# Elementary proof of algebraicity of Hecke eigenvalues in weight 1

It's "well known" that, for any weight $k$ and level $N$, the space $S_k(\Gamma_1(N))$ of cusp forms of that weight and level has a basis in which all the Hecke operators act by matrices with entries in $\mathbb{Z}$; consequently all the Hecke eigenvalues are algebraic numbers (indeed algebraic integers).

I was reflecting on how to prove this while teaching an undergraduate course on modular forms. For $k \ge 2$ it's not hard: there's the Eichler-Shimura machinery which relates it to a question about cohomology, and the cohomology with $\mathbb{Z}$ coefficients does the job. Alternatively, and more or less equivalently, you use the pairing with modular symbols. Both of these methods break down for $k = 1$; the only argument I know that works in this case is to use the fact that $X_1(N)$ has a model as an algebraic variety, and weight $k$ modular forms correspond to sections of the $k$-th power of a line bundle that has a purely algebraic definition. But that's not really something I can stand up and explain to a class of undergraduate students!

For cusp forms of weight $k = 1$, can the algebraicity of the Hecke eigenvalues be proved without quoting heavy machinery from arithmetic geometry?

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Let $S = S_{\mathbf{Q}} = M_{13}(\Gamma_1(N),\mathbf{Q})$, and $S_{\mathbf{C}} = S \otimes \mathbf{C}$ denote the corresponding space of modular forms over $\mathbf{C}$.

Let $V \subset S \times S$ be the subspace cut out by pairs of forms $(A,B)$ satisfying the following equation:

$$A \cdot E_{12} = B \cdot \Delta$$

As equations in the Fourier coefficients of $A$ and $B$ these are linear equations with coefficients in $\mathbf{Q}$. Since, by the $q$-expansion principle, a modular form can be recovered from some finite number of Fourier coefficients, $V$ is determined by the null space of some finite matrix with coefficients in $\mathbf{Q}$. Since a linear system over $\mathbf{Q}$ has the same rank over $\mathbf{C}$, it follows that $V_{\mathbf{C}} = V \otimes \mathbf{C}$, where $V_{\mathbf{C}}$ is the set of solutions in $S_{\mathbf{C}} \times S_{\mathbf{C}}$ of the same equations.

On the other hand, there is an isomorphism: $$V_{\mathbf{C}} \rightarrow M_{1}(\Gamma_1(N),\mathbf{C})$$ given by $$(A,B) \mapsto \frac{A}{\Delta} = \frac{B}{E_{12}}$$ The point is that $E_{12}$ and $\Delta$ do not have any common zeros, so the image of this map clearly consists of holomorphic forms. Hence the map is well defined. However, if $F$ has weight one, then $(A,B) = (F \cdot \Delta, F \cdot E_{12})$ maps to $F$, so the map is surjective. It is clearly injective, so it is an isomorphism.

It follows that the image of $V$ under this map gives a rational basis for $M_1(\Gamma_1(N))$. Since $V$ is then preserved by Hecke operators (as is obvious on $q$-expansions), the result follows.

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Of course! Very pretty. This one could certainly use in an undergrad course. –  David Loeffler May 9 '13 at 6:12
Here is another proof: Deligne and Serre have proven that the corresponding L-function equal the Artin L-function of a Galois representation. Deligne had proven similar facts known for weight $k \geq 2$ modular forms before that, and their proof essentially relies on the former results. This implies algebraicity and also is the only approach I know for the same result for Maass Hecke cusp forms of Laplace eigenvalue $1/4$, where the same algebraicity result is unknown.
Dear Marc, Rebecca Black is correct that this is circular. As noted in another recent comment I made on a question of yours, you can't begin to to do the Deligne--Serre argument without congruences between wt. 1 forms and higher wt. forms, and such congruences don't make sense without knowing that wt. 1 eigenforms have algebraic integer $q$-expansion coefficients (or some equivalent algebro-geometric statement of the kind that the OP was trying to avoid). Regards, Matthew –  Emerton May 10 '13 at 23:32