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

Rossmann, Tobias

doi:10.1007/978-3-319-70566-8_25

Cited by 3 publications

(5 citation statements)

References 42 publications

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“…Using a theorem of Voll we will furthermore see that the identity

{sans-serifZ}_{{prefixM}_{d \times e} false(frakturO false)}^{sans-serifask} false(T false) {false|}_{false(q, T false) \to false(q^{- 1}, T^{- 1} false)} = false(- q^{d} T false) \cdot {sans-serifZ}_{{prefixM}_{d \times e} false(frakturO false)}^{sans-serifask} false(T false)

is no coincidence (Theorem ). Our

p

‐adic formalism is also compatible with our previous computational work (summarised in ) which allows us to explicitly compute numerous further examples of

{sans-serifZ}_{M}^{sans-serifask} false(T false)

; see § 9 for some of these.…”

Section: Introductionmentioning

confidence: 83%

“…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%

“…The author's software package

sans-serifZeta

for Sage can compute numerous types of ‘generic local’ zeta functions in fortunate (‘non‐degenerate') cases. The techniques used by

sans-serifZeta

were developed over the course of several papers; see , in particular, and for an overview and references to other pieces of software that

sans-serifZeta

relies upon. When performing computations,

sans-serifZeta

proceeds by attempting to explicitly compute certain types of

p

‐adic integrals.…”

Section: Further Examplesmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Using a theorem of Voll we will furthermore see that the identity

{sans-serifZ}_{{prefixM}_{d \times e} false(frakturO false)}^{sans-serifask} false(T false) {false|}_{false(q, T false) \to false(q^{- 1}, T^{- 1} false)} = false(- q^{d} T false) \cdot {sans-serifZ}_{{prefixM}_{d \times e} false(frakturO false)}^{sans-serifask} false(T false)

is no coincidence (Theorem ). Our

p

‐adic formalism is also compatible with our previous computational work (summarised in ) which allows us to explicitly compute numerous further examples of

{sans-serifZ}_{M}^{sans-serifask} false(T false)

; see § 9 for some of these.…”

Section: Introductionmentioning

confidence: 83%

“…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%

“…The author's software package

sans-serifZeta

for Sage can compute numerous types of ‘generic local’ zeta functions in fortunate (‘non‐degenerate') cases. The techniques used by

sans-serifZeta

were developed over the course of several papers; see , in particular, and for an overview and references to other pieces of software that

sans-serifZeta

relies upon. When performing computations,

sans-serifZeta

proceeds by attempting to explicitly compute certain types of

p

‐adic integrals.…”

Section: Further Examplesmentioning

confidence: 99%

See 1 more Smart Citation

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

Rossmann

2018

Proc. London Math. Soc.

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show abstract

“…On the other hand, the conclusion of Corollary B does not hold for either M (c) or M (d). Indeed, using the package Zeta [24,27] for SageMath [32], we find that if O is a compact DVR with sufficiently large residue characteristic and residue field size q, then Z ask M (c) (T ) and Z ask M (d) (T ) are both of the form…”

Section: Examples Non-examples and Rank Distributionsmentioning

confidence: 99%

Linear relations with disjoint supports and average sizes of kernels

Carnevale¹,

Rossmann²

2020

Preprint

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We study the effects of imposing linear relations within modules of matrices on average sizes of kernels. The relations that we consider can be described combinatorially in terms of partial colourings of grids. The cells of these grids correspond to positions in matrices and each defining relation involves all cells of a given colour. We prove that imposing such relations arising from "admissible" partial colourings has no effect on average sizes of kernels over finite quotients of discrete valuation rings. This vastly generalises the known fact that average sizes of kernels of general square and traceless matrices of the same size coincide over such rings. As a group-theoretic application, we explicitly determine zeta functions enumerating conjugacy classes of finite p-groups derived from free class-3-nilpotent groups for p 5.

show abstract

“…On the other hand, the conclusion of Corollary B does not hold for either

M(\text{c})

M(\text{d})

. Indeed, using the package Zeta [25, 29] for SageMath [33], we find that if

\mathfrak {O}

is a compact DVR with sufficiently large residue characteristic and residue cardinality

q

, then

\mathsf {Z}^{\mathsf {ask}}_{M(\text{c})}(T)

and

\mathsf {Z}^{\mathsf {ask}}_{M(\text{d})}(T)

are both of the form

\begin{align*} \frac{ 1 + N q^{-1} T - 2 (N + 1) q^{-2} T + N q^{-3} T + q^{-4}T^2}{(1- q^{-1}T)(1 - T)^2} & = 1 + {\left(2 + \frac{N+1}{q} - 2\frac{N+1}{q^{2}} + \frac{N}{q^{3}}\right)}T + \ma...

…”

Section: Introductionmentioning

confidence: 99%

Linear relations with disjoint supports and average sizes of kernels

Carnevale

Rossmann

2022

Journal of London Math Soc

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We study the effects of imposing linear relations within modules of matrices on average sizes of kernels. The relations that we consider can be described combinatorially in terms of partial colourings of grids. The cells of these grids correspond to positions in matrices and each defining relation involves all cells of a given colour. We prove that imposing such relations arising from 'admissible' partial colourings has no effect on average sizes of kernels over finite quotients of discrete valuation rings. This vastly generalises the known fact that average sizes of kernels of general square and traceless matrices of the same size coincide over such rings. As a group-theoretic application, we explicitly determine zeta functions enumerating conjugacy classes of finite 𝑝-groups derived from free class-3-nilpotent groups for 𝑝 ⩾ 5.

show abstract

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

Cited by 3 publications

References 42 publications

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

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

Linear relations with disjoint supports and average sizes of kernels

Linear relations with disjoint supports and average sizes of kernels

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