1$. So $\sqrt{p \gamma}$ must also generate a discrete sublattice of $\mathbb{C}$. Looking at the previous list of cases, the only one that survives is $\gamma = -1$ and $\alpha = \sqrt{-p}$.

So the fixed points of Atkin-Lehner come from lattices $\Lambda_1$ such that $\sqrt{-p} \Lambda_1 \subset \Lambda_1$; for each such lattice $\Lambda_1$ we get the fixed point $(\Lambda_1, \sqrt{-p} \Lambda_1)$. Using the previous discussion, the number of such $\Lambda_1$'s is essentially the class number of $\mathbb{Q}(\sqrt{-p})$.

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