On the Bloch-Kato conjecture for Hecke \(L\)-functions
Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) with complex multiplication by the ring of integers of an imaginary quadratic field \(K\). Let \(\psi\) be the Größencharacter attached to \(E/K\); for \(k>j>0\), \(\psi^k \overline \psi^{-j}\) is a Größencharacter. The author studies the \(p\)-part of the conjecture of Bloch-Kato for the special value of the Hecke \(L\)-function \(L(\psi^{-k} \overline \psi^j,s)\) at \(s=0\) when the prime \(p\) splits in \(K\) and \(p>k+1\). The Block-Kato conjecture may be considered as a vast generalization, for motives, of the conjecture of Birch and Swinnerton-Dyer on elliptic curves. In the present case, the motive \({\mathcal M}= {\mathcal M}_{k,j}\) comes from the Größencharacter \(\psi^k \overline \psi^{-j}\) and its motivic \(L\)-function coincides with the Hecke \(L\)-function \(L(\psi^{-k} \overline \psi^j, s)\). If \(L(\psi^{-k} \overline \psi^j,0)\neq 0\), the author shows that the \(p\)-part of the Bloch-Kato conjecture giving the Tamagawa number of the motive \({\mathcal M}\) is true, and is in fact equivalent to a formula for the special values of \(L(\psi^{-k} \overline \psi^j,s)\) previously proved by him [Math. Ann. 297, 221-233 (1993; Zbl 0789.14018)].
elliptic curve with complex multiplication
special value of the Hecke \(L\)-function
scientific article; zbMATH DE number 874201