Singularities, Monodromy and Zeta Functions Blatt 4

Singularities, Monodromy and Zeta Functions Blatt 4

Exercises for presentation in the exercise class on 15.11.2018

Recall that a functionf :X →Qⁿp from a semi-algebraic setX ⊂Q^mp is defined to be a semi-algebraic (map) when, for everyr∈Nand every semi-algebraic setS ⊂Qⁿp×Q^rp, the set{(x, z)∈X×Q^rp|(f(x), z)∈S}is semi-algebraic.

Aufgabe 1:

(a) Prove, without using the Projection Theorem, that if X ⊂ Q^mp and f : X → Qⁿp are semi-algebraic, then the graph off is a semi-algebraic subset ofQ^mp ×Qⁿp.

(b) Assuming the Projection Theorem, prove the converse to (a). That is, prove that

(∗)Every functionf whose graph is a semi-algebraic subset ofQ^mp ×Qⁿp is a semi-algebraic map.

(c) Now assume (∗)and deduce the Projection Theoremfor sets of the formX×Y. Aufgabe 2:

Letm, l, k∈Z,X ⊂Q^mp a semi-algebraic set, andh, a1, a2, c:X →Qp semi-algebraic maps.

(a) Show that {x∈X|v(h(x))≡l mod (k)} is semi-algebraic.

(b) Show that

Y :={(x, t)∈X×Qp|v(a1(x))< v(t−c(x))< v(a2(x)) ∧ v(t−c(x))≡l mod (k)}

is a semi-algebraic subset ofX×Qp. (c) Show thatπ(Y)⊂Q^mp is semi-algebraic.

(d) Deduce, forν∈Zand

W :={(x, t)∈X×Qp|v(a1(x))< v(t−c(x))< v(a₂(x)) ∧ ν·v(t−c(x)) +v(h(x))≡0 mod (k)}, that the setπ(W)⊂Q^mp is semi-algebraic.

Course website:http://reh.math.uni-duesseldorf.de/~internet/Zeta_WS18/