On component groups of Jacobians of Drinfeld modular curves.
Given an abelian variety \(A\) over a local field \(K\) with Neron model \({\mathcal A}\), the component group \(\Phi_A\) of the special fiber of \({\mathcal A}\) is an important invariant. In number theoretical situations, if \(A\) is the Jacobian variety of a modular curve, important arithmetical information is encoded in \(\Phi_{A,p}\), the component group at a prime \(p\), and in its relation with other data, e.g., the cuspidal divisor class group \(C\) of \(A\). Let now \(A\) be the Jacobian \(J_0(N)\) of the Hecke modular curve \(X_0(N)\) with prime conductor \(N\). In this situation, there are three basic results: (1) (B. Mazur) The canonical maps \(C \to \Phi_{J_0(N),N}\) and \(C \to J_0(N)({\mathbb Q})_{\text{tor}}\) are bijective. (2) (Former conjecture of W. Stein, proved by M. Emerton) Let \(A_f\) be the optimal abelian variety quotient of \(J_0(N)\) corresponding to a normalized weight-2 newform of level \(N\). Then the specialization mappings from the cuspidal group \(C_f\) of \(A_f\) to the component group \(\Phi_{A_f,N}\) and to \(A_f({\mathbb Q})_{\text{tor}}\) are isomorphisms. (3) (Emerton) The natural map on component groups \(\Phi_{J_0(N),N} \to \Phi_{A_f,N}\) is surjective. The author first discusses these results (and their failure if \(N\) is no longer assumed prime) and then considers the analogous situation where \(J_0(N)\) is replaced by the Jacobian of a Drinfeld modular curve. Here, as the author writes, ``some effort has been made to transfer Mazur's results on the Eisenstein ideal to the context of Drinfeld modular curves \dots ''; ``One encounters very non-trivial technical difficulties while doing this, so the theory over function fields is not yet as satisfactory as over \({\mathbb Q}\). There is an analogue of the cuspidal group \(C \subset J_0(N)({\mathbb Q})\), and Gekeler proved \dots that the specialization map \(C_{\mathfrak n}:\: C \to \Phi_{J_0(\mathfrak n),\mathfrak n}\) is an isomorphism.'' ``Even though one should expect (analogues of (1), (2), (3)) to be true in this case, proving these will require additional efforts to develop the techniques of Mazur and Ribet over \dots function fields.'' A special feature of the function field situation is the existence of reductions at the place \(\infty\) at infinity, which have no immediate counterparts in the classical framework. Therefore it is worthwhile to investigate the above questions (1), (2), (3) at \(\infty\). It is known that the analogue of (2) at \(\infty\) fails. The present paper focusses on the \(\infty\)-adic analogue of (3), without any restrictions on the level \(\mathfrak n\). Let \(J_0(\mathfrak n)\) be the Drinfeld modular curve of Hecke type of level \(\mathfrak n\), let \(A\) be an optimal abelian variety quotient of \(J_0(\mathfrak n)\), and let \(B\) be the kernel of the quotient map. The main result is Theorem 1.5, a sufficient criterion for the exactness of the canonical sequence \[ 0 \to \Phi_{B,\infty} \to \Phi_{J_0(\mathfrak n), \infty} \to \Phi_{A,\infty} \to 0 \] of component groups of \(B\), \(J_0(\mathfrak n)\) and \(A\) at \(\infty\). It is satisfied whenever either \(A\) or \(B\) is an elliptic curve. As the author remarks, the component groups at \(\infty\) are much more mysterious than the corresponding groups at finite places, essentially, since \(\infty\) does not appear in the formulation of the moduli problem that defines the Drinfeld modular curve \(X_0(\mathfrak n)\), and hence it is less obvious how to use deformation theory to deduce the structure of the special fiber of \(X_0(\mathfrak n)\) at \(\infty\). The key ingredient in the proof of Theorem 1.5 is the study of the canonical polarization on \(A\). The author computes the order of its kernel (a finite group scheme) in two different ways: one algebraic, based on Grothendieck's monodromy pairing, the other rigid-analytic, based on the analytic description of \(J_0(\mathfrak n)\), from which Theorem 1.5 follows.
scientific article; zbMATH DE number 2162453
scientific article