Hida families and rational points on elliptic curves
Hida families and rational points on elliptic curves
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飞驒族和椭圆曲线上的有理点
DOI：
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发表时间：
2007
期刊：
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通讯作者：
H. Darmon
中科院分区：
文献类型：
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作者：
M. Bertolini;H. Darmon
设E为导体N = Mp的Q上的椭圆曲线，有一个素数p‖N的乘法约简。由于E是模的，它对应于Γ0(N)上的一个归一化权2特征形式，其q展开记为f =∑nanq。LetX:= homZ×p,Z×p)Z /(p−1)Z×Zp，其中通过将字符x→xk−2与k∈Z关联，将Z作为密集子集包含。用A(U)表示X的紧开子集U上的p值p进解析函数的环。Hida的理论将f联系到2∈X的邻域U（为简单起见，可以假设它包含在2模p−1的残馀类中）和一个形式的q展开
Let E be an elliptic curve over Q of conductor N = Mp, having a prime p‖N of multiplicative reduction. Because E is modular, it corresponds to a normalised weight two eigenform on Γ0(N), whose q-expansion is denoted f = ∑n anq . LetX := hom(Z×p ,Z×p ) Z/(p−1)Z×Zp, which contains Z as a dense subset by associating to k ∈ Z the character x → xk−2. Denote by A(U) the ring of Cp-valued p-adic analytic functions on a compact open subset U of X. Hida’s theory associates to f a neighborhood U of 2 ∈ X (which can be assumed, for simplicity, to be contained in the residue class of 2 modulo p − 1) and a formal q-expansion