A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--48.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....i hK i = hKi = hKOi: One can do a similar argument for KT , using the fact that KT RP 2 = K Sigma 3 K: Corollary 2.5. hEll i = hElli: Proof. The same proof as given above for E(n) works fine, except that it is not so obvious that hEll i hK(2)i. Here one can use Baker s results [1] that show that Ell= p; v 1 ) is a wedge of suspensions of K(2) 3. Other Cohomological Bousfield Classes In this section, we calculate the cohomological Bousfield classes of spectra of finite type, E n , and LK(n) S 0 . We also point out the connection between Conjecture 3.10 of [9] and the ....

A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. AMS 107 (1989) 537-548.

....i hK i = hKi = hKOi: One can do a similar argument for KT , using the fact that KTRP 2 = K Sigma 3 K: Corollary 2.5. hEll i = hElli: Proof. The same proof as given above for E(n) works fine, except that it is not so obvious that hEll i hK(2)i. Here one can use Baker s results [1] that show that Ell= p; v 1 ) is a wedge of suspensions of K(2) COHOMOLOGICAL BOUSFIELD CLASSES 11 3. Other Cohomological Bousfield Classes In this section, we calculate the cohomological Bousfield classes of spectra of finite type, E n , and LK(n) S 0 . We also point out the connection ....

A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. AMS 107 (1989) 537-548.

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A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--48.

....the Max Planck Institute, Bonn. This paper appeared in Jour. Lond. Math. Soc. 49 (1994) 581 93. Typeset by A M S T E X work. It is also worth mentioning the recent work of J. Francke [7] as of interest as a companion approach to these matters. As basic references on elliptic cohomology, we use [2], 3] 11] and [12] We also refer to [1] and [14] for essential ideas on formal groups and their relationship to complex oriented cohomology theories. For details of level N structures and the Weil pairing we cite [9] 10] and [16] I would like to thank F. Clarke, F. Hirzebruch, M. Hovey, ....

....by the class of u say, the constant term w say, in the minimal polynomial of u would also annihilate v 1 modulo p. But this could only happen if w and hence u were 0 modulo p. It is perhaps worth remarking that v 1 E p 1 (mod p) and so this is not usually a prime element in E## (p) see [2] for more on this) We have thus obtained our promised cohomology theories. Proposition (3.1) For each M SL 2 (Z) the functor (E## ) # ( is a multiplicative, complex oriented cohomology theory on finite CW complexes. Now the conjugation map from : M 1 # 1 (N)M 1 M 2 # 1 (N)M 1 ....

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A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--548.

....ALGEBRAS ACTING ON ELLIPTIC COHOMOLOGY Andrew Baker University of Glasgow Introduction. In our earlier papers [2,3,4,5,6], we investigated stable operations and cooperations in elliptic cohomology and its variants, relating these to known operations on rings of modular forms. The purpose of this article is to give an introduction to these stable operation algebras, in particular explaining the connections with Hecke ....

....for this paper and also to o#er felicitations and thanks to Mark Mahowald, who always knows a good operation. Recollections on elliptic cohomology. We refer to [17,18,19] for detailed expositions of the basic notions of elliptic cohomology; for discussions of reductions modulo prime ideals see [3,4]. For actions of Hecke algebras see [2,3] In the present exposition we will concentrate on elliptic cohomology of level 1, having coe#cient ring E## = Z[1 6] E 4 , E 6 , # 1 ] the ring of modular forms for the full modular group SL 2 (Z) which are meromorphic at infinity and have ....

[Article contains additional citation context not shown here]

A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--48.

....follows the situation for KU KU , which can be thought of as a sort of dual to the monoid ring Z[Z Gamma f0g] This approach to stable operations in elliptic cohomology becomes more manageable if we reduce modulo an invariant ideal in the coefficient ring E . Such ideals were considered in [7]. The most interesting examples are of the form J p;1 = p) and J p;1 ; r 1 J p;2 = p; E p Gamma1 ) and J p;2 ; s 1; 11.3) where p 3 is a prime. Actually the second example consists of ideals in the p localization (E ) p) since E (p Gamma1) may only exist p locally. We can form ....

....in [14] Clarke and Johnson have made some observations on the K theoretic part of the 1 line 1; 2 , using Serre s theory of p adic modular forms. This p adic theory is discussed in [41] and its elliptic cohomology version in [5] For the supersingular theory of modular forms, see [34] and also [7] for the topological version. 12. The operator of Halphen Fricke Ramanujan Swinnerton Dyer Serre The operator of the title has an interesting history; it plays a central role in the algebraic theory of the ring of modular forms. For our present purposes, it is an operator on the ring of ....

A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--548.

