By Garett P.

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Additional info for Automorphic Representations and L-functions

Example text

After a finite number of steps we get an element with Hopf invariant in the zero stem and stop the process. Of course there is some indeterminacy in desuspending but we can ignore it for now. We call this sequence of integers the serial number of α. In Fig. 9 we indicate each element of E1k,n = πn+k (S 2n−1 ) by its serial number. In almost all cases if pα = 0, its serial number differs from that of α itself. To get back to Fig. 9, we now have to determine the groups E1k,k−t = π2k−2 (S 2k−5 ) for k ≥ 4, which means examining the 3-stem in detail.

B) For p = 2 αt is so represented provided t ≡ 3 mod (4). If t ≡ 2 mod (4) then the element has order 4; otherwise it has order 2. It is in im J if t is even. 12) v 1 0 → Σq L/(p) −−→ L/(p) → L/(p, v1 ) → 0 is an short exact sequence of G-modules and there is a connecting homomorphism δ1 : H i (G; L/(p, v1 )) → H i+1 (G; L/(p)). 10 are 3. 13. Theorem. H 0 (G; L/(p, v1 )) = Z/(p)[v2 ] where v2 ∈ L has dimension 2(p − 1). 14. Definition. For t > 0 let βt = δ0 δ1 (v2t ) ∈ E2 . More work is required to show that these elements are nontrivial for p > 2, and β1 = 0 for p = 2.

Classification Theorem (Lazard [2]). (a) Two formal group laws defined over the algebraic closure of Fp are isomorphic iff they have the same height. 16, where Kis finite field. Now we come to Morava’s point of view. Let K = Fp , the algebraic closure of the field with p elements, and let GK ⊂ K[[x]] be the group (under composition) of power series with leading term x. We have seen that GK acts on Hom(L, K), the set formal group laws defined over K. Since L is a polynomial ring, we can think of Hom(L, K) as an infinite-dimensional vector space V over K; a set of polynomial generators of L gives a topological basis of V .