Superspecial Points on Shimura Curves

Abstract: Let $X$ be the Shimura curve attached to an indefinite quaternion $\mathbb{Q}$-algebra $B$ with a maximal order $O_B$. This paper investigates the reduction $X\otimes \mathbb{F}_p$ of $X$ modulo an arbitrary prime $p$, focusing particularly on its superspecial locus. We give an explicit criterion for the existence of superspecial $\mathbb{F}_q$-rational points on $X$. Furthermore, we compute both the number of geometric superspecial points and the number of $\mathbb{F}_p$-rational superspecial points, through the Eichler class number formula and the Selberg trace formula. As a key ingredient, we classify the Dieudonné modules attached to superspecial $O_B$-abelian surfaces, which generalizes Ribet's classification of admissible quaternion bimodules of rank $2$ by dropping the admissible hypothesis. These results generalize Deuring's explicit formula for supersingular elliptic curves over $\mathbb{F}_p$ and give the Shimura-curve analogue of the Ibukiyama-Katsura formulas for principally polarized superspecial abelian surfaces over $\mathbb{F}_p$.

• The paper establishes Honda–Tate criteria for superspecial points over finite fields and shows that, when the residue characteristic divides the discriminant, all points over odd-degree extensions are superspecial and their count is bounded.
• The paper classifies exactly five conjugacy classes of superspecial quaternionic multiplication structures in the bad-reduction case, identifying them with the superspecial locus, singular locus, and irreducible components of the special fiber.
• The paper derives explicit cardinality formulas using Eichler class numbers, optimal embeddings, and Selberg trace formulas, including a finite-field count that generalizes Deuring’s formulas and connects to quaternionic modular forms.

This paper, by Terakado, Xue, and Yu (2608.16036), studies the superspecial locus of the special fiber of a Shimura curve $X = X_{O_B}$ attached to a maximal order $O_B$ in an indefinite quaternion $\mathbb{Q}$-algebra $B$ of discriminant $\Delta$, and computes the number of superspecial points defined over finite fields. The main results generalize Deuring's formulas for supersingular elliptic curves over $\mathbb{F}_p$ and provide the Shimura-curve analogue of the Ibukiyama–Katsura formulas for principally polarized superspecial abelian surfaces.

Setting and moduli interpretation

The coarse moduli scheme $X$ over $\mathbb{Z}$ parametrizes $O_B$-abelian surfaces $(A, \iota)$ satisfying the determinant condition. When $O_B$0, the curve has good reduction and every supersingular point is superspecial, as for classical modular curves. When $O_B$1, the curve has bad reduction in the sense of Čerednik and Drinfeld: the special fiber $O_B$2 is a projective curve whose normalization is a disjoint union of rational curves, with only ordinary double points, and the entire special fiber is supersingular. In this case the paper asks how superspecial points sit inside the special fiber and how many are defined over a prescribed finite field $O_B$3.

Existence criterion via Honda–Tate theory

The first main result (Theorem thm:Fq-pts) gives an explicit criterion for the existence of an $O_B$4-rational superspecial point. The method combines the classification of supersingular Weil $O_B$5-numbers $O_B$6 with $O_B$7 (Proposition prop:summ-Weil-num), the condition that $O_B$8 splits $O_B$9 (equivalently, no prime $\mathbb{Q}$0 splits in $\mathbb{Q}$1), and a Serre-tensor-construction argument showing that any QM abelian surface over $\mathbb{Q}$2 is isogenous to one carrying an integral $\mathbb{Q}$3-action satisfying the determinant condition. A notable structural consequence: when $\mathbb{Q}$4 and $\mathbb{Q}$5 is odd, every $\mathbb{Q}$6-abelian surface over $\mathbb{Q}$7 is automatically superspecial, and $\mathbb{Q}$8 is independent of $\mathbb{Q}$9; hence $B$0 is finite — a bounded point count along all odd degrees, contrasting with the Weil bounds for smooth curves, and explained by the fact that all such points lie in the singular locus.

When $B$1 is even, supersingularity does not imply superspeciality: the paper constructs, via an explicit Dieudonné module $B$2 with $B$3, a supersingular but non-superspecial $B$4-abelian surface over $B$5 of Lie type $B$6 whenever a QM surface with Frobenius $B$7 ($B$8) exists. This yields a corollary characterizing exactly when $B$9 in terms of Kronecker symbols $\Delta$0 and $\Delta$1 at primes dividing $\Delta$2.

Classification of superspecial $\Delta$3-abelian surfaces

Over $\Delta$4, every superspecial abelian surface is isomorphic to $\Delta$5, so isomorphism classes of superspecial $\Delta$6-abelian surfaces correspond to $\Delta$7-conjugacy classes of embeddings $\Delta$8. If $\Delta$9, Morita equivalence gives a single genus. If $\mathbb{F}_p$0, the local problem is the classification of $\mathbb{F}_p$1-conjugacy classes of embeddings $\mathbb{F}_p$2, where $\mathbb{F}_p$3 is the maximal order of the quaternion division $\mathbb{F}_p$4-algebra.

