Computing topological zeta functions of groups, algebras, and modules, I

Rossmann, Tobias

doi:10.1112/plms/pdv012

Cited by 28 publications

(88 citation statements)

References 50 publications

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“…Then, it is necessary to show that definition (4.1) is independent of the resolution of singularities chosen, this fact was established by Denef and Loeser in [31], see also [32]. By using the explicit formula (B.5)-(B.6), Denef and Loeser showed that…”

Section: Jhep08(2018)043mentioning

confidence: 99%

“…Section 4 gives the description of the p → 1 limit of the p-adic string amplitudes. For this we use the formulation of topological zeta functions [31,32]. We also present the computation of N = 4 and N = 5 points Denef-Loeser amplitudes.…”

Section: Jhep08(2018)043mentioning

confidence: 99%

“…To make mathematical sense of the limit of Z (N ) (s, Q p ) as p → 1 we use the work of Denef and Loeser, see [31] and [32]. The first step is to pass from Q p to K e , e ∈ N, and consider Z (N ) (s, K e ) instead of Z (N ) (s, Q p ), and compute the limit of Z (N ) (s, K e ) as e → 0 instead of the limit of Z (N ) (s, Q p ) as p → 1.…”

Section: Topological Zeta Functionsmentioning

confidence: 99%

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On p-adic string amplitudes in the limit p approaches to one

Bocardo-Gaspar

García‐Compeán

Zúñiga–Galindo

2018

J. High Energ. Phys.

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In this article we discuss the limit p approaches to one of tree-level p-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that p-adic strings are related to the ordinary strings in the p → 1 limit. Previously, we established that p-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa's local zeta functions, consequently, they are convergent integrals that admit meromorphic continuations as rational functions. The meromorphic continuation of local zeta functions has been used for several authors to regularize parametric Feynman amplitudes in field and string theories. Denef and Loeser established that the limit p → 1 of a Igusa's local zeta function gives rise to an object called topological zeta function. By using Denef-Loeser's theory of topological zeta functions, we show that limit p → 1 of tree-level p-adic string amplitudes give rise to certain amplitudes, that we have named Denef-Loeser string amplitudes. Gerasimov and Shatashvili showed that in limit p → 1 the well-known non-local effective Lagrangian (reproducing the tree-level p-adic string amplitudes) gives rise to a simple Lagrangian with a logarithmic potential. We show that the Feynman amplitudes of this last Lagrangian are precisely the amplitudes introduced here. Finally, the amplitudes for four and five points are computed explicitly.

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Section: Jhep08(2018)043mentioning

confidence: 99%

Section: Jhep08(2018)043mentioning

confidence: 99%

Section: Topological Zeta Functionsmentioning

confidence: 99%

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On p-adic string amplitudes in the limit p approaches to one

Bocardo-Gaspar

García‐Compeán

Zúñiga–Galindo

2018

J. High Energ. Phys.

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“…Informally, the topological ask zeta function

ζ_{M}^{top} false(s false) \in Q false(s false)

M

is the constant term of

false(1 - q_{v}^{- 1} false) ζ_{M_{v}} false(s false)

as a series in

q_{v} - 1

; for a rigorous definition, combine the formalism developed in [, § 5] (and summarised in [, § 4.2]), Proposition and [, proof of Lemma 3.4]. For example, Proposition implies that

ζ_{{prefixM}_{d \times e} false(boldZ false)}^{top} false(s false) = \frac{s + e}{s (s - d + e)} .

We note that, as in the case of subobject [, Proposition 5.19] and representation zeta functions [, Proposition 4.3], the topological ask zeta function of

M

only depends on

M \otimes_{frakturo} \bar{k}

, where

\overset{k}{true}

is an algebraic closure of

k

.…”

Section: Rationality Of Sans-serifzmfalse(tfalse) and P‐adic Integrationmentioning

confidence: 99%

“…Supposing that Question indeed has a positive answer, if

G ⩽ {prefixU}_{d} \otimes_{boldZ} k

is an algebraic group over

k

with associated

frakturo

‐form

G ⩽ {prefixU}_{d} \otimes_{boldZ} o

(see Corollary ), then we may evaluate the meromorphic continuation of

{sans-serifZ}_{G ({frakturo}_{v})}^{sans-serifoc} false(q_{v}^{- s} false)

s = 0

for almost all

v \in {scriptV}_{k}

. Inspired by similar questions regarding the behaviour at zero of local subalgebra [, Conjecture IV], submodule [, Conjecture E] and representation [, Question 8.5] zeta functions, it would then be interesting to see if one can interpret the resulting rational numbers, say in terms of properties of the orbit space

{frakturo}_{v}^{d} / G false(o_{v} false)

.…”

Section: Further Examplesmentioning

confidence: 99%

The average size of the kernel of a matrix and orbits of linear groups

Rossmann

2018

Proc. London Math. Soc.

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Let frakturO be a compact discrete valuation ring of characteristic 0. Given a module M of matrices over frakturO, we study the generating function encoding the average sizes of the kernels of the elements of M over finite quotients of frakturO. We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules M. Using p‐adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro‐p groups.

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A Framework for Computing Zeta Functions of Groups, Algebras, and Modules

Rossmann

2017

Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

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Computing topological zeta functions of groups, algebras, and modules, I

Cited by 28 publications

References 50 publications

On p-adic string amplitudes in the limit p approaches to one

On p-adic string amplitudes in the limit p approaches to one

The average size of the kernel of a matrix and orbits of linear groups

A Framework for Computing Zeta Functions of Groups, Algebras, and Modules

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