Cuspidal divisor group conjecture for Drinfeld modular Jacobians

Let AA be the ring of functions regular away from a fixed place at infinity, and let nA\mathfrak n\lhd A be a non-zero square-free ideal. Let X0(n)X_0(\mathfrak n) be the smooth projective Drinfeld modular curve, let J0(n)J_0(\mathfrak n) be its Jacobian, let C(n)\mathcal{C}(\mathfrak n) be the subgroup generated by classes of divisors ccc-c' for cusps c,cc,c' of X0(n)X_0(\mathfrak n), and let T(n):=J0(n)(F)tor\mathcal{T}(\mathfrak n):=J_0(\mathfrak n)(F)_\mathrm{tor}, where FF is the function field of AA. For square-free n\mathfrak n, the cusps are FF-rational, so C(n)T(n)\mathcal{C}(\mathfrak n)\subseteq\mathcal{T}(\mathfrak n). Drinfeld modular cuspidal torsion conjecture. For every square-free ideal nA\mathfrak n\lhd A, the two groups are equal:

C(n)=T(n).\mathcal{C}(\mathfrak n)=\mathcal{T}(\mathfrak n).

This is proposed as the Drinfeld-modular analogue of the generalized Ogg conjecture. The source gives no resolution status for the assertion.

Sources & referencesView supporting material

Primary source

Mihran Papikian and Fu-Tsun Wei, “The rational torsion subgroups of Drinfeld modular Jacobians and Eisenstein pseudo-harmonic cochains”, arXiv:1512.00586 (2015).

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