On the non/existence of p-adic points on certain Shimura curve quotients

Pete L. Clark

March 7, 2002

1 Introduction

Let p, q be distinct primes numbers, and N a positive integer that is prime to pq. Let B/ℚ be the (unique, up to isomoprhism) indefinite rational quaternion algebra ramified precisely at the primes p and q. Fix also O_{B} a maximal order of B (all such are conjugate) and t an element of O_{B} such that t^{2} = −pq. Denote by x ↦ x^{∘} the canonical involution on B, i.e. the one defined by x + x^{∘} =trace(x). We define a new involution x ↦ x^{∗} = t^{−1}x^{∘}t. Associated to the algebraic group B^{×}/ℚ we have an inverse system of Shimura varieties: by taking no level structure, Γ_{0}(N)-structure and Γ_{1}(N)-structure respectively, we get algebraic curves X^{pq}(1), X_{0}^{pq}(N), X_{1}^{pq}(N) all defined over ℚ. These are (at least) coarse moduli schemes for the following moduli problems: for simplicity, we restrict to schemes S over Spec ℤ[\frac{1}{pq}]:

X^{pq}(1) : S ↦ pairs (A, ι), where A/S is an abelian scheme of relative dimension two and ι : O_{B} ↪ End_{S}(A) is (by definition) a quaternionic structure. It is known [Milne in Corvallis] that A admits a unique principal polarization compatible with ι in the sense that the Rosati involution on End(A) induced by the polarization restricts to x ↦ x^{∗} on O_{B}. We call a pair (A, ι) a QM-surface (in the literature, one also sees the term false elliptic curve, but this sounds to the author a bit pejorative!) We recall that X^{pq}(1)(ℂ) is given as a double coset

space: B^{×}\B^{×}(𝔸)/ℂ^{×}\begin{matrix} \\ O_{B} & × \end{matrix}.

X_{0}^{pq}(N): Since we have assumed that (N, pq) = 1, for all primes l dividing N, we have B_{l}^{×} = B^{×}(ℚ_{l}) = GL_{2}(ℚ_{l}), and we can give, prime-by-prime, the definition of level N structure familiar from the elliptic modular case: namely, we put { \atop Γ_{0}(N)} := the set of x in \begin{matrix} \\ O_{B} & × \end{matrix} such that for all l dividing N, x_{l} is congruent

to [ \begin{matrix}a & b \\ 0 & d \end{matrix} ] modulo l^{ord_{l}(N)}. Here is the moduli problem: S ↦ (A, ι, C), where

(A, ι) is a QM structure as above, and C ≤ A is a finite flat S-subgroup scheme of A, everywhere of rank N^{2}, which is stable under O_{B} and fppf-locally cyclic as an O_{B}-module. Equivalently, over an algebraically closed field k, A[N](k) is a free rank 1 O_{B} ⊗ ℤ/Nℤ = M_{2}(ℤ/Nℤ)-module; as such, corresponding to a choice of orthogonal idempotents e_{1} + e_{2} = 1, we get a decompostion of C into C = C_{1} ⊕ C_{2}, where each C_{i} is a finite flat group scheme of rank N which is locally cyclic in the usual sense. (See [Buzzard, 1996] for some treatment of this.) The data (A, ι, C) is caled a Γ_{0}(N)-structure on the QM-surface A/S.

X_{1}^{pq}(N): Similarly, put { \atop Γ_{1}(N)} := the set of x in \begin{matrix} \\ O_{B} & × \end{matrix} such that for all l

dividing N, x_{l} is congruent to [ \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ] modulo l^{ord_{l}(N)}. and X_{1}^{pq}(N)(ℂ) =

B^{×}\B^{×}(𝔸)/ℂ^{×}\begin{matrix} \\ Γ_{1}(N) & × \end{matrix}. Here is the moduli problem: S ↦ (A, ι, C, P), where (A, ι, C) is as above, and P ∈ C(S) is an O_{B}-module generator for C. Equivalently, under C = C_{1} ⊕ C_{2}, we specify P_{1} as a generator of C_{1} in the usual sense. The data (A, ι, C, P) is called a Γ_{1}(N)-structure on the QM-surface A/S.

