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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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