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On the component groups and the Shimura subgroup of \(J_ 0(N)\). (English) Zbl 0691.14009

Sémin. Théor. Nombres, Univ. Bordeaux I 1987-1988, Exp. No. 6, 10 p. (1988).
Let N be a positive integer, and let p be a prime number which is prime to N. Consider the usual modular curves \(X_ 0(N)\) and \(X_ 0(pN)\) over \({\mathbb{Q}}\). Recall that there are two natural degeneracy maps \(X_ 0(pN)\rightrightarrows X_ 0(N)\). We may interpret these maps as follows: An elliptic curve with a \(\Gamma_ 0(pN)\)-structure may be thought of as a triple \((E,C_ N,C_ p)\), where E is an elliptic curve, and \(C_ N\) and \(C_ p\) are cyclic subgroups on E, of order N and p, respectively. Forgetting the subgroup \(C_ p\), we obtain the pair \((E,C_ N)\), which is an elliptic curve with a \(\Gamma_ 0(N)\)- structure. This construction defines one of the two degeneracy maps. To define the other, we divide E by its subgroup \(C_ p\), thereby obtaining a second elliptic curve \(E'\). The image of \(C_ N\) on \(E'\) is a cyclic subgroup \(C'_ N\) on \(E'\), and the pair \((E',C'_ N)\) is the image of \((E,C_ N,C_ p)\) under the second degeneracy map. Let J and \(J'\) be the Jacobians \(J=Pic^ 0(X_ 0(N))\), \(J'=Pic^ 0(X_ 0(pN))\). The two degeneracy maps introduced above induce by Pic functoriality a pair of degeneracy maps \(\alpha\),\(\beta\) : \(J\rightrightarrows J'\), where \(\alpha\) corresponds to the first construction discussed above, and \(\beta\) to the second. The purpose of this note is to illustrate this theme in two contexts. First, recall the natural covering of modular curves \(\pi: X_ 1(N)\to X_ 0(N).\) By Pic functoriality, this covering induces a map \(\pi^*: J\to J_ 1(N).\) The kernel \(\Sigma_ N\) of this homomorphism is the Shimura subgroup of \(J_ 0(N).\)
Theorem 1. The relation \(\alpha =\beta\) holds on \(\Sigma_ N\). In particular, \(\Sigma_ N\) is Eisenstein in the sense of Mazur’s article [B. Mazur, Publ. Math. 47 (1977), 33-186 (1978; Zbl 0394.14008)].
In the second result of this note, we suppose that N is a product Mq, where q is a prime number which is prime to M. Consider the fibers over \({\mathbb{F}}_ q\) of the Néron models of J and \(J'\). These “special fibers” are commutative group schemes which are not (necessarily) connected. Let \(\Phi\) and \(\Phi'\) be the groups of components of the special fibers of J and \(J'\), respectively. The groups \(\Phi\) and \(\Phi'\) are finite abelian groups which can be expressed in terms of the supersingular points of \(X_ 0(M)\) and \(X_ 0(pM)\) in characteristic q. The maps \(\alpha\) and \(\beta\) induce by functoriality homomorphisms \(\alpha_*,\beta_*: \Phi \rightrightarrows \Phi'.\)
Theorem 2. The maps \(\alpha_*\) and \(\beta_*\) are equal. In particular, the group \(\Phi\) is Eisenstein.

MSC:

14G25 Global ground fields in algebraic geometry
14H52 Elliptic curves
14H45 Special algebraic curves and curves of low genus

Citations:

Zbl 0394.14008
Full Text: EuDML