On the arithmetic of the curves \(y^2=x^l+A\). II
This paper deals with the arithmetic of the curves \(C_A: y^2 = x^\ell + A\) and their Jacobians \(J_A\), where \(\ell>2\) is a prime, \(A\) is an integer prime to \(\ell\). It is a sequel to a paper by the same author [J. Reine Angew. Math. 501, 171-189 (1998; Zbl 0902.11024)]. The main fact giving leverage in this situation is that \(J_A\) has complex multiplication by the cyclotomic field \(K = \mathbb{Q}(\zeta_\ell)\). The first part of the paper extends a result from op. cit. concerning the Selmer group of \(J_A\). It roughly says that \[ \dim_{\mathbb{F}_\ell} \text{Sel}^{(1 - \zeta_\ell)}(K, J_A) = d_\ell(A) + 2c_\ell(A), \] where \(d_\ell(A)\) is about \((\ell-1)/4\) and depends only on \(A\) mod \(\ell^2\) and the sign of \(A\), whereas \(c_\ell(A)\) depends only on the Galois module structure of the \(\ell\) torsion in the class group of \(K(\sqrt{A})\). By the work of \textit{G. Shimura} [Abelian varieties with complex multiplication and modular functions (1998; Zbl 0908.11023)] and \textit{J. S. Milne} [Invent. Math. 17, 177-190 (1972; Zbl 0249.14012)], one knows that the \(L\)-function of \(C_A\), or \(J_A\), is the \(L\)-function of a Hecke character \(\eta_A\) of \(K\). The second part of the paper calculates \(\eta_A\). The method is interesting. The author calculates, using Jacobi sums, the number of points on the reduction of \(C_A\) modulo primes not dividing \(2\ell A\). This dictates the nature of \(\eta_A\) at almost every prime. The author provides a Hecke character \(\eta\) whose behavior at almost every prime is as needed. This implies \(\eta = \eta_A\). In this part of the paper, the author also finds the root number of the \(L\)-series. The third part of the paper establishes a Birch-Swinnerton-Dyer type of result: assuming that \(\text{ Ш}({\mathbb{Q}}, J_A)[\ell]\) is finite, it is proven that \[ \text{ord}_{s=1}L(J_A, s) \equiv \operatorname {rank} J_A(\mathbb{Q}) \pmod{2}. \]
scientific article; zbMATH DE number 1765873
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