Drinfeld modular rational cuspidal torsion conjecture

Let Fq\mathbb{F}_q be a finite field, let A=Fq[T]\mathcal{A}=\mathbb{F}_q[T], and let K=Fq(T)K=\mathbb{F}_q(T). For a monic nA\mathfrak{n}\in\mathcal{A}, let X0(n)X_0(\mathfrak{n}) be the Drinfeld modular curve over KK, let J0(n)J_0(\mathfrak{n}) be its Jacobian, let Cn\mathcal{C}_{\mathfrak{n}} be the cuspidal subgroup, and let Cn(K)=CnJ0(n)(K)\mathcal{C}_{\mathfrak{n}}(K)=\mathcal{C}_{\mathfrak{n}}\cap J_0(\mathfrak{n})(K). Let C(n)\mathcal{C}(\mathfrak{n}) be the rational cuspidal divisor class group, consisting of classes of KK-rational divisors fixed by the absolute Galois group of KK. Drinfeld modular rational cuspidal torsion conjecture. For every monic nA\mathfrak{n}\in\mathcal{A},

C(n)=Cn(K)=J0(n)(K)tors.\mathcal{C}(\mathfrak{n})=\mathcal{C}_{\mathfrak{n}}(K)=J_0(\mathfrak{n})(K)_{\operatorname{tors}}.

The inclusions C(n)Cn(K)J0(n)(K)tors\mathcal{C}(\mathfrak{n})\subseteq\mathcal{C}_{\mathfrak{n}}(K)\subseteq J_0(\mathfrak{n})(K)_{\operatorname{tors}} are established in the context, and this conjecture predicts equality throughout. Its resolution is not specified in the supplied source.

Sources & referencesView supporting material

Primary source

Mar Curcó-Iranzo, “Rational torsion of generalised Drinfeld modular Jacobians of prime power level”, arXiv:2412.14313 (2025).

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