Stability results for local zeta functions of groups algebras, and modules

Rossmann, Tobias

doi:10.1017/s0305004117000585

Cited by 9 publications

(8 citation statements)

References 25 publications

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“…We note that Theorems and both behave well under local base extensions; cf. [, Theorem 2.3; , Remark 1.6].…”

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

confidence: 99%

“…Remark (i)The operation ‘

q_{v} \to q_{v}^{- 1}

’ can be unambiguously defined in terms of an arbitrary formula of the form ; see and cf. [, Corollary 4.3]. If

{sans-serifZ}_{M_{v}} false(T false) = W false(q_{v}, T false)

for almost all

v \in {scriptV}_{k}

and some

W (X, T) \in Q (X, T)

, then Theorem asserts that

W false(X^{- 1}, T^{- 1} false) = false(- X^{d} T false) \cdot W false(X, T false)

; see [, § 4] and cf.…”

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

confidence: 99%

“…[, Corollary 4.3]. If

{sans-serifZ}_{M_{v}} false(T false) = W false(q_{v}, T false)

for almost all

v \in {scriptV}_{k}

and some

W (X, T) \in Q (X, T)

, then Theorem asserts that

W false(X^{- 1}, T^{- 1} false) = false(- X^{d} T false) \cdot W false(X, T false)

; see [, § 4] and cf. . (ii)Using Theorem and , Theorem also establishes, for almost all

v \in {scriptV}_{k}

, a functional equation for

{sans-serifZ}_{M \otimes_{frakturo} {frakturK}_{v} false[[z] false]} false(T false)

; cf.…”

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

confidence: 99%

“…Proof Corollary and [, § 4] allow us to recover

{prefixdim}_{k} false(boldG false)

from

{(Z_{sans-serifG false(o_{v} false)}^{cc} (T))}_{v \in V_{k} ∖ S}

for any finite set

S \subset {scriptV}_{k}

□

…”

Section: Orbits and Conjugacy Classes Of Linear Groupsmentioning

confidence: 99%

“…can be unambiguously defined in terms of an arbitrary formula of the form (4.4); see [23,91] and cf. [76,Corollary 4.3]. If Z Mv (T ) = W (q v , T ) for almost all v ∈ V k and some W (X, T ) ∈ Q(X, T ), then Theorem 4.18 asserts that…”

Section: Local Functional Equations and Global Analytic Propertiesmentioning

confidence: 99%

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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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“…We note that Theorems and both behave well under local base extensions; cf. [, Theorem 2.3; , Remark 1.6].…”

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

confidence: 99%

“…Remark (i)The operation ‘

q_{v} \to q_{v}^{- 1}

’ can be unambiguously defined in terms of an arbitrary formula of the form ; see and cf. [, Corollary 4.3]. If

{sans-serifZ}_{M_{v}} false(T false) = W false(q_{v}, T false)

for almost all

v \in {scriptV}_{k}

and some

W (X, T) \in Q (X, T)

, then Theorem asserts that

W false(X^{- 1}, T^{- 1} false) = false(- X^{d} T false) \cdot W false(X, T false)

; see [, § 4] and cf.…”

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

confidence: 99%

“…[, Corollary 4.3]. If

{sans-serifZ}_{M_{v}} false(T false) = W false(q_{v}, T false)

for almost all

v \in {scriptV}_{k}

and some

W (X, T) \in Q (X, T)

, then Theorem asserts that

W false(X^{- 1}, T^{- 1} false) = false(- X^{d} T false) \cdot W false(X, T false)

; see [, § 4] and cf. . (ii)Using Theorem and , Theorem also establishes, for almost all

v \in {scriptV}_{k}

, a functional equation for

{sans-serifZ}_{M \otimes_{frakturo} {frakturK}_{v} false[[z] false]} false(T false)

; cf.…”

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

confidence: 99%

“…Proof Corollary and [, § 4] allow us to recover

{prefixdim}_{k} false(boldG false)

from

{(Z_{sans-serifG false(o_{v} false)}^{cc} (T))}_{v \in V_{k} ∖ S}

for any finite set

S \subset {scriptV}_{k}

□

…”

Section: Orbits and Conjugacy Classes Of Linear Groupsmentioning

confidence: 99%

Section: Local Functional Equations and Global Analytic Propertiesmentioning

confidence: 99%

See 3 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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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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Enumerating graded ideals in graded rings associated to free nilpotent Lie rings

Lee

Voll

2018

Math. Z.

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We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free d-generator Lie rings f c,d of nilpotency class c for all c ď 2 and for pc, dq P tp3, 3q, p3, 2q, p4, 2qu. We apply our computations to obtain information about p-adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.Date: June 15, 2016. 2000 Mathematics Subject Classification. 17B70, 11M41, 11S40. Key words and phrases. Graded ideal zeta functions, free nilpotent Lie rings, local functional equations.Ÿgr f 2,d pOq psq for all d ě 2 in Section 4 and those for pc, dq P tp3, 2q, p4, 2qu in Section 5 of the current paper.The paper's most involved result is the computation, in Section 3, of the graded ideal zeta function of f 3,3 pOq. To this end we compute an explicit formula for ζ Ÿgr f 3,3 poq psq, valid for all finite extension o of the p-adic integers Z p , where p is a prime, viz. a local ring of the form o " O p for a nonzero prime ideal p of O lying above p. Theorem 1.1. There exists an explicitly determined rational function W Ÿgr 3,3 P QpX, Y q such that, for all primes p and all finite extensions o of Z p , with residue cardinality q, ζ Ÿgr f 3,3 poq psq " W Ÿgr 3,3 pq, q´sq. It may be written as Wa polynomial of degree 115 in X and 131 in Y , and N 3,3 P QrX, Y s is a polynomial of degree 81 in X and 108 in Y .The rational function W Ÿgr 3,3 satisfies the functional equationWe note that the Witt function W 3 (cf. (1.2)) takes the values pW 3 p1q, W 3 p2q, W 3 p3qq " p3, 3, 8q and that 115´81 " 34 "`3 2˘``3 2˘``8 2˘a nd 131´108 " 23 " 3¨3`2¨3`1¨8; cf. Conjecture 6.2.Our proof of Theorem 1.1 yields W Ÿgr 3,3 as a sum of 15 explicitly given summands, listed essentially in Section 3.2. We do not reproduce the "final" outcome of this summation here, as the numerator N 3,3 of W Ÿgr 3,3 fills several pages. We do record, however, several corollaries of the explicit formula for W Ÿgr 3,3 . The first corollary concerns analytic properties of the global zeta function ζ Ÿgr f 3,3 pOq psq.Corollary 1.2. The global graded ideal zeta function ζ Ÿgr f 3,3 pOq psq converges on ts P C | Repsq ą 3u and may be continued meromorphically to ts P C | Repsq ą 14{9u.

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Stability results for local zeta functions of groups algebras, and modules

Cited by 9 publications

References 25 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

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

Enumerating graded ideals in graded rings associated to free nilpotent Lie rings

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