## Number Theory Seminar

Seminar information archive ～09/10｜Next seminar｜Future seminars 09/11～

Date, time & place | Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | Naoki Imai, Shane Kelly |

### 2023/05/10

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Swan exponent of Galois representations and fonctoriality for classical groups over p-adic fields (English)

**Guy Henniart**(Paris-Sud University)Swan exponent of Galois representations and fonctoriality for classical groups over p-adic fields (English)

[ Abstract ]

This is joint work with Masao Oi in Kyoto. Let F be a p-adic field for some prime number p,

F^ac an algebraic closure of F, and G_F the Galois group of F^ac/F. A continuous finite dimensional

representation σ (on a complex vector space W) has a Swan exponent s(σ), a non-negative integer

which measures how "wildly ramified" σ is. Langlands functoriality makes it of interest

to compare s(σ) and s(r o σ) when r is an algebraic representation of Aut_C(W). The first cases

for r are the determinant, the adjoint representation, the symmetric square representation and

the alternating square representation. I shall give some relations (inequalities mostly, with

equality in interesting cases) between the Swan exponents of those representations r o σ. I shall

also indicate how such relations can be used to explicit the local Langlands correspondence of

Arthur for some simple cuspidal representations of split classical groups over F.

This is joint work with Masao Oi in Kyoto. Let F be a p-adic field for some prime number p,

F^ac an algebraic closure of F, and G_F the Galois group of F^ac/F. A continuous finite dimensional

representation σ (on a complex vector space W) has a Swan exponent s(σ), a non-negative integer

which measures how "wildly ramified" σ is. Langlands functoriality makes it of interest

to compare s(σ) and s(r o σ) when r is an algebraic representation of Aut_C(W). The first cases

for r are the determinant, the adjoint representation, the symmetric square representation and

the alternating square representation. I shall give some relations (inequalities mostly, with

equality in interesting cases) between the Swan exponents of those representations r o σ. I shall

also indicate how such relations can be used to explicit the local Langlands correspondence of

Arthur for some simple cuspidal representations of split classical groups over F.