But these are not quite the curves we want to study, for the following reason: our main interest is in studying QM surfaces over number fields, especially over ℚ. But, analogously to the situation in which there exist CM elliptic curves defined over ℚ, but we can never have all the endomorphisms defined over ℚ, we will never have a QM surface (A, ι)/ℚ as defined above: indeed, if O_{B} ↪ End_{ℚ}(A), then consider the action of O_{B} on the two-dimensional subspace W ≤ H^{1}(A(ℂ), ℚ) fixed by complex conjugation. This gives a represenation B ↪ End(W) ≅ M_{2}(ℚ) and a contradiction. So, let us say a potentially QM-surface A/k is a principally polarized abelian surface such that A/\overline{k} can be endowered with a QM structure compatible with its polarization. It turns out that that the map which takes a QM surface and forgets the QM-structure but retains the associated principal polarization is a 2-1 branched cover given by a certain Atkin-Lehner involution.

For every positive integer d dividing the discriminant pq, we have a modular involution w_{d} : X^{pq}(1) → X^{pq}(1) which we will now recall. First, by Noether-Skolem, every automorphism of O_{B} is inner, so Aut(O_{B}) = N(O_{B}^{×})/O_{B}^{×}, where N(O_{B}) is the normalizer inside GL_{2}(ℝ)^{+} (here we have implicitly fixed an isomorphism B ⊗ ℝ ≅ M_{2}(ℝ)). One knows that the group N(O_{B}^{×})/O_{B}^{×} has order 2^{2r}, where 2r is the number of ramified primes of B (i.e. 2 in our situation, but the statement is true for any B/ℚ.) Moreover, representatives for the quotient are given by elements γ_{d} for all d dividing the discriminant, where γ_{d}^{2} = −d. From this one already sees how the w_{d} act on the complex points of the Shimura curve. We can also give a description in terms of moduli: w_{d}(A, ι) := (A, γ_{d}^{−1} ∘ ι ∘ γ_{d}), i.e. w_{d} preserves the underlying abelian surface but changes the quaternionic structure via the associated automorphism of O_{B}. In view of the above, it is natural to ask which involutions w_{d} change the quaternionic structure but preserve the associated principal polarization, and the answer is that it is only the main involution w_{pq} which has this property [Jordan’s thesis]. It follows that X^{pq}/w_{pq} is the moduli space for PQM surfaces, and similarly X_{0}^{pq}(N)/w_{pq}, X_{1}^{pq}(N)/w_{pq} are the moduli spaces for PQM surfaces with Γ_{0}(N), Γ_{1}(N)-structure.

So our objects of study are the curves X^{pq}(1)/w_{pq}, X_{0}^{pq}(N)/w_{pq}, and X_{1}^{pq}(N)/w_{pq}. It is easy to see that whenever the maximum of N, p, q is large enough, the genus of these curves is at least 2 (and indeed goes to infinity with max(N, p, q)), all but the first few of these curves will have only finitely many ℚ-points; of course this gives a fortiori a finiteness result for PQM-surfaces defined over ℚ. In contrast to the elliptic modular case in which there is always at least one “obvious” ℚ-point – i.e., a rational cusp – there are not necessarily any “easy” ℚ-points on these curves, and indeed the author conjectures that there exist infinitely many values of p and q such that X^{pq}(1)(ℚ) is empty. This lack of cuspidal points is in some contexts extremely annoying (e.g. one cannot use the machinery of q-expansions to study automorphic forms on these curves), but it also raises the possibility of answering global questions by local means, namely by showing e.g. that X^{pq}(1)(ℚ_{l}) is empty for some prime l. The aim of this paper is to present the following local results:

Theorem 1 X^{pq}(1)/w_{pq}(ℚ_{p}) is nonempty.

Corollary 1 The curve X^{pq}(1)/w_{pq} has points everywhere locally, i.e. at Q_{l} for all primes l and at ℝ.

