Configuration Spaces: Geometry, Combinatorics and Topology

YOUNG SEMINAR: The explicit forms of the Igusa local zeta functions of certain plane curves

speaker: Denis Ibadula (Universitatea)

abstract: We explore the Igusa local zeta functions associated to the nondegenerate homogeneous polynomials of degree three in two variables (plane cubics) over ℚ_p, for p≠ 2,3. We identify two plane cubics F and F^′ if there exists a matrix g∈ GL₂(ℚ_p) such that (g· F^′)(x_1,x2)=F(x_1,x2), where (g· F^′)(x_1,x2)=F^′((x_1,x2)g^t) and g^t denotes the transposed of the matrix g. Hence, the isomorphism classes are orbits of GL₂(ℚ_p)-action.

First, we determine explicitly the representative nondegenerate plane cubics over ℚ_p of the orbits of GL₂(ℚ_p)-action. Then, we explore the Igusa local zeta functions of the GL₂(ℚ_p)-orbit of each representative. We prove that it suffices to reduce the investigation of Igusa local zeta functions of the GL₂(ℚ_p)-orbit of any plane cubic F to the GL₂(ℚ_p)mod ℚ_p^×GL₂(ℤ_p)− orbit of F. Thus, the idea is to use the tree X:=GL₂(ℚ_p)ℚp^× GL₂(ℤp) and to write an arbitrary plane cubic F as g· Fi, with g∈ GL₂(ℚp) and Fi one of the representatives of the orbits of the GL₂(ℚp)-action. The key idea in our determination is to calculate ZFig, where g runs through a set of representatives of X and where Fig denote the primitive polynomial associated to g· Fi.

timetable:
Thu 27 May, 17:00 - 18:00, Sala Conferenze Centro De Giorgi
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