# Galois Properties of the Values of Modular Forms

Let $f \in S_0(N)$ be a normalized Hecke eigenform. It is well known that its coefficients are algebraic integers, and $f^\sigma$ lies in $S_0(N)$ for $\sigma \in G_{\mathbb{Q}}$. At CM points $z \in \mathbb{H}$, $f$ takes an algebraic value as well.

Can anything be said about the relation between $f(z)$ and $f^\sigma(z)$ or $f(z^{\sigma})$?

I am aware that there is a nice relation in the $p$-adic setting, due to the $p$-adic convergence of cusp forms, and that this is behind the remarkable usefulness of the Tate curve, but I am more interested in the global behavior.

Perhaps this is wishful thinking, but are there any ($p$-adic or global) results for non-cusp forms as well?

• "..I am aware that there is a nice relation in the p-adic setting, due to the p-adic convergence of cusp forms, and that this is behind the remarkable usefulness of the Tate curve,.." can you elaborate what you mean here?
– DBS
Jul 10, 2013 at 4:42
• For an elliptic curve $E/K$, $K$ a p-adic field, there is a uniformization, as Galois modules, $E(\overline{K}) \cong \overline{K}^*/q^{\mathbb{Z}},$ for some $q \in K^*$. It is a Galois isomorphism, because for the p-adic power series which define it, the Galois action is continuous, so you can apply it to every term in the sequence, the same as applying to the infinite sum. Jul 10, 2013 at 14:54

When working with modular forms as analytic objects, the Galois structure is somewhat invisible and has to be rediscovered using the Hecke algebra. However, there is a purely algebraic notion of modular form, due to Katz, which makes the Galois structure (among other things) a lot more transparent. Katz's construction works well for level structures for which the corresponding moduli functor is representable, and unfortunately this excludes the case of $$\Gamma_0(N)$$. However, for level $$\Gamma = \Gamma_1(N)$$ or $$\Gamma = \Gamma(N)$$ (with $$N>4$$), the moduli problem is representable by an affine curve $$Y_\Gamma$$ over $$\mathbf Q$$, and using Katz's theory, the following result becomes almost tautological:

1. Let $$f$$ be a modular form of weight $$k$$ and level $$\Gamma$$. Suppose that the $$q$$-expansion of $$f$$ has coefficients in $$\overline{\mathbf Q}$$. Let $$\sigma \in G_{\mathbf Q}$$. Then $$f^\sigma$$, obtained by applying $$\sigma$$ to the $$q$$-expansion of $$f$$, is a modular form of the same weight and level as $$f$$.

2. Let $$p \in Y_\Gamma(\overline{\mathbf Q})$$ be a $$\overline{\mathbf Q}$$-point of $$Y_\Gamma$$. Let $$f$$, as in (1), be a modular form with coefficients in $$\overline{\mathbf Q}$$. Then $$f(p) \in \overline{\mathbf Q}$$ and $$f^\sigma(p) = f(p^\sigma) = f(p)^\sigma.$$

From the theory of complex multiplication, it is known that a CM point $$p \in Y_\Gamma(\mathbf C) = \mathbf H/\Gamma$$ is in fact a $$\overline{\mathbf Q}$$-point of $$Y_\Gamma$$ (more precisely, it is defined over the Hilbert class field of the imaginary quadratic field to which it belongs). (Warning: just because a point of the upper-half plane is algebraic, does not mean that it corresponds to an algebraic point of $$Y_\Gamma$$. The identification of $$\mathbf H/\Gamma$$ with $$Y_\Gamma(\mathbf C)$$ is transcendental. It is a special property of CM points that they are algebraic "on both sides", but in the present context, this is more of a pitfall than anything else.) It follows from this and from (2) that:

Theorem. Let $$f$$ be a modular form of weight $$k$$ and level $$\Gamma$$ whose $$q$$-expansion has coefficients in $$\overline{\mathbf Q}$$. Let $$p$$ be a CM point in the upper-half plane. Then $$f(p)$$ is algebraic, and $$f^\sigma(p) = f(p)^\sigma$$ for every $$\sigma \in G_{\mathbf Q}$$.

However, keep the above-stated pitfall in mind: the equality with $$f(p^\sigma)$$ makes no sense from the classical point of view because the Galois action on $$p$$ takes place in $$Y_\Gamma(\overline{\mathbf Q})$$, not in the upper half-plane. In fact, in the naive sense, either $$p^\sigma=p$$ or $$p^\sigma = \overline{p}$$ does not even lie on the upper half-plane, so the expression $$f(p^\sigma)$$ is meaningless from the classical point of view.

Over $$\Gamma_0(N)$$, the same results are true, but to prove it in this way, we have to pull everything back to $$\Gamma(N)$$ and push everything back down at the end.