On the square root of special values of certain \(L\)-series
The main result of the paper is an explicit formula for the value at \(s=1\) of the \(L\)-series \(L(\psi,s)\) attached to certain Hecke characters \(\psi\) of an imaginary quadratic field of discriminant \(p\) where \(p\) is a prime \(\equiv 7\pmod 8\). The two main ingredients of the proof are (i) a formula of Hecke expressing the above \(L\)-series at \(s=1\) as a sum of binary theta series and (ii) a certain factorization lemma that gives the values of these theta series at complex multiplication points in the upper half-plane as the product of values of Jacobi theta functions. The author then applies his formula to the study of the elliptic curve \(A(p)\) over \(\mathbb{Q}(j(1+\sqrt{-p})/2)\) investigated by \textit{B. H. Gross} [Compos. Math. 45, 155-164 (1982; Zbl 0541.14010)] whose \(L\)-series can be expressed in terms of the series \(L(\psi,s)\) as above. He obtains an explicitly computable nonzero rational integer whose square by the Birch- Swinnerton-Dyer conjecture is the predicted order of the Tate-Shafarevich group of \(A(p)\). Finally it is pointed out that results of the type presented in this paper also follow from the work of \textit{J.-L. Waldspurger} [J. Math. Pures Appl., IX. Sér. 60, 375-484 (1981; Zbl 0431.10015)].
theta series at complex multiplication points
scientific article; zbMATH DE number 59787
scientific article