Theorem 2 Assume for simplicity that q is at least 5. If q ≡ 1 modulo 4, then X_{0}^{pq}(N)(ℚ_{p}) is nonempty only if N a norm from ℚ(\sqrt{−q}). If q ≡ −1 modulo 4, then X_{0}^{pq}(N)(ℚ_{p}) is nonempty only if N or 2N is a norm from ℚ(\sqrt{−q}).

Theorem 3 If X_{1}pq(N)/w_{pq}(ℚ_{p}) is nonempty only if N ≤ q + 1, unless q is 2 or 3. If q = 2, the bound is N ≤ 5¡ and if q = 3, the bound is N ≤ 7.

Corollary 2 For fixed p, q, there exist only finitely many positive integers N such that there exists a PQM-surface A/ℚ and a point P ∈ A[N](ℚ) which generates an O_{B}-cyclic subgroup of rank N^{2}.

The basic strategy of the proofs of all three theorems is the same: we use the p-adic uniformization techniques developed by Mumford, Cerednik-Drinfeld, Kurihara, and Jordan-Livne to reduce the problem to an analysis of certain finite graphs. Following Jordan-Livne, we see that in principle the answers to our problems can be read off certain combinatorial data of these graphs. On the other hand, inspired by work of Ribet, we repeatedly rephrase these graphtheoretic conditions in terms of properties of supersingular elliptic curves over 𝔽_{q} (the switch from ℚ_{p} to 𝔽_{q} is characteristic of Cerednik-Drinfeld’s interchange of local invariants). Once we have phrased the problem in this form, the actual proofs are quite short.

2 Proof of Theorem 1

Notation: When our choice of level structure is clear from the context, we will allow ourselves to write X for any of: X^{pq}(1), X_{0}^{pq}(N), X_{1}^{pq}(N), X^{(p)} for any of: X^{pq}(1)/w_{p}, X_{0}^{pq}(N)/w_{p}, X_{1}^{pq}(N)/w_{p}, and X^{(pq)} for any of: X^{pq}(1)/w_{p}q, X_{0}^{pq}(N)/w_{p}q, X_{1}^{pq}(N)/w_{p}q. What is worse: in case an argument goes through equally well for Γ_{0}(N)-structure or Γ_{1}(N)-structure, we may write e.g. X_{•}^{pq}(N) to mean either one.

First we need to recall some of [Jordan-Livne 1]: work of Cerednik-Drinfeld gives us canonical Z_{p}-models for all of our curves, such that the special fibres are admissible curves in the sense of Jordan-Livne: they are (reduced) semistable curves /𝔽_{p}, every irreducible component of which has geometric genus 0. To such a curve we can associate a finite graph, its dual graph, as follows: the vertex set of the graph corresponds to the irreducible components of the curve, and edges e : v_{1} → v_{2} correspond to intersection points of the components corresponding to v_{1}, v_{2}. Recall also that these curves are not Mumford curves; rather thery are twists of Mumford curves under Frob_{p} ↦w_{p}. That is, a component (respectively, a singular point) is defined over 𝔽_{p} iff it is fixed under the action of w_{p}. We will use:

Proposition 3 (Hensel’s Lemma): Let X/K be a projective curve over a field K which is complete with respect to a non-Archimedean valuation. Let X /O_{K} be any regular model for X. Then X(K) is non-empty iff X (O_{K}/𝔪O_{K}) has a smooth point.

For a proof, see [Jordan-Livne 1]. But in fact, the canonical models X /ℤ_{p} need not be regular. Indeed, let P ∈ X /ℤ_{p}^{∞} be a point lying on the closed fiber whose reduction into X (\overline{𝔽_{p}}) is singular. Then there is an analytic neighborhood of P in X /ℤ_{p}^{∞} isomorphic to ℤ_{p}^{∞}[[X, Y]]/(XY − p^{a}). It is not hard to check that this local ring is regular iff a = 1. In general, the integer a is associated to the edge e and called its length. Thus, we have in all the structure of a “finite l-graph,” i.e. a finite graph in which each unoriented edge is associated a positive integer. To perform the regularization of the arithmetic surface X, we repeatedly blowup at the points on the special fibre corresponding to edges e with length ¿ 1. In terms of l-graphs, this corresponds to replacing the single edge of length m with m edges of length 1. Thus, from the data of an l-graph we can construct the special fibre of a regular model for X/K (and indeed, we could go on to construct the minimal model, if we needed it; see [Jordan-Livne 1] for details.)