The central classification result (Theorem thm:5-emb) states that there are exactly five such conjugacy classes, with explicit representatives $\mathbb{F}_p$5. This drops the admissibility hypothesis in Ribet's classification of quaternion bimodules (Ribet's admissible classes correspond to $\mathbb{F}_p$6). The proof reformulates embeddings as $\mathbb{F}_p$7-bimodule structures on $\mathbb{F}_p$8, i.e., left modules over $\mathbb{F}_p$9 with $X$0 an Eichler order of level $X$1 in $X$2; the extension data is a 1-cocycle $X$3 with $X$4 subject to an integrality condition, and orbit counting over the three $X$5-lattice types gives $X$6. Distinctness of the five classes is verified by computing Lie types and critical indices (in the sense of Zink):

Embedding	Lie type	Critical indices
$X$7	$X$8	$X$9
$\mathbb{Z}$0	$\mathbb{Z}$1	$\mathbb{Z}$2
$\mathbb{Z}$3	$\mathbb{Z}$4	$\mathbb{Z}$5
$\mathbb{Z}$6	$\mathbb{Z}$7	—
$\mathbb{Z}$8	$\mathbb{Z}$9	—

The five genera acquire geometric meaning through Zink's and Ribet's work: $O_B$0, the singular locus is $O_B$1 (Lie type $O_B$2, critical indices $O_B$3), and the irreducible components of $O_B$4 are parametrized by $O_B$5 (Lie types $O_B$6 and $O_B$7, via division by $O_B$8-subgroups). Frobenius $O_B$9 swaps $(A, \iota)$0 and $(A, \iota)$1, and fixes $(A, \iota)$2 pointwise on $(A, \iota)$3-rational points: $(A, \iota)$4.

The endomorphism rings $(A, \iota)$5 are centralizers in $(A, \iota)$6: residually inert Bass orders of discriminant $(A, \iota)$7 for $(A, \iota)$8; Eichler orders of level $(A, \iota)$9 for $O_B$00; maximal orders for $O_B$01. Each genus is identified adelically with an ideal class set $O_B$02 of a definite quaternion order, so geometric counting reduces to class numbers.

Explicit counting formulas

The main theorem gives closed formulas. With $O_B$03 if $O_B$04 and $O_B$05 if $O_B$06, part (1) gives, for good reduction,

$O_B$07

and part (2) gives analogous formulas for each $O_B$08 in the ramified case, with the $O_B$09 formulas carrying factors $O_B$10 and correction terms $O_B$11 depending on $O_B$12 and $O_B$13. The derivation uses the Eichler class number and mass formulas together with local optimal embedding numbers, the latter supplying the correction terms needed because, absent level structure, nontrivial automorphisms (units of $O_B$14 and $O_B$15) make the mass differ from the unweighted class number.

The count of $O_B$16-rational points is the most technically involved part. Via the adelic description, Frobenius acts on $O_B$17 by right multiplication by the Frobenius endomorphism $O_B$18, so

$O_B$19

where $O_B$20 is an Atkin–Lehner type Hecke operator on the space of algebraic modular forms on the definite quaternion algebra $O_B$21. Evaluating $O_B$22 by the Selberg trace formula for compact quotients, the orbital integrals reduce to sums over optimal embeddings of the quadratic orders $O_B$23 generated by supersingular Weil $O_B$24-numbers, yielding modified Hurwitz class numbers. The final formula for $O_B$25 is, for $O_B$26,

$O_B$27

with separate elementary product formulas for $O_B$28 and $O_B$29. The trace-formula approach handles the ramified case uniformly; in the split case the classical Eichler trace formula for Brandt matrices also applies. A consistency check recovers the Deuring relation $O_B$30, and for $O_B$31 the formula degenerates to Deuring's $O_B$32 for supersingular elliptic curves over $O_B$33.

Relation to prior work and limitations

The paper's criterion for rational superspecial points is moduli-theoretic and complements the Jordan–Livné criterion for rational points on Shimura curves over local fields, which uses the dual graph of the special fiber; the two approaches are complementary rather than overlapping. The counting formulas depend on the fixed maximal order $O_B$34 only through the discriminant $O_B$35, as expected. The paper leaves open the generalization of the five-class classification beyond $O_B$36 (noting that Xue and Yang have treated arbitrary $O_B$37 by different methods), and the corresponding analysis for Shimura curves with nontrivial level structure or for higher-dimensional quaternionic Shimura varieties is not addressed here. The authors also note that the correspondence between the five bimodule classes of Lemma lem:mod-str-counting and the embeddings $O_B$38 is established but its details are omitted as it is not used elsewhere.

Conclusion

The paper completes the arithmetic picture of superspecial points on Shimura curves at an arbitrary prime $O_B$39: it gives a Honda–Tate criterion for their existence over $O_B$40, classifies the five genera of superspecial $O_B$41-abelian surfaces via a non-admissible extension of Ribet's bimodule theory, identifies these genera with the superspecial locus, singular locus, and irreducible components in bad reduction, and computes all associated cardinalities through Eichler class number formulas and the Selberg trace formula. The resulting formulas extend Deuring's classical counts and provide the unpolarized, quaternionic analogue of the Hashimoto–Ibukiyama–Katsura theory of superspecial abelian surfaces.