p -adic Rankin L -series and rational points on CM elliptic curves (original) (raw)
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Journal of Number Theory, 1976
An elementary proof is given for the existence of the Kubota-Leopoldt padic L-functions. Also, an explicit formula is obtained for these functions, and a relationship between the values of the padic and classical L-functions at positive integers is discussed.
$p$-adic interpolation of special values of Hecke L-functions
1992
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p-Adic Aspects of Modular Forms, 2016
These are the expanded notes of a mini-course of four lectures by the same title given in the workshop "p-adic aspects of modular forms" held at IISER Pune, in June, 2014. We give a brief introduction of p-adic L-functions attached to certain types of automorphic forms on GLn with the specific aim to understand the p-adic symmetric cube L-function attached to cusp forms on GL 2 over rational numbers. Contents 1. What is a p-adic L-function? 2 2. The symmetric power L-functions 11 3. p-adic L-functions for GL 4 16 4. p-adic L-functions for GL 3 × GL 2 22 References 27 The aim of this survey article is to bring together some known constructions of the p-adic L-functions associated to cohomological, cuspidal automorphic representations on GL n /Q. In particular, we wish to briefly recall the various approaches to construct p-adic L-functions with a focus on the construction of the p-adic L-functions for the Sym 3 transfer of a cuspidal automorphic representation π of GL 2 /Q. We note that p-adic L-functions for modular forms or automorphic representations are defined using p-adic measures. In almost all cases, these p-adic measures are constructed using the fact that the L-functions have integral representations, for example as suitable Mellin transforms. Candidates for distributions corresponding to automorphic forms can be written down using such integral representations of the L-functions at the critical points. The well-known Prop. 2 is often used to prove that they are indeed distributions, which is usually a consequence of the defining relations of the Hecke operators. Boundedness of these distributions are shown by proving certain finiteness or integrality properties, giving the sought after p-adic measures. In Sect. 1, we discuss general notions concerning p-adic L-functions, including our working definition of what we mean by a p-adic L-function. As a concrete example, we discuss the construction of the p-adic L-functions that interpolate critical values of L-functions attached to modular forms. Manin [47]
A relation between p-adic L-functions and the Tamagawa number conjecture for Hecke characters
Archiv der Mathematik, 2004
We prove that the submodule in K-theory which gives the exact value (up to Z * (p) ) of the L-function by the Beilinson regulator map at noncritical values for Hecke characters of imaginary quadratic fields K with cl(K) = 1(p-local Tamagawa number conjecture) satisfies that the length of its coimage under the local Soulé regulator map is the p-adic valuation of certain special values of p-adic L-functions associated to the Hecke characters. This result yields immediately, up to Jannsen's conjecture, an upper bound for #H 2 et (OK [1/S], Vp(m)) in terms of the valuation of these p-adic L-functions, where V p denotes the p-adic realization of a Hecke motive. *
2015
Let z ∈ Q and let γ be an `-adic path on P1Q̄\{0, 1,∞} from → 01 to z. For any σ ∈ Gal(Q̄/Q), the element x−κ(σ)fγ(σ) ∈ π1(P1Q̄ \{0, 1,∞}, → 01)pro−`. After the embedding of π1 into Q{{X,Y }} we get the formal power series ∆γ(σ) ∈ Q{{X,Y }}. We shall express coefficients of ∆γ(σ) as integrals over (Z`) with respect to some measures Kr(z). The measures Kr(z) are constructed using the tower ( P1Q̄ \({0,∞}∪μ`n ) n∈N of coverings of P 1 Q̄ \{0, 1,∞}. Using the integral formulas we shall show congruence relations between coefficients of the formal power series ∆γ(σ). The congruence relations allow the construction of `-adic functions of non-Archimedean analysis, which however rest mysterious. Only in the special case of the measures K1( → 10) and K1(−1) we recover the familiar Kubota-Leopoldt `-adic L-functions. We recover also `-adic analogues of Hurwitz zeta functions. Hence we get also `-adic analogues of L-series for Dirichlet characters.
Special values of hecke L-functions and abelian integrals
In this article we attempt to explain the formalism of Deligne's rationality conjecture for special values of motivic L-functions (see [DI]) in the particular case of L-functions attached to algebraic Hecke characters ("Gr~Bencharaktere of type A0"). In this case the conjecture is now a theorem by virtue of two complementary results, due to D. Blasius and G. Harder, respectively: see §5 below. For any "motive" over an algebraic number field, Deligne's conjecture relates certain special values of its L-function to certain periods of the motive. Most of the time when motives come up in a geometric situation, we tend to know very little about their L-functions. In the special case envisaged here, however, the situation is quite different: The Lfunctions of algebraic Hecke characters are among those for which Hecke proved analytic continuation to the whole complex plane and functional equation. But the "geometry" of the corresponding motives has eme...