Local systems with tame, and a unipotent, local monodromy

Yuval Z. Flicker; Engin Özkan

doi:10.1007/s40863-023-00374-8

Local systems with tame, and a unipotent, local monodromy

Yuval Z. Flicker, Engin Özkan

Department of Mathematics

Ariel University

Research output: Contribution to journal › Article › peer-review

Abstract

We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two Q¯ _ℓ -smooth sheaves on X₁- S₁ , where X₁ is a smooth projective absolutely irreducible curve of genus g over a finite field F_q and S₁ is a reduced divisor, with pre-specified tamely ramified monodromy data at S, including precisely one point of principal unipotent monodromy, twisted by a tame character. Equivalently, we compute the number of the corresponding automorphic representations. The approach is based on using an explicit form of the trace formula for GL (2) , extending the work “Counting local systems with tame ramification” to include a Steinberg (= special) component, twisted by a tame character, by employing a pseudo-coefficient thereof.

Original language	English
Pages (from-to)	465-482
Number of pages	18
Journal	Sao Paulo Journal of Mathematical Sciences
Volume	17
Issue number	2
DOIs	https://doi.org/10.1007/s40863-023-00374-8
State	Published - Dec 2023

Keywords

Automorphic representations
Counting
Curves over finite fields
Explicit trace formula
Local systems
Pseudo-coefficients
Tame and Steinberg representations
ℓ-adic representations

All Science Journal Classification (ASJC) codes

Statistics, Probability and Uncertainty
Computational Theory and Mathematics
General Mathematics

Access to Document

10.1007/s40863-023-00374-8

Cite this

@article{f6ad8efb49f041b69071a6485720e959,

title = "Local systems with tame, and a unipotent, local monodromy",

abstract = "We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two Q¯ ℓ -smooth sheaves on X1- S1 , where X1 is a smooth projective absolutely irreducible curve of genus g over a finite field Fq and S1 is a reduced divisor, with pre-specified tamely ramified monodromy data at S, including precisely one point of principal unipotent monodromy, twisted by a tame character. Equivalently, we compute the number of the corresponding automorphic representations. The approach is based on using an explicit form of the trace formula for GL (2) , extending the work “Counting local systems with tame ramification” to include a Steinberg (= special) component, twisted by a tame character, by employing a pseudo-coefficient thereof.",

keywords = "Automorphic representations, Counting, Curves over finite fields, Explicit trace formula, Local systems, Pseudo-coefficients, Tame and Steinberg representations, ℓ-adic representations",

author = "Flicker, {Yuval Z.} and Engin {\"O}zkan",

note = "Publisher Copyright: {\textcopyright} 2023, Instituto de Matem{\'a}tica e Estat{\'i}stica da Universidade de S{\~a}o Paulo.",

year = "2023",

month = dec,

doi = "10.1007/s40863-023-00374-8",

language = "الإنجليزيّة",

volume = "17",

pages = "465--482",

journal = "Sao Paulo Journal of Mathematical Sciences",

issn = "1982-6907",

publisher = "Springer Nature",

number = "2",

}

TY - JOUR

T1 - Local systems with tame, and a unipotent, local monodromy

AU - Flicker, Yuval Z.

AU - Özkan, Engin

PY - 2023/12

Y1 - 2023/12

N2 - We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two Q¯ ℓ -smooth sheaves on X1- S1 , where X1 is a smooth projective absolutely irreducible curve of genus g over a finite field Fq and S1 is a reduced divisor, with pre-specified tamely ramified monodromy data at S, including precisely one point of principal unipotent monodromy, twisted by a tame character. Equivalently, we compute the number of the corresponding automorphic representations. The approach is based on using an explicit form of the trace formula for GL (2) , extending the work “Counting local systems with tame ramification” to include a Steinberg (= special) component, twisted by a tame character, by employing a pseudo-coefficient thereof.

AB - We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two Q¯ ℓ -smooth sheaves on X1- S1 , where X1 is a smooth projective absolutely irreducible curve of genus g over a finite field Fq and S1 is a reduced divisor, with pre-specified tamely ramified monodromy data at S, including precisely one point of principal unipotent monodromy, twisted by a tame character. Equivalently, we compute the number of the corresponding automorphic representations. The approach is based on using an explicit form of the trace formula for GL (2) , extending the work “Counting local systems with tame ramification” to include a Steinberg (= special) component, twisted by a tame character, by employing a pseudo-coefficient thereof.

KW - Automorphic representations

KW - Counting

KW - Curves over finite fields

KW - Explicit trace formula

KW - Local systems

KW - Pseudo-coefficients

KW - Tame and Steinberg representations

KW - ℓ-adic representations

UR - http://www.scopus.com/inward/record.url?scp=85174417158&partnerID=8YFLogxK

U2 - 10.1007/s40863-023-00374-8

DO - 10.1007/s40863-023-00374-8

M3 - مقالة

SN - 1982-6907

VL - 17

SP - 465

EP - 482

JO - Sao Paulo Journal of Mathematical Sciences

JF - Sao Paulo Journal of Mathematical Sciences

IS - 2

ER -

Local systems with tame, and a unipotent, local monodromy

Abstract

Keywords

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Cite this