Algebraic Geometry Iv Linear Algebraic Groups Invariant by A.N. Parshin

By A.N. Parshin

Two contributions on heavily similar matters: the idea of linear algebraic teams and invariant concept, through famous specialists within the fields. The e-book could be very priceless as a reference and study consultant to graduate scholars and researchers in arithmetic and theoretical physics.

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2) when we use the main theorem of Tate ([Ta]) since D 0 and D E clearly correspond to ordinary p-divisible groups. Now the q-expansion principle implies that dimFp X[m ] ≤ 1 where X = {H 0 (Σµ1 , Ω1 ) ⊕ H 0 (Σ´1et , Ω1 )} and m is defined by embedding T/m → Fp and setting m = ker : T⊗Fp → Fp under the map t ⊗ a → at mod m. Also T acts on Pic0 Σµ1 × Pic0 Σ´1et , the abelian variety part of the closed fibre of the Neron model of J1 (N p)/O , and hence also on its cotangent space X. 2 below. For similar versions in slightly simpler contexts see [Wi3, §6] or [Gro, §12].

This has distinct eigenvalues ±1 on ]ρm so we may decompose D[p] into eigenspaces for τ : D[p] = D[p]+ ⊕ D[p]− . 4) up to order. So in the decomposition D[m] = D[m]+ ⊕ D[m]− one of the eigenspaces is isomorphic to Tm and the other to (Tm /p)[m]. But since ρm is irreducible it is easy to see by considering D[m]⊕Hom(D[m], det ρ m ) that τ has the same number of eigenvalues equal to +1 as equal to −1 in D[m], ∼ whence #(Tm /p)[m] = #(T/m). This shows that D[m]+ → D[m]− T/m as required. Now we consider the case where ∆(p) is trivial mod m.

To check this, suppose that λ1 (x) = −λ1 (y). Then λ1 (x) ∈ H 1 (Γ1 (N q r ) ∩ Γ(q), Qp /Zp )∆q . So λ1 (x) is the restriction of an x ∈ H 1 Γ1 (N q r−1 ), Qp /Zp 1 whence x − res1 (x ) ∈ ker λ1 = 0. It follows also that y = −res (x ). Now conjugation by the matrix ( 0q 01 ) induces isomorphisms Γ1 (N q r ) Γ1 (N q r ), Γ1 (N q r ) ∩ Γ(q) Γ1 (N q r , q r+1 ). 13) yields the exact sequence of the lemma, except that we have to change from group cohomology to the cohomology of the associated complete curves.

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