Page 429 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 429
NUMBER THEORY (MS-63)
Composition series of a class of induced representations
Igor Ciganovic´, igor.ciganovic@math.hr
University of Zagreb, Faculty of Science, Croatia
We determine compostion series of a class of parabolically induced representation
δ([ν−bρ, ν cρ]) × δ([ν 1 ρ, νaρ]) σ of p-adic symplectic group in terms of Mœglin Tadic´ clasi-
2
fication. Here 1 ≤ a < b < c ∈ Z + 1 are half integers, ν = |det|F where F is a p-adic field, ρ
2 2
is a cuspidal representation of a general linear group, σ is a cuspidal representation of a p-adic
symplectic group such that ν 1 ρ σ reduces and δ([νxρ, νyρ]) → νyρ × · · · × νxρ is a discrete
2
series representation for x ≤ y ∈ Z + 1 .
2
An epsilon constant conjecture for higher dimensional representations
Alessandro Cobbe, alessandro.cobbe@unibw.de
Universität der Bundeswehr München, Germany
Coauthor: Werner Bley
The equivariant local epsilon constant conjecture was formulated in various forms by Fontaine
and Perrin-Riou, Benois and Berger, Fukaya and Kato and others. If N/K is a finite Galois
extension of p-adic fields and V a p-adic representation of GK, then the above conjecture de-
scribes the epsilon constants attached to V in terms of the Galois cohomology of T , where T is
a GK-stable Zp-sublattice T such that V = T ⊗Zp Qp.
Here we will discuss the case when N/K is at most weakly ramified (this includes the case
of tame ramification) and T = Zrp(χcyc)(ρnr), i.e. the Zp-module Zrp with the trivial action of GK
twisted by the cyclotomic character and by an unramified representation ρnr : GK → Glr(Zp).
The main results generalize previous work by Izychev, Venjakob, Bley and the author. This is a
joint work with Werner Bley.
On the Galois module structure of integers of p-adic fields. The question
of the minimal index
Ilaria Del Corso, ilaria.del.corso@unipi.it
Università di Pisa, Italy
Coauthors: Fabio Ferri, Davide Lombardo
Let L/K be a Galois extension with Galois group G. The Normal Basis Theorem shows that
L is a free K[G]-module of rank 1. When L/K is a number field or a local field extension, it
is natural to consider the question of determining the structure of the ring of integers OL as a
OK[G]-module. It is well-known that OL contains free OK[G]-submodules of finite index, but,
in general, it is not free.
In this talk, after a brief overview of the main classical results in this context, I will present
some recent results on the minimal index of a free OK[G]-submodule into OL, in the case of
p-adic fields.
427
Composition series of a class of induced representations
Igor Ciganovic´, igor.ciganovic@math.hr
University of Zagreb, Faculty of Science, Croatia
We determine compostion series of a class of parabolically induced representation
δ([ν−bρ, ν cρ]) × δ([ν 1 ρ, νaρ]) σ of p-adic symplectic group in terms of Mœglin Tadic´ clasi-
2
fication. Here 1 ≤ a < b < c ∈ Z + 1 are half integers, ν = |det|F where F is a p-adic field, ρ
2 2
is a cuspidal representation of a general linear group, σ is a cuspidal representation of a p-adic
symplectic group such that ν 1 ρ σ reduces and δ([νxρ, νyρ]) → νyρ × · · · × νxρ is a discrete
2
series representation for x ≤ y ∈ Z + 1 .
2
An epsilon constant conjecture for higher dimensional representations
Alessandro Cobbe, alessandro.cobbe@unibw.de
Universität der Bundeswehr München, Germany
Coauthor: Werner Bley
The equivariant local epsilon constant conjecture was formulated in various forms by Fontaine
and Perrin-Riou, Benois and Berger, Fukaya and Kato and others. If N/K is a finite Galois
extension of p-adic fields and V a p-adic representation of GK, then the above conjecture de-
scribes the epsilon constants attached to V in terms of the Galois cohomology of T , where T is
a GK-stable Zp-sublattice T such that V = T ⊗Zp Qp.
Here we will discuss the case when N/K is at most weakly ramified (this includes the case
of tame ramification) and T = Zrp(χcyc)(ρnr), i.e. the Zp-module Zrp with the trivial action of GK
twisted by the cyclotomic character and by an unramified representation ρnr : GK → Glr(Zp).
The main results generalize previous work by Izychev, Venjakob, Bley and the author. This is a
joint work with Werner Bley.
On the Galois module structure of integers of p-adic fields. The question
of the minimal index
Ilaria Del Corso, ilaria.del.corso@unipi.it
Università di Pisa, Italy
Coauthors: Fabio Ferri, Davide Lombardo
Let L/K be a Galois extension with Galois group G. The Normal Basis Theorem shows that
L is a free K[G]-module of rank 1. When L/K is a number field or a local field extension, it
is natural to consider the question of determining the structure of the ring of integers OL as a
OK[G]-module. It is well-known that OL contains free OK[G]-submodules of finite index, but,
in general, it is not free.
In this talk, after a brief overview of the main classical results in this context, I will present
some recent results on the minimal index of a free OK[G]-submodule into OL, in the case of
p-adic fields.
427
