## Quaternion Algebras and Modular Forms

I know I promised a post  on modular curves, but I had to devote more time to my end of semester project. Since it’s strongly related to the topic of modular curves and I present on it tomorrow, I decided to make this post in the seminar talks category.

First we say something about modular forms and modular curves, but maybe with a bit of added generality. Consider a discrete subgroup $\Gamma$ of $SL_2(\mathbf{R})$. By the action of $M_2(\mathbf{C})$ on $\mathbb{P}^1(\mathbf{C})$, $\Gamma$ acts on the upper half plane $\mathcal{H}$ since it must preserve $\mathbb{P}^1(\mathbf{R})$ and not switch the upper and lower half-planes.

A modular form for $\Gamma$ of weight $k$ is then a holomorphic function $f$ on the upper-half plane which satisfies two properties:

• If $\gamma =\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\in \Gamma$ then $f(\gamma z) = (cz + d)^kf(z)$.
• $f$ is holomorphic at limiting points called cusps.

We pay little attention to the second condition in our case of modular forms on quaternion algebras. In general, a modular form will have a wealth of information found in the limiting points, which give a Fourier expansion for the modular form. In our case, we will be dealing with modular forms which live on compact manifolds.

One might ask oneself here, what do modular forms have to do with manifolds? The answer is that we can look at the upper half plane modulo the action of the discrete group $\Gamma$ and by adding the cusps to this quotient manifold, we get a compact complex 1-manifold(or if you prefer, a compact Riemann Surface). The modular forms of weight $2n$ which vanish at all cusps play a very important role as if $f$ is such a “cusp form” then $f(z)\mathrm{d}z^n$ is a holomorphic differential on $\Gamma \setminus \mathcal{H}^*$(or if you prefer, a global section of the sheaf of holomorphic differentials). Indeed the map described gives an isomorphism of complex vector spaces.

So now what are quaternion algebras and what can they do for us? Well there’s the correct definition which I will give in my talk tomorrow and I will surely revisit here at some point when I want to talk about the Brauer group and the “good enough” definition which is correct as long as the characteristic of your field is not 2, which I give now.

Definition:A quaternion algebra over a field $k$ of characteristic not 2 is a 4-dimensional algebra $(a,b)_k = k \oplus ik \oplus jk \oplus ijk$ such that $i^2 =a\in k, j^2 = b\in k, ij = -ji$.

Example: The only quaternion algebra you are guaranteed to have over any field $k$ is the 2×2 matrices $M_2(k)$.

Defining a quaternion algebra in this manner gives a natural involution where if $\alpha = a + bi + cj + dij$ then $\overline{\alpha} = a -bi - cj - dij$. Thus $(X - \alpha)(X - \overline{\alpha}) = X^2 - (\alpha + \overline{\alpha})X + \alpha\overline{\alpha}\in k[X]$ because $t(\alpha) =(\alpha + \overline{\alpha})$ and $n(\alpha)= \alpha\overline{\alpha}$ are both invariant under the involution, and thus must belong to $k$. We denote $t(\alpha), n(\alpha)$ to respectively be the reduced trace and reduced norm of $\alpha$.

One natural thing to consider is how a quaternion algebra changes when we extend the field of definition. For instance, $(2,3)_\mathbf{Q}$ is not isomorphic to $M_2(\mathbf{Q})$ but $(2,3)_\mathbf{R}$ is, because $a,b$ only matter up to squares and both 2 and 3 are squares in $\mathbf{R}$.

Definition: A rational quaternion algebra $B$ is said to be indefinite if when we extend the base field to $\mathbf{R}$ it becomes isomorphic to $M_2(\mathbf{R})$ and definite otherwise. This choice of terminology comes from the theory of quadratic forms, indeed $(a,b)_\mathbf{Q}$ is definite/indefinite if the norm form $aX^2 +bY^2 -abZ^2$ is a definite/indefinite quadratic form.

Indefinite quaternion algebras over the rational numbers correspond to modular forms by way of orders.

Definition:A $\mathbf{Z}$-order $R$ in a rational quaternion algebra $B$ is a $\mathbf{Z}$-algebra of elements with integral reduced trace and reduced norm inside of $B$ such that $R\otimes_\mathbf{Z} \mathbf{Q} \cong B$. We always have at least one, $\mathbf{Z} \oplus i\mathbf{Z}\oplus j\mathbf{Z}\oplus ij\mathbf{Z}$.

If $B$ is an indefinite quaternion algebra, any order $R$ of $B$ can be realized as a discrete subalgebra of the real split quaternions, indeed $R^\times \subset GL_2(\mathbf{R} )$ and the invertible elements of reduced norm(one can check that this is now identified with the determinant) 1 embed as a discrete subgroup of $SL_2(\mathbf{R})$ so we can consider modular forms for $R$.

Moreover, classical modular forms for congruence subgroups of $SL_2(\mathbf{Z})$ arise in exactly this way because $M_2(\mathbf{Z})$ is a $\mathbf{Z}$-order in $M_2(\mathbf{Q})$. To see this directly, consider the congruence subgroup $\Gamma(N) = \ker(SL_2(\mathbf{Z}) \to SL_2(\mathbf{Z}/N\mathbf{Z}))$.

We can realize this as the invertible elements of reduced norm 1 in the order $\left(\begin{array}{cc} 1 + N\mathbf{Z} & N\mathbf{Z} \\ N\mathbf{Z} & 1 + N\mathbf{Z}\end{array}\right)$

Likewise $\Gamma_0(N)$ can be realized as the invertible elements of reduced norm 1 in $\left(\begin{array}{cc} \mathbf{Z} & \mathbf{Z} \\ N\mathbf{Z} & \mathbf{Z}\end{array}\right)$.

One very special thing we can do with $\Gamma_0(N)$ is to create a modularity map from the modular curve $X_0(N)$ to an elliptic curve $E$ using a special sort of cusp form of weight 2, called a newform.  For details on this construction, the reader is heartily recommended to check out A First Course in Modular Forms.It is the infamous Taniyama-Shimura Conjecture (now proved due to Breuil-Conrad-Diamond-Taylor-Wiles) that every elliptic curve may be realized in this way.

Well it turns out that using a result of Zhang from 2001, you can do the same with any newform of weight 2 on any quaternionic modular curve.

I have left out a number of things I plan to talk about, but this is a good start. After the talk I will post the notes I’ve written up for it.

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### 4 Responses to Quaternion Algebras and Modular Forms

1. Maurizio says:

Nice post, and very interesting. I look forward to read you notes!

2. Jim Stankewicz says:

Finally, the notes. The talk went well, although I had to blaze through just about everything that isn’t on the first 3 pages

http://math.uga.edu/~jstankew/quatalg.pdf

3. Minor detail (near the top): only the invertible 2×2 matrices act on P^1.

4. nice job. please write more about these quaternion algebras over the rationals. be well.