Now we consider the problem of examining the l-graph G to determine whether X(ℚ_{p}) is empty. A smooth 𝔽_{p}-point on the special fibre of the regularized surface correponds either to:

a) a w_{p}-fixed vertex v of G, or

b) an edge e of even length, which is flipped by w_{p} (i.e. such that w_{p} : v → w ↦ w → v).

(We remark that the length must be even so that the w_{p}-Frobenius action fixes the middle vertex in the chain created by blowing up the single edge e.) In fact, our first step is that we need not worry about b) in our situation. Let us agree for now that G, G^{(p)}, G^{(pq)} stand for the dual graphs of the special fibres of the curves X^{pq}(1), X^{pq}(1)/w_{p}, X^{pq}(1)/w_{pq} respectively.

Proposition 4 There is no two-sided even length edge e of G^{(pq)} which is flipped by w_{p}.

For the proof, we need:

Lemma 5 The sets l(G^{(p)}), l(G), l(G^{(pq)}) of lengths of the dual graphs are all the same. Moreover, when counted with multiplicity, the multisets of lengths of oriented eges of G^{(p)} and G^{(pq)} coincide.

Proof of the lemma: The graph G admits a natural bipartition coming from the bipartition of the Bruhat-Tits tree Δ given by considering vertices of even and odd distance from any given vertex. From e.g. [Kurihara], one knows that w_{p} interchanges the two subsets of the bipartition (and in particular acts freely on the vertices and oriented edges of G) and w_{q} preserves each subset of the bipartition, so clearly w_{pq} again interchanges the two subsets and acts freely on vertices and oriented edges. From this we see that the at-most-(2-1) maps G → G^{(p)}, G → G^{(pq)} are precisely 2-1 on the sets of oriented edges of G. Since an equivalent characterization of the length of an edge G is the cardinality of the stablizer of that edge inside the discrete subgroup of GL_{2}(ℚ_{p}) associated to the curve [Jordan-Livne 1], this lack of ramification immediately implies the lemma.

Proof of the proposition: From [Kurihara], we see that the only edges of even length in G^{(p)} are those of length 2, and moreover the number of such oriented edges is either 0,1 or 2. In case there is only one oriented edge, it must be one-sided. Its lift to G is therefore an edge e = v → w such that w_{p}(e) = \overline{e} = w → v. Since all the Atkin-Lehner involutions preserve the lengthstructure of the graphs, we must have w_{q}{v, w} = {v, w}; since moreover v and w are adjacent vertices and w_{q} preserves the bipartition, we must have w_{q}(v) = v and w_{q}(w) = w. Thus w_{pq}(e) = \overline{e} as well, so that in G^{pq} the unique edge of length 2 is again one-sided. But according to the recipe for computing the special fibre of an admissible curve from its dual graph, one-sided edges are removed [Kurihara,Jordan-Livne 1], showing the proposition in this case. In case there are two oriented edges of length 2 in G^{(p)} and they are both onesided, the above argument again shows that the corresponding pair of length 2 edges in G^{(pq)} are one-sided. In the remaining case, we can consult [Kurihara] to see that there will be a unique vertex admitting edges of length 2, so the two geometric edges of length 2 are the two orientations of a loop at that distinguished vertex. Choosing an orientation, this edge lifts to two geometric edges in G, e_{1} : v → w and e_{2} : w → v such that w_{p}(e_{1}) = e_{2}. As above, we find that w_{q}(v) = v and w_{q}(w) = w , so that the unique length 2 edge of G^{(pq)} is again a loop at a single vertex; this edge can be represented in the quotient graph as a pair {e_{1}, e_{2}}, and evidently w_{p}({e_{1}, e_{2}}) = {e_{2}, e_{1}} = {e_{1}, e_{2}}, so the edge is fixed by w_{p} and not flipped by it.

