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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 \(K\).

MSC:

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