Note on Fourier expansions at cusps
Abstract.
This was originally an appendix to our paper ‘Fourier expansions at cusps’ [1]. The purpose of this note is to give a proof of a theorem of Shimura on the action of on modular forms for from the perspective of algebraic modular forms. As the theorem is wellknown, we do not intend to publish this note but want to keep it available as a preprint.
In this note we give a new proof of the following theorem which is originally due to Shimura, see [5, Theorem 8] and [6, Lemma 10.5]. It gives the interaction between the action and the action on spaces of modular forms on the group . These actions on a modular form of weight are defined as follows:
for and . For any integer we denote .
Theorem 1.
This theorem immediately implies that if a modular form of level has Fourier coefficients in a field , then the Fourier coefficients of for any will lie in . In [1] we obtain improved results if is a modular form for or and in the case of newforms on , we determine the number field generated by the Fourier coefficients of explicitly.
We recall the theory of algebraic modular forms, in order to give a new proof of Theorem 1. For more details on this theory, see [3, Chap. II] and the references therein.
Definition 2.
Let be an arbitrary commutative ring, and let be an integer. A test object of level over is a triple where is an elliptic curve, is a nowhere vanishing invariant differential, and is a level structure on , that is an isomorphism of group schemes
satisfying for every . Here is the scheme of th roots of unity, and is the Weil pairing on ^{1}^{1}1Our definition of the Weil pairing is the reciprocal of Silverman’s definition [7, III.8]. With our definition, we have on the elliptic curve with ..
If is a ring morphism, we denote by the base change of to along .
The isomorphism classes of test objects over are in bijection with the set of lattices in endowed with a symplectic basis of [3, 2.4]. Another example is given by the Tate curve [2, §8]. It is an elliptic curve over endowed with the canonical differential and the level structure . The test object is defined over .
Definition 3.
An algebraic modular form of weight and level over is the data, for each algebra , of a function
satisfying the following properties:

for every ;

is compatible with base change: for every morphism of algebras and for every test object of level over , we have .
We denote by the module of algebraic modular forms of weight and level over .
Evaluating at the Tate curve provides an injective linear map
called the expansion map. The expansion principle states that if is a subring of , then an algebraic modular form belongs to if and only if the expansion of has coefficients in .
Algebraic modular forms are related to classical modular forms as follows. To any algebraic modular form , we associate the function defined by
with .
Proposition 4.
The map induces an isomorphism between and the space of weakly holomorphic modular forms on (that is, holomorphic on and meromorphic at the cusps). Moreover, the expansion of coincides with that of .
We now interpret the action of on modular forms in algebraic terms. Let with , and let . A simple computation shows that
(3) 
where the level structure is given by
(4) 
Let be the isomorphism defined by . Let us identify the level structure (resp. ) with the map (resp. ). Then (4) shows that
(5) 
What we have here is the right action of on the row space , which induces a left action on the set of level structures. As we will see, all this makes sense algebraically. For any algebra , we denote by the image of under the structural morphism .
Lemma 5.
If is a algebra, then there is an isomorphism of group schemes sending to .
Proof.
Note that and . If , then because all irreducible representations of have dimension 1. This isomorphism is given by the Fourier transform, and both and have coefficients in with respect to the natural bases. It follows that in general and this isomorphism sends to . ∎
Let be a algebra. We have an isomorphism of group schemes
given by . The group acts from the right on the row space by automorphisms, and for we define
(6) 
Using , we transport this to a left action of on the set of level structures of an elliptic curve over . Given a test object over , we define . For any , we define by the rule for any test object over any algebra . The computation (3) then shows that the right action of on corresponds to the usual slash action on .
Remark 6.
The action of on algebraic modular forms over algebras has the following consequence: if a classical modular form has Fourier coefficients in some subring of , then for any , the Fourier expansion of lies in .
We now interpret the action of in algebraic terms (see [4, p. 88]). Let . For any algebra , we define , which means that for all , . We endow with the structure of a algebra using the map . We denote by the map defined by . The map is a ring isomorphism, but one should be careful that is not a morphism of algebras, as it is only linear. For any test object over , we denote by its base change to using the ring morphism .
Let be an algebraic modular form. For any algebra , we define
One may check that the collection of functions satisfies the conditions (1) and (2) above, hence defines an algebraic modular form . Moreover, since the Tate curve is defined over , one may check that the map corresponds to the usual action of on the Fourier expansions of modular forms: for every and every , we have .
We finally come to the proof of Theorem 1.
Proof.
Let with corresponding algebraic modular form . Let and . We take as test object over . Since a modular form is determined by its Fourier expansion, and unravelling the definitions of and , it suffices to check that the test objects and over are isomorphic. Since acts only on the level structures of the test objects, we have to show that
(7) 
For any scheme over , let denote its base change to along . Since is a ring isomorphism, the canonical projection map is an isomorphism of schemes, and we also denote by the inverse map.
Put and . Let . By functoriality, the level structure is given by the following commutative diagram
(8) 
Let us compute the dotted arrow . Since is linear, we have . It follows that
(9) 
so that . We may thus express in terms of by
(10) 
Let us make explicit both sides of (7). By (6) and (10), the left hand side is given by
(11) 
Let us now turn to the right hand side of (7). By (6), we have . Applying the commutative diagram (8) with replaced by , we get
(12) 
Finally, we note that .
∎
References
 [1] François Brunault and Michael Neururer. Fourier expansions at cusps. 2019. Preprint, https://arxiv.org/abs/1807.00391.
 [2] P. Deligne. Courbes elliptiques: formulaire d’après J. Tate. pages 53–73. Lecture Notes in Math., Vol. 476, 1975.
 [3] Nicholas M. Katz. adic interpolation of real analytic Eisenstein series. Ann. of Math. (2), 104(3):459–571, 1976.
 [4] Masami Ohta. On the adic EichlerShimura isomorphism for adic cusp forms. J. Reine Angew. Math., 463:49–98, 1995.
 [5] Goro Shimura. On some arithmetic properties of modular forms of one and several variables. Ann. of Math. (2), 102(3):491–515, 1975.
 [6] Goro Shimura. Arithmeticity in the theory of automorphic forms, volume 82 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000.
 [7] Joseph H. Silverman. The Arithmetic of Elliptic Curves, volume 106 of Graduate Texts in Mathematics. Springer, Dordrecht, second edition, 2009.