In fact it is not the proposition itself that we need in this section (it points in the direction of the emptiness of the ℚ_{p}-points, not the non-emptiness), but rather its analogue in the case of level structure:

Corollary 6 There are no one-sided edges of even length in the dual graphs G(X_{•}^{pq}(N)/w_{pq}) which are flipped by w_{p}.

In order to prove the corollary we must recall work of [Cerednik-Drinfeld], made explicit in our context by [Ribet] that interprets all our graphs in terms of structures on supersingular elliptic curves. It is convenient to work adelically: let U ≤ B^{×}(𝔸_{f}) be a compact open subgroup which is maximal at p and q. Following [Ribet], we introduce the p-adic space X_{U} := Π_{l≠p} U_{l}\D̂^{×}/D^{×}, where D/ℚ is the quaternion algebra obtained by “interchanging the invariants at p and ∞,” i.e. D is definite of discriminant q. Notice that the spaces D̂^{×} and B̂^{×} differ only at their p-components; thus, the prime-to-p part of U is an acceptable prime-to-p-level structure for the zero-dimensional Shimura variety D^{×}. Now we have two key observations:

Proposition 7 (Ribet, (4.3))

a) GL_{2}(ℤ_{p})\X_{Γ_{•}(N)} is canonically isomorphic to the vertex set of G(X_{•}^{pq}(N)/w_{p}). Moreover the vertex set of G(X_{•}^{pq}(N)) is canonically given as the disjoint union of two copies of GL_{2}(ℤ_{p})\X_{Γ_{•}(N)}.

b) The edge set of G(X_{•}^{pq}(N)) is canonically isomorphic to Γ_{0}(p)\X_{Γ_{•}(N)}. Moreover, the attaching map for the graph is as folows: to an element e ∈ Γ_{0}(p)\X_{Γ_{•}(N)}, we associate its initial vertex e(0) via the natural projection π_{1} : Γ_{0}(p)\X_{Γ_{•}(N)} ⟶ (GL_{2}(ℤ_{p})\X_{Γ_{•}(N)})^{1}, and its terminal vertex via π_{2}∘(m^{−1}x), whereπ_{2} : Γ_{0}(p)\X_{Γ_{•}(N)} ⟶ (GL_{2}(ℤ_{p})\X_{Γ_{•}(N)})^{2} is the same map as before but formally landing in the second copy, and m ∈ D̂^{×} is an idele which is everywhere locally trivial except at

p, and such that m_{p} = [ \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ].

Proposition 8

a) GL_{2}(ℤ_{p})\X_{Γ_{•}(N)} is in bijection with the set of supersingular elliptic curves with Γ_{•}(N)-structure, i.e. with X_{•}(N)(\overline{𝔽_{q}})^{ss}.

b) Γ_{0}(p)\X_{Γ_{•}(N)} is in bijection with the set of supersingular elliptic curves with Γ_{0}(p) + Γ_{•}(N)-structure.

c) Under these correspondences, w_{q} on the dual graph of X_{•}^{pq}(N)/w_{p} correponds to the q-power Frobenius morphism Frob_{q} on X_{•}(N)(\overline{𝔽_{q}})^{ss}. In particular, the set of w_{q}-fixed vertices corresponds to the locus X_{•}(N)(𝔽_{q})^{ss}. Proof: a),b) are well-known from the theory of Shimura varieties. For c), see [Ribet, (?.?)].

Proof of the corollary: Let ẽ ∈ G(X_{•}^{pq}(N)/w_{pq}) be a two-sided edge of even length that is flipped by w_{p}, and let e ∈ G(X_{•}^{pq}/w_{pq}) be the image of ẽ under the quotient map. I claim that the quotient map π : G(X_{•}^{pq}(N)/w_{pq}) → G(X^{pq}/w_{pq}) preserves the orientation of edges; equivalently, G(X_{•}^{pq}(N)) → G(X^{pq}) is orientation-preserving. Indeed, the initial vertex of the edge ˜ ẽ = (E, C_{N}orP, C_{P}) is evidently just (E, C_{N}orP), so that the initial vertex of the image of ˜ ẽ in G(X^{pq}) is unambiguously E (and indeed there are no loops in these bipartite graphs, so this analysis suffices). Now, it is clear that since w_{p} flips ẽ, it must either flip or fix the image vertex e. Suppose first that it does both, i.e. that e is one-sided. Then, because π is orientation-preserving, ẽ must be one-sided as well, contrary to our assumption. Otherwise, the orientationpreserving nature of π implies that since ẽ gets flipped upstairs, e is a two-sided edge of even length that gets flipped downstairs, contradicting Proposition 7.

