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The Drinfeld modular Jacobian \(J_1 (N)\) has connected fibers. (English) Zbl 1165.11047
Let \(n\in A:= {\mathbb F}_q[T]\) be a prime element. Let \(X_1(n)\) be the smooth projective curve over \({\mathbb F}_q(T)\) associated to the moduli problem of classifying pairs \((\phi,P)\), where \(\phi\) is a rank 2 Drinfeld \(A\)-module over an \({\mathbb F}_q(T)\)-scheme and \(P\) is a nowhere vanishing point of its \(n\)-torsion.
Let \(J_1(n)\) be the Jacobian of \(X_1(n)\). The main result of this paper is that the closed fibre of the Néron model of \(J_1(n)\) over \(A_{(n)}\) has trivial geometric component group. It follows that the Néron model of \(J_1(n)\) over \({\mathbb P}^1_{{\mathbb F}_q} - \infty\) has connected fibres, hence the title. (The component group of the fibre above \(\infty\) is a much more complicated beast, but is not part of the moduli problem). This work is a function field analogue of results of B. Conrad, B. Edixhoven and W. Stein [Doc. Math., J. DMV 8, 331–408 (2003; Zbl 1101.14311)], and the proof strategy is similar, although the translation is far from trivial.
To prove the result, it suffices to show that \(X_1(n)\) has a regular proper model over \(A_{(n)}\) with geometrically integral special fibre. The author does this by first constructing a model with a unique non-regular point (corresponding to \((\phi,P)\), where \(j(\phi)=0\) and \(P\) is the kernel of Frobenius). This is a cyclic quotient singularity, which is resolved using the Jung-Hirzebruch resolution developed in [loc. cit.]. Contraction of the special fibre of this resolution then yields the desired integral model of \(X_1(n)\).
A result of independent interest obtained along the way is a function field analogue of the theory of Igusa curves. The paper is rather technical, but very well written.
MSC:
11F52 Modular forms associated to Drinfel’d modules
14L05 Formal groups, \(p\)-divisible groups
11G09 Drinfel’d modules; higher-dimensional motives, etc.
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