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$$p$$-adic $$L$$-functions for CM fields. (English) Zbl 0417.12003
The values $$\zeta(-k)$$ of the Riemann zeta-function $$\zeta$$ at negative integers $$-k$$ are known to be rational, from Euler. Furthermore, for any fixed prime number $$p$$, the numbers $$\zeta_a^{(p)} (-k) = (1-a^{k+i})(1-p^k)\zeta(-k)$$ are $$p$$-integral for all integers $$k\geq 0$$ and integers $$a$$ prime to $$p$$ and satisfy the congruence conditions:
$\sum c_r \zeta_a^{(p)}(-r)\equiv 0 \pmod{p^N},$ whenever $$\sum c_rx^r \equiv 0\pmod{p^N}$$, for every $$p$$-adic unit $$x$$ and integers $$c_r$$. The totality of these Kummer congruences (for $$m\geq 1$$gives rise to a measure $$\mu_a$$, on the group $$\mathbb Z_p^\times$$ of $$p$$-adic units (with values in the ring $$\mathbb Z_p$$ of $$p$$-adic integers) such that for all integers $$k\geq 0$$
$\int_{\mathbb Z_p^\times} x^k\,d\mu_a = (1-a^{k+i})(1-p^k) \zeta(-k).$
The corresponding rationality results for values of the Dedekind zeta-function and $$L$$-series for totally real algebraic number fields $$K$$ were established by Siegel and Klingen. The analogues by way of the Kummer congruences and the measure $$\mu_a$$ are due to Coates-Sinnott for quadratic $$K$$ and to Deligne-Ribet for any (totally real) $$K$$. If $$K$$ is not totally real, the values of the $$L$$-series at negative integers are $$0$$ (under the influence of $$\Gamma$$-factors in their functional equation!). One may, however, investigate the values at $$0$$ of Hecke $$L$$-series for grössencharacters of type $$A_0$$ (in the sense of Weil).
If $$L$$ is a totally complex quadratic extension of a totally real field $$K$$ and $$\chi$$, a grössencharacter of type $$A_0$$ with a power of $$p$$ as conductor (viewing $$\chi$$ as a $$p$$-adic character of $$G=\text{Gal}(L(p^\infty)/L)$$, where $$L(p^\infty)$$ is maximal abelian and unramified outside $$p$$ over $$L$$), the author constructs (under suitable conditions) a $$p$$-adic measure $$\mu$$ on $$G$$ such that the Mellin transform $$L_\mu(\chi)$$ is closely related to the value $$L(0,\chi)$$ at $$0$$ of the $$L$$-series associated with $$L$$ and $$chi$$ (and provides a $$p$$-adic interpolation of the Hecke $$L$$-function); a functional equation for the $$p$$-adic function $$L_\mu$$, compatible with the usual functional equation for Hecke $$L$$-series with grössencharacter is also obtained.
[The case when the CM-field $$L$$ is of degree 2 over $$\mathbb Q$$ was treated by the author earlier in Ann. Math. (2) 104, 459–571 (1976; Zbl 0354.14007), using the abundant information specially available in this case when the abelian varieties are just elliptic curves].
The construction is based on the fact that the complex numbers $$L(0, \chi)$$ are finite sums of values of (non-analytic) Eisenstein series for (congruence) subgroups of the Hilbert modular group (over $$K$$) at points corresponding to abelian varieties admitting $$L$$ for their complex multiplications. An application of suitably defined $$p$$-adic differential operators to holomorphic Eisenstein series yields $$p$$-adic Eisenstein series whose values have good interpolation properties.

##### MSC:
 11S40 Zeta functions and $$L$$-functions 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 12H25 $$p$$-adic differential equations 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11G15 Complex multiplication and moduli of abelian varieties
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