Proof of Theorem 1: For the remainder of the section, we work with no level structure, and we write G, G^{(p)}, G^{(pq)} for the three dual graphs in question. Because of Proposition 7, we must show that X^{(p)}(𝔽_{p}) is nonempty. First observe that a w_{p}-fixed vertex of G^{(pq)} can be viewed as a pair {v,w_{pq}v} such that w_{p}{v, w_{pq}v} = {v, w_{pq}v}, i.e. {w_{p}v, w_{q}v} = {v, w_{pq}v}. So either v = w_{p}v or v = w_{q}v. The former is impossible, since w_{p} is fixedpointfree on the vertex set of G; hence v = w_{q}v. That is, fixed points of w_{p} on G^{(pq)} correspond to pairs of fixed points of w_{q} on G. Similarly, under the quotient map G → G^{(p)}, each pair of w_{q}-fixed points gets mapped to a single w_{q}-fixed point of G^{(p)}, which by Proposition 11c), corresponds to an element of X(1)(𝔽_{q})^{ss}, i.e. to a supersingular elliptic curve defined over the prime subfield 𝔽_{q}. It is well-known that this set is non-empty; e.g., by Tate-Honda theory the Weil q-number \sqrt{−q} provides us with at least one such element. (Indeed, more precisely, one knows that the number of such elements is given by h(−q) + h(−4q), where h(n) is the classnumber of the quadratic order of discriminant n or 0 if n is not the discriminant of a quadratic order; moreover this is the number of fixed points of the classical Atkin-Lehner involution w_{q} on the elliptic modular curve X_{0}(q)/ℂ.) We see that there are certainly components of X^{(pq)}(𝔽_{p}) which are defined over 𝔽_{p}. The only way that such a component could not yield a smooth 𝔽_{p}-point is if the component had the maximum number of singular points – namely p + 1, and if all of these singularities were themselves defined over 𝔽_{p}. Since singular points on the special fibre correspond to edges in the dual graph, it will be enough to show the following

Claim: For any w_{q}-fixed vertex v of G^{(p)} such that p + 1 edges emanate from v, the w_{q}-action on this set of edges is nontrivial. Sufficiency of the claim: if e is an edge of a w_{q}-fixed vertex, then w_{q}(e(1)) = e(1) iff the terminal vertex of the associated vertex in G^{(pq)} is w_{q}-fixed. (To paraphrase, we are exploiting – not for the first time – the following trick: the graph G^{(p)} is the one we understand the best, thanks to Propositions 10 and 11. On the other hand, we are interested in the graph G^{(pq)}, for which we are not offering a direct interpretation in terms of supersingular elliptic curves. However, all the questions we ask pertain to w_{q}-fixed points, and, since w_{pq} = w_{p} ∘ w_{q}, locally about any w_{q}-fixed point the two graphs are isomorphic.)

Proof of the claim: Because of Propositions 10 and 11, we can rephrase in terms of supersingular elliptic curves: namely, it is enough to show that if E/𝔽_{q} is a supersingular elliptic curve, it is not the case that the Galois (=Frobenius)- action on the set of order p-subgroups of E is trivial. First assume that q > 3, so the characteristic polynomial of Frobenius must be X^{2} + q, and the trace of Frobenius acting on E[p] is zero, hence the matrix is nonscalar and accordingly moves some one-dimensional 𝔽_{p}-subspace, establishing the claim in this case. Now assume that q is 2 or 3, so the characterstic polynomial of Frobenius, if not X^{2} + q, is x^{2} \frac{+}{−}qX + q, whose rational canonical form as an endomorphism of T_{p}(E) is [[0 − q][1\frac{+}{−}q]], so that the action on E[p] is again non-scalar. This completes the proof of the claim, and hence of Theorem 1.

