# Birch's conjecture from Representation Theory

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.

• Can you give a reference for this conjecture of Birch? Jan 20, 2016 at 18:03
• Birch, B.J. "Hecke actions on classes of ternary quadratic forms". Computational Number Theory: Proceedings of the Colloquium on Computational Number Theory held at Kossus Lajos University. de Gruyter, 1991.books.google.com/… Jan 21, 2016 at 1:46
• There are a couple of things that don't make sense. "The nonsplit SO(3)" - it is not unique if you're working over global field. $D^\times$ is an infinite cover of an SO(3). Anyway, why don't you just work with $PD^\times$? Feb 2, 2016 at 22:17
• @Kimball I think I mentally included only the units in my $D^{x}$, and picked $x^2+y^2+z^2$ as the "right thing". The reason I want to do this is that the paramodular forms Jeffery Hein and I computed are the ones that happen in an analogous situation in $GU_{2}(D)$ vs $O(5)$. Feb 3, 2016 at 2:44
• Well then, if I understand what you're asking, then yes you can do it representation theoretically, or just in terms of level/holomorphicity for squarefree level. I wrote this as an answer, but let me know if I misunderstood your question. Feb 3, 2016 at 3:12

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).
• @WatsonLadd Well, SO(3) is $PD^\times$. I added this to the answer. Does this answer your question now? Feb 3, 2016 at 23:26