arXiv:2606.05820v1

Drinfeld modules in rank 2 with CM and S-unit j-invariants

Liam Baker, Fabien Pazuki, Patricio Perez Pina

math.NT11G0911G1511R3711Y4014G1714G4011J93

Paper metadata

Original submission: June 4, 2026
Claimed MSC: 11G09, 11G15, 11R37, 11Y40, 14G17, 14G40, 11J93
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗

Abstract

We prove the finiteness of the set of $j$ -invariants of Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$ which are CM and $S$ -units, for $S$ the infinite set of primes with even degrees. The proof is based on the study of ordinary reduction and supersingular reduction of Drinfeld modules, and on the splitting behaviour of primes dividing the difference of two Drinfeld singular moduli. We also provide an algorithm to compute a polynomial with coefficients in $\mathbb{F}_q[T]$ and roots the $j$ -invariants having CM by a given order, and use it to compute some explicit examples, providing for instance counterexamples to a conjecture of Dorman. For a maximal order $\mathcal{O}$ , we prove by a universality argument that our algorithm computes the Hilbert modular polynomial $H_\mathcal{O}$ .

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The finiteness theorem for rank-two CM Drinfeld $j$ -invariants that are units outside the even-degree primes, the ordinary reduction-and-lifting correspondence, and the effective construction of CM polynomials are supported under the hypotheses stated in the paper.

Theorem A and Propositions 2.4–2.5Correct

$S_0$ -unit finiteness for CM $j$ -invariants

Pages 2 and 6–7 · arXiv:2606.05820v1

Here $S_0$ is the infinite set of finite primes of even degree. For a CM field different from $\mathbb F_{q^2}(T)$ , Theorem C shows that a prime dividing both the given singular modulus and the modulus $j=0$ must be inert in both CM fields; such a prime has odd degree and hence is outside $S_0$ . Thus an $S_0$ -unit singular modulus is actually a unit. For the exceptional CM field $\mathbb F_{q^2}(T)$ , Proposition 2.5 shows that every degree-one prime divides the modulus, excluding the $S_0$ -unit case. Theorem A then follows from the cited finiteness theorem for CM units.

Theorem DCorrect

Reduction and lifting of CM classes

Pages 3 and 5–6 · Theorem D and its proof · arXiv:2606.05820v1

For a fixed ordinary reduction, Proposition 2.1 identifies precisely how the characteristic prime changes the conductor. Restricting to $J_{\mathfrak p}$ forces the lifted and reduced endomorphism orders to agree. The cited function-field Deuring lifting theorem gives surjectivity. After fixing an order $O$ , Proposition 1.1 and the simple transitive class-group action of Theorem 2.2 give cardinality $\#\operatorname{Cl}(O)$ on both sides, so the restricted reduction map is injective as well.

Theorem ECorrect

Effective construction of the CM polynomial

Pages 3 and 9–12 · Theorem E and its proof · arXiv:2606.05820v1

Writing an endomorphism as an additive polynomial and imposing $\varphi\Phi_T=\Phi_T\varphi$ gives a finite recursive system for its coefficients. Eliminating the auxiliary coefficient produces a monic polynomial $P_O$ whose roots are exactly the singular moduli with CM by orders containing $O$ . Galois symmetry places $P_O$ in $A[X]$ in the separable case, while squaring does so in the inseparable case. The universal rank-one Drinfeld-module argument, with level structure when needed, proves multiplicity one for a maximal order and hence identifies the resulting polynomial with its Hilbert class polynomial.

02Proofs1 reported findingCorrect

The splitting-and-conductor reduction, class-group lifting, $S_0$ -unit reduction, and coefficient-elimination arguments are correct and complete. The maximal-order hypothesis is used only for the final multiplicity-one identification in Theorem E, not for Theorem D.

Proofs of Theorems A, D and ECorrect and complete

Reduction, lifting, finiteness, and effective CM-polynomial chains

Pages 5–12 · arXiv:2606.05820v1

Theorem C separates the split and nonsplit reduction cases and tracks conductor divisibility explicitly. Theorem D restricts to $J_{\mathfrak p}$ before comparing class-group torsors. Propositions 2.4–2.5 reduce the infinite set $S_0$ to the previously proved unit case without discarding the exceptional CM field. In Theorem E, the finite coefficient recursion proves both directions of the root characterization, and the universal-family argument is invoked only where the order is maximal.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.05820v1
Authors listed: Liam Baker, Fabien Pazuki, Patricio Perez Pina
Audit date: August 18, 2026

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Drinfeld modules in rank 2 with CM and S-unit j-invariants

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Abstract

Audit summary

Detailed mathematical audit

S0S_0S0​-unit finiteness for CM jjj-invariants

Reduction and lifting of CM classes

Effective construction of the CM polynomial

Reduction, lifting, finiteness, and effective CM-polynomial chains

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$S_0$ -unit finiteness for CM $j$ -invariants