Universität Regensburg Mathematik

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1 Universität Regensburg Mathematik Beta-elements and divided congruences Jens Hornbostel and Niko Naumann Prerint Nr. 12/2005

2 Beta-elements and divided congruences Jens Hornbostel and Niko Naumann Abstract The f-invariant is an injective homomorhism from the 2-line of the Adams-Novikov sectral sequence to a grou which is closely related to divided congruences of ellitic modular forms. We comute the f-invariant for two infinite families of β-elements and exlain the relation of the arithmetic of divided congruences with the Kervaire invariant one roblem. 1. Introduction One of the most successful tools for studying the stable homotoy grous of sheres is the Adams- Novikov sectral sequence (ANSS) E s,t 2 = Ext s,t MU MU (MU, MU ) π s t s(s 0 ). The corresonding filtration on π s := π s (S 0 ) defines a succession of invariants of framed bordism, each being defined whenever all of its redecessors vanish, the first one of which is simly the degree d : F 0, /F 1, = π s 0 E 0,0 2 = Z which is an isomorhism. The next invariant, defined for all n > 0, is the e-invariant e : π s n = F 1,n+1 E 1,n+1 2 /Z, c.f. [Sw, Chater 19]. Though defined urely homotoy-theoretic here, the e-invariant is well-known to encode subtle geometric information. For its relation to index theory via the η-invariant see [APS, Theorem 4.14]. The e-invariant vanishes for all even n = 2k 2, thus giving rise to the f-invariant f : π s 2k = F2,2k+2 E 2,2k+2 2. The understanding of this invariant is fragmentary at the moment. In articular, there is no indextheoretic interretation of it comarable to the one available for the e-invariant. As a first ste towards understanding the f-invariant, G. Laures [L1] showed how ellitic homology can be used to consider the f-invariant 2000 Mathematics Subject Classification 5510, 55T15, 55N34, 11F33 Keywords: beta elements, f-invariant, divided congruences

3 Jens Hornbostel and Niko Naumann f : π s 2k E2,2k+2 2 D k+1 /Z as taking values in a grou which is closely related to divided congruences of modular forms. Note that this is similar to the role taken by comlex K-theory in the study of the e-invariant. Strictly seaking, at this oint we had better switched from MU to BP. In fact, we will always work locally at a fixed rime in the following. This surrising connection of stable homotoy theory with something as genuinely arithmetic as divided congruences certainly motivates to ask for a thorough understanding of how these are related by the f-invariant. The main urose of this aer is to make this relation exlicit. We also include a fairly self-contained review of G. Laures above version of the f-invariant to hel the reader who might be interested in making his own comutations. We now review the individual sections in more detail. In section 2, we first remind the reader of the β-elements which generate the 2-line of the ANSS (with a little excetion at the rime 2). We then construct, for suitable Hof algebroids, a comlex which is quasi-isomorhic to the cobar comlex and which will facilitate later comutations. Finally, we show how to use ellitic homology to obtain the f-invariant as above. In section 3, we recall some fundamental results of N. Katz on the arithmetic of divided congruences and oint out an interesting relation between BP-theory and the mod Igusa tower (Theorem 5). Next, we give some secific comutations of modular forms and divided congruences for Γ 1 (3) which we will need to study the f-invariant of the Kervaire elements β 2 n,2 n Ext2,2n+2 [BP] at the rime = 2. In section 4, we first comute the f-invariants of the infinite family of β-elements β t for t 1 not divisible by (Theorem 10). Then we exlain how to aroach the roblem of comuting the Chern numbers determining the β 2 n,2 n (see [L2, Corollary 4.2.5] for the case of β 1 at the rime 3). We do this by exlicit comutations in BP-theory for dimension 2 and 6 (Theorem 18). The comutations get very comlicated in higher dimensions. In order to use divided congruences, we then comute the f-invariants of the family β s2 n,2 n for n 0 and s 1 odd (Theorem 19). We hoe that clever use of divided congruences will enable us to comute the Chern numbers determining the β 2 n,2 n for all n. See Corollary 21 for a quick summary of what we can and cannot do at the moment. Acknowledgements The idea that it might be ossible to roject to the element β 2 n,2n using divided congruences - and consequently rehrase the Kervaire invariant one roblem - was communicated to us by G. Laures. 2. The construction of the f-invariant We remind the reader of the construction of β-elements in section 2.1. In section 2.2, we construct a comlex which is quasi-isomorhic to the cobar comlex and in which we will comute reresentatives for some of the β-elements. This is used in section 2.3 where we exlain how to exress the f-invariant of elements in Ext 2,2k [BP] in terms of divided congruences. 2