....flatness in proving such results was influenced by Hopkins [9] I would like to thank Francis Clarke, Mark Hovey, Keith Johnson, Gerd Laures and Hal Sadofsky for keeping me informed about their work and commenting on mine. x1 Recollections on elliptic cohomology. We refer to our earlier papers [1,3,4] for notation and other background material. We take the coefficient ring of elliptic cohomology to be the graded ring of meromorphic modular forms of level 1 with q coefficients in Z[1=6] E = Z[1=6] Q; R; Delta Gamma1 ] where Q = E 4 ; R = E 6 ; and Delta = 1 1728 (Q 3 Gamma R 2 ....

.... is then defined to be the functor on finite CW complexes given by E ( E Omega MU MU ( where E n = E Gamman and the tensor product is formed using the level 1 genus MU Gamma E of [2] For a prime p 3, we discussed the reductions E = p) and E = p; A) in [1,3], where A = E p Gamma1 2 (E ) p) THE ADAMS E2 TERM FOR ELLIPTIC COHOMOLOGY 3 is the Hasse invariant (at the prime p) In particular we showed that there is an associated cohomology theory (E = p; A) with coefficient ring E = p; A) and moreover there is a natural multiplicative ....

[Article contains additional citation context not shown here]

A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--48. ANDREW BAKER

....complex cobordism comodules. We hope to investigate further the precise relationships between these approaches in future work. It is also worth mentioning the recent work of J. Francke [7] as of interest as a companion approach to these matters. As basic references on elliptic cohomology, we use [2], 3] 11] and [12] We also refer to [1] and [14] for essential ideas on formal groups and their relationship to complex oriented 1991 Mathematics Subject Classification. 55N22 14L05 10D05. Key words and phrases. Elliptic cohomology, level N structure, Weil pairing. The author would like to ....

....by the class of u say, the constant term w say, in the minimal polynomial of u would also annihilate v 1 modulo p. But this could only happen if w and hence u were 0 modulo p. It is perhaps worth remarking that v 1 j E p Gamma1 (mod p) and so this is not usually a prime element in E = p) see [2] for more on this) We have thus obtained our promised cohomology theories. ELLIPTIC GENERA OF LEVEL N AND ELLIPTIC COHOMOLOGY 11 Proposition (3.1) For each M 2 SL 2 (Z) the functor (E M Gamma 1 (N)M Gamma1 ) is a multiplicative, complex oriented cohomology theory on finite CW ....

[Article contains additional citation context not shown here]

A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--548.

.... given by the Johnson Wilson spectra E(n) 10] and their I n adic completions [ E(n) see below) but many of our arguments apply also to a much wider class of theories, such as complex K theory and its p adic completion, or elliptic cohomology and its supersingular completion, as defined in [1]. To each standard cohomology theory, E ( Gamma) say, that we consider here, there are representing Omega spectra, i.e. sets of H spaces fE r ; r 2 Zg with natural isomorphisms (of groups) E r (X) X; E r ] where [X; E r ] denotes the set of homotopy classes of maps from the ....

....and of countable rank over some subring of the rationals. Theorem (3. 3) thus has an analogue for any completion of a suitable Landweber exact spectrum defined as a homotopy limit like [ E(n) for example, p adic K theory or the first author s completion of elliptic cohomology c EllP , described in [1]. Remark 3.12 We have computed colim k Hom(H ( E(n) I k n ) r ) without actually computing much about the individual algebras H ( E(n) I k n ) r ) The case of k = 1, namely Morava K theory, has already been computed in [19] Strictly speaking, the algebraic object H ( E(n) I k n ) is ....

A. J. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. A. M. S., 107 (1989),537-548.

....of modular forms with both p and E p Gamma1 killed is considered for example in [8] This seems a worthwhile area of study since it is much more likely that genuinely v 2 periodic phenomena will be found than in the situation with E p Gamma1 v 1 inverted. We consider this supersingular case in [3]. c) It would be interesting to construct other operations in E ( for example the operator of [10] which is a derivation on E and increases weight by 2) may very well be the restriction of an operation. In particular ( Delta N ) 0 and hence respects the periodicity. ....

A.Baker, "On the homotopy type of the spectrum representing elliptic cohomology", preprint (1988).

....in terms of isogenies and morphisms in certain enlarged isogeny categories. We are particularly inspired by number theoretic work of G. Robert, whose work we reformulate and generalize in our setting. Introduction In previous work we investigated supersingular reductions of elliptic cohomology [5], stable operations and cooperations in elliptic cohomology [3, 4, 6, 8] and in [9, 10] gave some applications to the Adams spectral sequence based on elliptic (co)homology. In this paper we investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular ....

....morphism Fr Gammak . 3. Recollections on elliptic cohomology A general reference on elliptic cohomology is provided by the foundational paper of Landweber, Ravenel Stong [25] while aspects of the level 1 theory which we use can be found in Landweber [24] as well as our earlier papers [4, 5, 6]. Let p 3 be a prime. We will denote by E the graded ring of modular forms for SL 2 (Z) meromorphic at infinity and with q expansion coefficients lying in the ring of p local integers Z (p) Here E 2n consists of the modular forms of weight n. We have Theorem 3.1. As a graded ring, E ....

