Tresses, monodromie et le groupe symplectique

Monodromy groups of critical point of functions

“…c~--2 for a skew-symmetric form of a critical point of a function in two variables. A similar statement was conjectured by N. A'Campo [1]. …”

“…The symmetric monodromy group for a simple critical point is the one of Weyl groups of classical Lie algebras ([2, 4]). Papers [1,6, 7] are devoted to a description of skew-symmetric monodromy groups of functions in two variables. …”

The integral monodromy of hyperelliptic and trielliptic curves

“…Let C z,1 and C z,2 denote the two components of the formal completion of C ψ(z) at z. First, ι 0 (1) stabilizes each branch C z,i ; second, the characters of the action of ι 0 on the tangent spaces of C z,1 and C z,2 at z are inverses.…”

“…Specifically, we require that Lie(X) be a locally free Z[ζ 3 ] ⊗ O S -module of signature (2, 1) and that ι take complex conjugation on Z[ζ 3 ] to the Rosati involution on End(X). Then Sh (2,1) is the Shimura variety associated to the reductive group G (2,1) .…”

“…More precisely, ξ is a Z[ζ 3 ]-linear isomorphism ξ : X [ℓ] → (V (2,1) ⊗ Z/ℓ) ⊗ O S compatible with the given pairings. The forgetful functor Sh (2,1),ℓ → Sh (2,1) induces a Galois cover of stacks, with covering group SG (2,1) (Z/ℓ) [13, 1.4]. Note that, as in [11,IV.6.1] but in contrast to [3,21], we have implicitly chosen an isomorphism µ ℓ → Z/ℓ.…”