3 Proofs of Theorems 2 and 3

Having done all the necessary analysis in the preceding section, we can immediately reduce to the realm of Γ_{•}(N)-structures on supersingular elliptic curves over 𝔽_{q}. Indeed, using the discussion at the beginning of Section 2 and using Corollary 9, a sufficient condition for X_{•}^{pq}(N)/w_{pq}(ℚ_{p}) to be empty is for the action of w_{p} on the dual graph G(X_{•}^{pq}(N)/w_{pq}) to have no fixed points. By the same argument used at the end of the proof of Theorem 1, if v is a w_{q}-fixed vertex of G^{(p)}, then either the cardinality of the star of v is less than p + 1 or the action of w_{q} on the set of edges emanating from v will be nontrivial, implying that at least one of the p + 1 𝔽_{p}-rational points on the rational curve corresponding to v is smooth, whence we have established:

Proposition 9 X_{•}^{pq}(N)/w_{pq}(ℚ_{p}) is empty iff the action of w_{q} on G^{(p)} is fixed point-free.

Translating to the language of supersingular elliptic curves, we get:

Proposition 10 X_{0}^{pq}(N)/w_{pq}(ℚ_{p}) is empty iff there is no pair (E, C)/𝔽_{q}, E/𝔽_{q} a supersingualr elliptic curve, C ≤ E a cyclic order N (hence etale!) subgroup scheme of E defined over 𝔽_{q}. X_{1}^{pq}(N)/w_{pq}(ℚ_{p}) is empty iff there is no pair (E, P)/𝔽_{q}, where E/𝔽_{q} is a supersingular elliptic curve and P ∈ E[N](𝔽_{q}) is a point of exact order N.

Proof of Theorem 2: Assume first that q ≡ 1 modulo 4, so in particular q > 3. Suppose there exists a pair (E, C)/𝔽_{q} as in the proposition. Since q > 3, the characteristic polynomial of Frobenius is necessarily X^{2} + q, so End_{𝔽_{q}}(E) = ℤ[\sqrt{−q}], and this is the maximal order in the CM quadratic field ℚ[\sqrt{−q}]. By Deuring’s lifting lemma (see e.g. [Gross-Zagier]), there exists Ẽ/\overline{ℚ_{q}} an elliptic curve with CM by Z[\sqrt{−q}] and (Ẽ, \sqrt{−q}) reduces to (E, F rob_{q}) modulo q. Moreover, using the isomorphism of prime-to-q- adic Tate modules T^{q̸}(Ẽ) → T^{q̸}(E), we can lift C to a subgroup C̃ ≤ Ẽ. Since this isomorphism respects the CM-structures, we get moreover that C̃ is stable under End(Ẽ) = ℤ[\sqrt{−q}]. This implies that ℤ[\sqrt{−q}] acts on the quotient Ẽ/C̃; since this is the maximal order, it must be that Ẽ/C̃ has precisely ℤ[\sqrt{−q}]-CM. By the theory of complex multiplication, there exist integral ideals 𝔞, 𝔟 of ℤ[\sqrt{−q}] so that over ℂ the isogeny Ẽ → Ẽ/C̃ may be realized as ℂ/𝔞 → ℂ/𝔞𝔟^{−1}. The degree of this isogeny is on the one hand N and on the other hand the norm of 𝔟, which is what we wanted to show.

