ppp-adic interpolation of special values of Hecke L-functions (original) (raw)
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]
31 p-adic L-functions for modular forms
2019
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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.
The Iwasawaμ-invariant ofp-adic HeckeL-functions
Annals of Mathematics, 2010
For an odd prime p, we compute the -invariant of the anticyclotomic Katz p-adic L-function of a p-ordinary CM field if the conductor of the branch character is a product of primes split over the maximal real subfield. Except for rare cases where the root number of the p-adic functional equation is congruent to 1 modulo p, the invariant vanishes.
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.