Divisibility of anticyclotomic \(L\)-functions and theta functions with complex multiplication. (English) Zbl 1111.11047

Let \(K\) be an imaginary quadratic field of discriminant \(-D\), let \(p\) be an odd prime that splits in \(K\), and let \(E\) be an elliptic curve with complex multiplication by the ring of integers of \(K\). Let \(\Omega_{\infty}\) be a suitably chosen complex period of \(E\). Let \(\ell\) be a prime other than \(p\). A Hecke character \(\lambda\) is said to be anticyclotomic of infinity type \((-k, k-1)\) if its infinity component is \(\lambda_{\infty}(x)=x^k\bar{x}^{1-k}\) and the restriction of \(\lambda\) to \(\mathbb A_{\mathbb Q}^{\times}\) is the quadratic character for \(K/\mathbb Q\) times the idèlic absolute value. Let \(W(\lambda)\) and \(W_{\ell}(\lambda)\) be the global and local root numbers for the Hecke \(L\)-function \(L(s,\lambda)\). The present paper proves that if \(k>0\) and \(d>0\) are fixed and if \(\ell\) splits in \(K\), then for all anticyclotomic Hecke characters of type \((-k, k-1)\), of conductor dividing \(dDp^{\infty}\), and with \(W(\lambda)=1\), the \(\ell\)-adic valuation of \(\Omega_{\infty}^{1-2k} (k-1)!(2\pi/\sqrt{D})^{k-1} W_{\ell}(\lambda) L(0,\lambda)\) is greater than or equal to a simple expression, with equality for all but finitely many \(\lambda\). If \(\ell\) is inert or ramified in \(K\) and we restrict to infinity type \((-1,0)\), then the \(\ell\)-adic valuation of \(\Omega_{\infty}^{-1}D^{1/4} L(0,\lambda)\) is greater than or equal to a simple expression, with equality for all but finitely many \(\lambda\) (with the possible exception of \(K=\mathbb Q(\sqrt{-3})\) and \(\ell=3\)). The theorem is the anticyclotomic analogue of a theorem of the reviewer [Invent. Math. 49, 87–97 (1978; Zbl 0403.12007)] for cyclotomic \(\mathbb Z_p\)-extensions of abelian number fields, and of a theorem of R. Gillard [Math. Ann. 279, No. 3, 349–372 (1988; Zbl 0646.12003)] for the \(\mathbb Z_p\)-extensions of \(K\) in which only one of the primes above \(p\) ramifies. The proof of the present paper is based on W. Sinnott’s [Journées arithmétiques, Besançon/France 1985, Astérisque 147/148, 209–224 (1987; Zbl 0616.12004)] algebraic proof of the reviewer’s result, Gillard’s version of this method for elliptic curves with complex multiplication, and a result of Tonghai Yang [J. Reine Angew. Math. 485, 25–53 (1997; Zbl 0867.11037)] connecting anticyclotomic \(L\)-values to special values of theta functions on such elliptic curves. H. Hida [The Iwasawa \(\mu\)-invariant of \(p\)-adic Hecke \(L\)-functions; preprint] has obtained similar results by different techniques in the case where \(\ell\) splits in \(K\).


11R23 Iwasawa theory
11Fxx Discontinuous groups and automorphic forms
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11R29 Class numbers, class groups, discriminants
Full Text: DOI