[Article contains additional citation context not shown here]

A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--48.

....ALGEBRAS ACTING ON ELLIPTIC COHOMOLOGY Andrew Baker University of Glasgow Introduction. In our earlier papers [2,3,4,5,6], we investigated stable operations and cooperations in elliptic cohomology and its variants, relating these to known operations on rings of modular forms. The purpose of this article is to give an introduction to these stable operation algebras, in particular explaining the connections with Hecke ....

....this paper and also to offer felicitations and thanks to Mark Mahowald, who always knows a good operation. x1 Recollections on elliptic cohomology. We refer to [17,18,19] for detailed expositions of the basic notions of elliptic cohomology; for discussions of reductions modulo prime ideals see [3,4]. For actions of Hecke algebras see [2,3] In the present exposition we will concentrate on elliptic cohomology of level 1, having coefficient ring E = Z[1=6] E 4 ; E 6 ; Delta Gamma1 ] the ring of modular forms for the full modular group SL 2 (Z) which are meromorphic at infinity and have ....

[Article contains additional citation context not shown here]

A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--48.

....flatness to prove such results was influenced by Hopkins [9] I would like to thank Francis Clarke, Mark Hovey, Keith Johnson, Gerd Laures and Hal Sadofsky for keeping me informed about their work and commenting on mine. x1 Recollections on elliptic cohomology. We refer to our earlier papers [1,3,4] for notation and other background material. We take the coefficient ring of elliptic cohomology to be the graded ring of meromorphic modular forms level 1 with q coefficients in Z[1=6] E = Z[1=6] Q; R; Delta Gamma1 ] where Q = E 4 ; R = E 6 ; and Delta = 1 1728 (Q 3 Gamma R 2 ) ....

.... is the defined to be the functor on finite CW complexes given by E ( E Omega MU MU ( where E n = E Gamman and the tensor product is formed using the level 1 genus MU Gamma E of [2] For a prime p 3, we discussed the reductions E = p) and E = p; A) in [1,3], where A = E p Gamma1 2 (E ) p) is the Hasse invariant (at the prime p) In particular we showed that there is an associated cohomology theory (E = p; A) with coefficient ring E = p; A) and moreover there is a natural multiplicative equivalence of functors (E = p; A) ....

[Article contains additional citation context not shown here]

A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--48.

....K(1) Lifting this result gives an equivalence b K [ E(1) which is known to arise before p adic completion. Elliptic cohomology. Let E be the spectrum representing the version of elliptic cohomology whose coefficient ring is the ring of modular forms for SL 2 (Z) meromorphic at infinity (see [2]) localised at a prime p 3. Then in [2] we showed that if E p Gamma1 denotes the (p Gamma 1) st Eisenstein function, then there is an equivalence of ring spectra E = p; E p Gamma1 ) ff Sigma 2f(ff) K(2) where E = p; E p Gamma1 ) is the reduction of E modulo the ideal (p; E p Gamma1 ....

....b K [ E(1) which is known to arise before p adic completion. Elliptic cohomology. Let E be the spectrum representing the version of elliptic cohomology whose coefficient ring is the ring of modular forms for SL 2 (Z) meromorphic at infinity (see [2] localised at a prime p 3. Then in [2] we showed that if E p Gamma1 denotes the (p Gamma 1) st Eisenstein function, then there is an equivalence of ring spectra E = p; E p Gamma1 ) ff Sigma 2f(ff) K(2) where E = p; E p Gamma1 ) is the reduction of E modulo the ideal (p; E p Gamma1 ) in an appropriate sense. This lifts to a ....

A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--48.

....follows the situation for KU KU , which can be thought of as a sort of dual to the monoid ring Z[Z Gamma f0g] This approach to stable operations in elliptic cohomology becomes more manageable if we reduce modulo an invariant ideal in the coefficient ring E . Such ideals were considered in [7]. The most interesting examples are of the form J p;1 = p) and J r p;1 ; r 1 J p;2 = p; E p Gamma1 ) and J s p;2 ; s 1; 11.3) where p 3 is a prime. Actually the second example consists of ideals in the p localization (E ) p) since E (p Gamma1) may only exist p locally. We can form ....

....[14] Clarke and Johnson have made some observations on the K theoretic part of the 1 line E 1; 2 , using Serre s theory of p adic modular forms. This p adic theory is discussed in [41] and its elliptic cohomology version in [5] For the supersingular theory of modular forms, see [34] and also [7] for the topological version. 12. The operator of Halphen Fricke Ramanujan Swinnerton Dyer Serre The operator of the title has an interesting history; it plays a central role in the algebraic theory of the ring of modular forms. For our present purposes, it is an operator on the ring of ....

A. Baker, On the homotopy type of the spectrum representing elliptic cohomology, Proc. Amer. Math. Soc. 107 (1989), 537--548. Operations and Cooperations in Elliptic Cohomology, I 73

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