Birch's conjecture from Representation Theory

Question

Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic representations of $PGL(2)$ are representations of $GL(2)$ with trivial central character, and Jacquet-Langlands describes the ones that come from representations of $D^{\times}$ where $D$ is a definite quaternion algebra. (Supposedly: I've not found where in the book that result is) Then using the fact that $D^{\times}$ is a double cover of the nonsplit $SO(3)$, we should be able to prove Birch's conjecture. However, this requires a description of the representations of $D^{\times}$ that includes the central characters as a representation of $D^{\times}$. Does this appear somewhere in Jacquet-Langlands, or do people have some other source? Or is this line of reasoning insufficient to get Birch's conjecture?

In the case of squarefree levels we can avoid supercuspidals, so a description that only works for the principal series and the Steinberg (and twists of Steinberg) suffices.

Answer 1

I haven't looked at Birch's conjecture, but yes, Jacquet and Langlands characterize the image of of their transfer. Here is what it is in terms of representation theory. Fix a quaternion division algebra $D$ over a number field $F$. Let $S$ be the set of places at which $D/F$ is ramified. Then a cuspidal automorphic representation $\pi$ of $GL_2(\mathbb A_F)$ lies in the image of the Jacquet-Langlands correspondence from $D^\times(\mathbb A_F)$ if and only if $\pi_v$ is discrete series at each $v \in S$ (for finite places, this means a twist of Steinberg or supercuspidal). The correspondence preserves central characters, so this descends to a correspondence from $PD^\times = D^\times/F^\times$ to $PGL(2)$. Note $PD^\times$ is a form of $SO(3)$.

In particular if $F=\mathbb Q$ and you want to work with squarefree level, locally on $PGL(2)$ you can only get Steinberg or a quadratic twist at primes dividing the level. Let $M$ be the product of the finite primes where $D$ ramifies. If $D$ is definite, then you get all holomorphic cusp forms of level $MN$, where $N$ is prime to $M$. If $D$ is indefinite, you get both holomorphic and nonholomorphic forms. If $N$ is not prime to $M$, there's no simple way to determine which holomorphic forms come from $D$ except in terms of representation theory (because you'd need to distinguish between ramified principal series versus special and supercuspidal).

There are various surveys that describe this, e.g., Gelbart's book on GL(2), or his Lectures on the Arthur-Selberg trace formula.