How to compute modular forms of weight one on Shimura curves?

Question

Classical modular forms of weight $k\geq 1$ can be computed explicitly by exhibiting their $q$-expansion. When the weight is at least two, the most standard method uses modular symbols. When the weight is one, we can not use modular symbols directly because such modular forms are not cohomological, but one can still compute their $q$-expansion by using ideas of Buhler, Frey, Serre, Buzzard and others.

Say now that we want to compute explicitly modular forms of weight $k\geq 1$ on a Shimura curve associated to an order in an indefinite quaternion algebra over $\mathbb{Q}$. I can think of at least two ways of exhibiting such modular forms: either by (1) computing their Taylor expansion about a CM point or by (2) invoking Cerednik-Drinfeld theory and computing the harmonic cocycle on the Bruhat-Tits tree associated to it.

Method (1) is in principle available for all weights $k\geq 1$ but I'm not sure how well it works systematically in practice, as there is no canonical choice of CM point.

Method (2) seems more canonical and robust to me, but it has the drawback that it can only be used directly for $k\geq 2$.

My question is: how would you compute in practice quaternionic modular forms of weight $1$?

I seek an answer which should allow us to play around with these modular forms and do basic operations like for instance multiplying two forms of weight one and recognize the output as an explicit linear combination of eigenforms of weight two. This is beyond what one can gather from Jacquet-Langlands, I would say.

Answer 1

I wonder whether you mean something else, but if you want to know what the answer is (dimensions of spaces, Hecke action etc) then all of this can be read off from Jacquet-Langlands and the analogous calculations for $GL(2)$, which as you point out can already be done (although in practice it's difficult to do them explicitly once the level gets to over 2000).

Note that you'll need some level structure at all primes dividing the discriminant, because classical weight 1 forms can never be Steinberg at a finite place (because e.g. the associated Hecke eigenvalue would not be an algebraic integer, or you can use local Langlands and local-global and the fact that the global image of Galois is finite to deduce that the local image must be finite).

Answer 2

Great question! You have a few options.

1. You could use the method of power series expansions (https://math.dartmouth.edu/~jvoight/articles/ModFormPaper-090613.pdf). We replace $q$-expansions with a normalized power series expansions at a point (CM points of class number 1, when they exist, give the nicest expansions); we solve for the undetermined coefficients by writing down linear equations implied by being modular of weight 1 (and, to get a good set of equations, it is best to the Cauchy integral formula). You can then also compute the action of Hecke operators and multiply to your heart's delight. This method is analytic/experimental, exponentially convergent (roughly speaking, each Arnoldi iteration gives you an extra digit of precision in your answer); depending on your application, you may need an extra step to ensure that whatever you need from the output (e.g. multiplication table) is rigorous. For this algorithm, working over a totally real field is just as easy.
2. Paul Nelson gives an algorithm to evaluate the Shimizu lift explicitly (https://arxiv.org/pdf/1210.1243v1.pdf), providing an exponentially convergent series expansion of a Jacquet-Langlands transfer. Paul only looks at even weight, but I don't think this is a serious restriction. Forms in this method are recorded by their values at (sufficiently many) points, and using this you can also multiply.
3. For Shimura curves that arise from congruence subgroups of triangle groups, one can do everything explicitly in terms of the ${}_2F_1$-hypergeometric function (https://math.dartmouth.edu/~jvoight/articles/shimcm-fixed-errata.pdf). Roughly speaking, there is Hauptmodul $t$ for the full triangle group, a modular equation relating $t(z)$ and $t(Nz)$ for level $N$, and canonical generators for the ring of fractional weight modular forms with character to play with. Some of this is work in progress (with Baba-Granath-Nelson-V-Yang), but I would be happy to discuss with you offline if it turns out this is what you need.

Answer 3

I seek an answer allowing us to perform basic operations like for instance multiplying two modular forms of weight one amd recognizing the output as an explicit linear combination of weight two eigenforms. This is, in my understanding, beyond Jacquet-Langlands.