Now assume q ≡ −1 modulo 4. Again the ring generated by the Frobenius map inside End^{0}(E) is ℤ[sqrt−q], and by Deuring we can lift to get a pair (Ẽ, C̃), where Ẽ has precisely ℤ[\sqrt{−q}]-CM and C̃ is stable under this ring. So again Ẽ/C̃ has at least ℤ[\sqrt{−q}]-CM, but since this is no longer the maximal order, a priori it could have more. We thus distinguish two cases: in the first case, the quotient Ẽ/C̃ has precisely ℤ[\sqrt{−q}]-CM, so running through the above argument we get that N is the norm of an ideal in the ring ℤ[\sqrt{−q}]. Pushing this ideal forward to the full ring of integers, we get that either N or 2N is a norm from ℚ(\sqrt{−q}). In the second case, Ẽ/C̃ has CM by the full ring of integers. We can write Ẽ = ℂ/𝔞 and define a new elliptic curve F = ℂ/𝔞ℤ[\frac{1+\sqrt{−q}}{2} ], the “improvement” of Ẽ to an elliptic curve with maximal CM. Obviously the

quotient map ϕ : E → F is a degree 2 isogeny; the composite F →^{ϕ̂} Ẽ → Ẽ/C̃ is then an isogeny between elliptic curves with CM by the maximal order, showing that 2N is a norm from ℚ(\sqrt{−q}).

Proof of Theorem 3: This is the easiest case. Indeed, if (E, P)/𝔽_{q} is as in (3.2) above, the existence of P implies that #E(𝔽_{q}) ≥ N. We can use the Weil bound on the number of points of E(𝔽_{q}) to get N ≤ q + 1 + \sqrt{2}q. And indeed since E is supersingular, we can do better: e.g. if q < 3, we get N ≤ q + 1, and in case q = 2 or q = 3, we get N ≤ q + 1 + q.

4 Some final thoughts

In this section we collect some disparate remarks and parting thoughts.

The Corollary to Theorem 1 is interesting in light of the conjecture (suggested by Mazur to the author) that for all but finitely many Shimura curves, the only ℚ-rational points should be the rational CM points. The latter are easy to work with – for each of the nine class number one CM fields K, there exists a rational point with CM by K on X^{pq}/w_{pq} iff p and q are both nonsplit in K. It follows that the “probability” that there are no rational CM points at all is (3/4)^{9}, i.e. this situation occurs in a positive density set of all quaternionic discriminants pq. Thus, by the corollary, Mazur’s conjecture implies the

Conjecture 11 There exist infinitely many Shimura curves X^{pq}/w_{pq} which violate the Hasse principle, i.e. have points everywhere locally but not globally.

If true, it would certainly be proof of the more lively Diophantine properties enjoyed by these Shimura curve quotients in contrast to elliptic modular curves.

The reader will have noticed that the statement of Theorem 1 is the strongest possible statement, whereas Theorems 2 and 3 both leave room for improvement, i.e. they do not give the necessary condition for the ℚ_{p}-points to be empty. In the case of Theorem 3 it would be interesting to look at some actual examples of PQMs A/ℚ – other thesis work of the author classifies the discriminants (ramified at any number primes) of rational quaternion algebras B/ℚ such that there exist infinitely many B-PQM abelian surfaces: there are 47 such discriminants, so there is no lack of examples. The estimate on N in Theorem 3 is simple-minded but seemingly not so bad (keep in mind that we can “optimize” over both ramified primes, but it is surely not sharp. In contrast, something like the converse to Theorem 2 must be true and easy to prove, limited only by the author’s impatience in writing up these notes. It might also be fun to use Brandt matrices to actually exhibit dual graphs in the Γ_{0}(N) case which give examples of curves having and not having ℚ_{p}-points.

Finally, it is natural to ask to generalize these three problems to the case of an arbitrary quaternionic discriminant – it is hard to believe that the results would be different. It is certainly a strange fact of life, however, that any indefinite division quaternion algebra B/ℚ occurs as the endomorphism algebra of some abelian surface A/\overline{ℚ}, but that only the definite quaternion algebras of prime discriminant arise as endomorphism algebras of supersingular elliptic curves. It is even stranger that the non-rigid setting, namely that of Γ_{0}(N)-structures, looks more tractable: in this case, one can interpret the dual graph in terms of the category of rank 1 projective modules over a level N-Eichler order in the interchanged quaternion algebra D of discriminant disc(B)/p. This category, though somewhat more awkward to deal with than the category of supersingular elliptic curves, should yield to a similar analysis leading to a generalization of Theorem 2. It is not so clear what should replace the Eichler orders in the Γ_{1}(N) case.