On the algebraicity of some products of special values of Barnes' multiple gamma function

Kashio, Tomokazu

doi:10.48550/arxiv.1510.01141

Cited by 2 publications

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“…(9) was proved in [Ka3, Proof of Theorem 3.1] by using Lemmas 1, 2, 3, and [Ka3, Lemma 3.4]. We can repeat the same argument for the p-adic analogue (10) by Lemma 4: Let D, ν, X t , t (t ∈ T , |T | < ∞) be as in [Ka3,Lemma 3.4]. Then we have…”

Section: The Case When a Real Place Splits Completelymentioning

confidence: 72%

“…We also provide their p-adic analogues by quite similar arguments. There are two applications of these properties: First, in §5, we prove the algebraicity of some products of exp p (X p (c, ι))'s, which is the p-adic analogue of the main results in [Ka3]. Then we can clarify the relation between Stark units and the ratios [exp(X(c, ι)) : exp p (X p (c, ι))].…”

Section: Introductionmentioning

confidence: 82%

“…Theorem 4 is one of the main results in this paper. The Archimedean part (9) was proved in [Ka3]. The p-adic part (10) is a new result, although it is proved by quite similar arguments.…”

Section: The Case When a Real Place Splits Completelymentioning

confidence: 92%

“…The following Theorem is essentially due to Yoshida: Yoshida transformed Shintani's formula (1) in [Shin2] into a similar form. Yoshida and the author modified the W -term slightly in [KY1], [Ka3] into the present form.…”

Section: The Case Of a Totally Positive Dmentioning

confidence: 99%

“…Proof. We use a similar argument to the proof of [Ka3,Lemma 3.6]. We can reduce the problem to a refinement of a cone:…”

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confidence: 99%

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On the ratios of Barnes' multiple gamma functions to the p-adic analogues

Kashio

2019

Journal of Number Theory

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Let F be a totally real field. For each ideal class c of F and each real embedding ι of F , Hiroyuki Yoshida defined an invariant X(c, ι) as a finite sum of log of Barnes' multiple gamma functions with some correction terms. Then the derivative value of the partial zeta function ζ(s, c) has a canonical decomposition ζ (0, c) = ι X(c, ι), where ι runs over all real embeddings of F . Yoshida studied the relation between exp(X(c, ι))'s, Stark units, and Shimura's period symbol. Yoshida and the author also defined and studied the p-adic analogue X p (c, ι): In particular, we discussed the relation between the ratios [exp(X(c, ι)) : exp p (X p (c, ι))] and Gross-Stark units. In a previous paper, the author proved the algebraicity of some products of exp(X(c, ι))'s. In this paper, we prove its p-adic analogue. Then, by using these algebraicity properties, we discuss the relation between the ratios [exp(X(c, ι)) : exp p (X p (c, ι))] and Stark units.

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Section: The Case When a Real Place Splits Completelymentioning

confidence: 72%

Section: Introductionmentioning

confidence: 82%

“…Theorem 4 is one of the main results in this paper. The Archimedean part (9) was proved in [Ka3]. The p-adic part (10) is a new result, although it is proved by quite similar arguments.…”

Section: The Case When a Real Place Splits Completelymentioning

confidence: 92%

Section: The Case Of a Totally Positive Dmentioning

confidence: 99%

“…Proof. We use a similar argument to the proof of [Ka3,Lemma 3.6]. We can reduce the problem to a refinement of a cone:…”

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confidence: 99%

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On the ratios of Barnes' multiple gamma functions to the p-adic analogues

Kashio

2019

Journal of Number Theory

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On a common refinement of Stark units and Gross-Stark units

Kashio

2017

Preprint

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The purpose of this paper is to formulate and study a common refinement of a version of Stark's conjecture and its p-adic analogue, in terms of Fontaine's p-adic period ring and p-adic Hodge theory. We construct period-ring-valued functions under a generalization of Yoshida's conjecture on the transcendental parts of CMperiods. Then we conjecture a reciprocity law on their special values concerning the absolute Frobenius action. We show that our conjecture implies a part of Stark's conjecture when the base field is an arbitrary real field and the splitting place is its real place. It also implies a refinement of the Gross-Stark conjecture under a certain assumption. When the base field is the rational number field, our conjecture follows from Coleman's formula on Fermat curves. We also prove some partial results in other cases.

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On the algebraicity of some products of special values of Barnes' multiple gamma function

Cited by 2 publications

References 0 publications

On the ratios of Barnes' multiple gamma functions to the p-adic analogues

On the ratios of Barnes' multiple gamma functions to the p-adic analogues

On a common refinement of Stark units and Gross-Stark units

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