p-adic L-functions for Rankin–Selberg convolutions over number fields

Januszewski, Fabian

doi:10.1007/s40316-016-0061-y

Cited by 10 publications

(21 citation statements)

References 18 publications

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“…Following the terminology of [17,Section 3], adapted to the Shalika case, we call an absolutely irreducible rational G-module M over E/Q arithmetic, if it is essentially self-dual over Q, i.e.…”

Section: Arithmeticity Conditionsmentioning

confidence: 99%

On period relations for automorphic 𝐿-functions I

Januszewski

2018

Trans. Amer. Math. Soc.

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We study Hecke algebras for pairs (g, K) over arbitrary fields E of characteristic 0, define the Bernstein functor and give another definition of the Zuckerman functor over E. Building on this and the author's previous work on rational structures on automorphic representations, we show that hard duality remains valid over E and apply this result to the study of rationality properties of Sun's cohomologically induced functionals. Our main application are period relations for the special values of standard L-functions of automorphic representations of GL(2n) admitting Shalika models.

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Section: Arithmeticity Conditionsmentioning

confidence: 99%

On period relations for automorphic 𝐿-functions I

Januszewski

2018

Trans. Amer. Math. Soc.

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“…See Kazhdan-Mazur-Schmidt [KMS], Mahnkopf [Mah], Raghuram [Rag1], Kasten-Schmidt [KS], Januszewski [Jan1], RS2], and Schmidt [Sch1,Sch2]. For more recent works using Theorem A (and Theorem C of Section 6), see Grobner-Harris [GH], Raghuram [Rag2], and Januszewski [Jan2,Jan3,Jan4].…”

Section: Introductionmentioning

confidence: 99%

The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions

Sun

2016

J. Amer. Math. Soc.

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“…For the left hand side, there has been an extensive study of arithmetic properties of critical values for L-functions of GL n × GL n−1 . See, for example, [29], [30], [31], [32], [34], [46], [54], [55], and [59]. To study the p-adic interpolation of these critical values, amongst the above references, Schmidt [59] and Januszewski [31] are particularly relevant.…”

Section: Via L-functions For Glmentioning

confidence: 99%

“…See, for example, [29], [30], [31], [32], [34], [46], [54], [55], and [59]. To study the p-adic interpolation of these critical values, amongst the above references, Schmidt [59] and Januszewski [31] are particularly relevant. We will explicate this theme in § 4 below.…”

Section: Via L-functions For Glmentioning

confidence: 99%

“…For simplicity, we only study the cohomology groups with constant coefficients, and our exposition is based on Schimdt [59]. The reader is referred to Kazhdan, Mazur and Schmidt [34], as well as recent papers of Januszewski ([29] [30] [31]) for generalization to GL n × GL n−1 over a general number field and for representations having cohomology for more general coefficient systems.…”

Section: Interpolation Of L(mentioning

confidence: 99%

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p-adic L-functions for GL(n)

Banerjee,

Raghuram

2015

Preprint

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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 27The 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]

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p-adic L-functions for Rankin–Selberg convolutions over number fields

Abstract: We unconditionally construct cyclotomic p-adic L-functions for Rankin-Selberg convolutions for GL(n + 1) × GL(n) over arbitrary number fields, and show that they satisfy an expected functional equation.

Cited by 10 publications

References 18 publications

On period relations for automorphic 𝐿-functions I

On period relations for automorphic 𝐿-functions I

The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions

p-adic L-functions for GL(n)

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