4 2.1 Beta-elements in stable homotoy Beta-elements and divided congruences We review some facts on Brown-Peterson homology BP at the rime and β-elements. See [MRW] and [R] for more details. The Brown-Peterson sectrum BP has coefficient ring BP = Z () [v 1, v 2,...] with v i in dimension 2( i 1). The universal -tyical formal grou law is defined over this ring. The coule (A, Γ) := (BP, BP BP) becomes a Hof algebroid in a standard way and we have BP BP = BP Z () [t 1, t 2,...] such that the left unit η L of the Hof algebroid (A, Γ) is the standard inclusion. The right unit η R is determined over by the formula in [R, Theorem A ]. Choosing the Hazewinkel generators [R, A.2.2.1] for the v i, a short comutation yields η R (v 1 ) = v 1 + t 1 and η R (v 2 ) = v 2 + v 1 t 1 v 1 t 1 mod. We have the chromatic resolution of BP as a left BP BP-comodule BP M 0 M 1... which gives rise to the chromatic sectral sequence Ext s, BP BP (BP, M t ) Ext s+t, BP BP (BP, BP ). This allows the construction of elements in Ext, BP (BP BP, BP ), the so called Greek-letter elements. Strictly seaking, these elements arise from comodule sequences 0 N n M n N n+1 0, but for our comutations we will need the related comodule sequences (1) and (2) below. See [MRW, Lemma 3.7 and Remark 3.8] for the relationshi between them. We abbreviate H n ( ) := Ext n, BP (BP BP, ) in the following. To construct the β-elements [MRW,. 476/477], choose integers t, s, r 1 such that ( r, v1 s, xt n) BP is an invariant ideal where t = n t, (, t ) = 1 and x n is a homogeneous olynomial in v 1, v 2 and v 3 considered as an element of v2 1 BP /( r, v1 s ) (see [R,. 202] or [MRW,. 476]), for examle x 0 = v 2. Consider the two short exact sequences of BP BP-comodules (1) 0 BP r BP BP /( r ) 0 (2) 0 Σ 2s( 1) BP /( r ) vs 1 BP /( r ) BP /( r, v1) s 0. Using the induced boundary mas δ : H 0 (BP /( r, v s 1)) H 1 (BP /( r )) and δ : H 1 (BP /( r )) H 2 (BP ) we define β t,s,r := δ δ(x t n). It is known [MRW, Theorem 2.6], [Sh] for which indices (t, s, r) the elements β t,s,r are non-zero in H 2 (BP ). In this case the order of β t,s,r is r. By construction, we have β t,s,r H 2,2t(2 1) 2s( 1) (BP ). Examle 1. For = 2 and n 1, the element β 2 n,2 n := β 2 n,2 n,1 Ext 2,2n+2 BP BP (BP, BP ) is called the Kervaire element. It is maed via the Thom reduction [R, Theorem 5.4.6] to the element h 2 n+1 Ext 2,2n+2 HZ/2 HZ/2 (HZ/2, HZ/2 ). The latter element survives to a non-zero element of π 2 n+2 2(S 0 ) in the Adams sectral sequence if and only if (as W. Browder [B, Theorem 7.1] has shown) there exists 3

5 Jens Hornbostel and Niko Naumann a framed manifold of dimension 2 n+2 2 with non-vanishing Kervaire invariant. Whether or not this is the case is unknown for n 5, for n 4 see [BJM], [KM]. 2.2 The rationalised cobar comlex The standard way of dislaying elements in Ext n := Ext n Γ (A, A) of a flat Hof algebroid (A, Γ) is by means of the cobar comlex. In this section we shall give another descrition of this Ext grou as a subquotient of (A ) n needed to comute f-invariants (Proosition 3). The results of this section are a more algebraic version of [L1, Section 3.1]. Let (A, Γ) be a Hof algebroid with structure mas η L, η R, ɛ and. This determines a cosimlicial abelian grou Γ as follows: Set Γ n := Γ An with cofaces i : Γ n Γ n+1 ; 0 (γ 1... γ n ) := 1 γ 1... γ n ; i (γ 1... γ n ) := γ 1... (γ i )... γ n (1 i n) and n+1 (γ 1... γ n ) := γ 1... γ n 1 for n 1 and 0 := η R, 1 := η L for n = 0 and codegeneracies σ i : Γ n+1 Γ n, σ i (γ 0... γ n ) := γ 0... ɛ(γ i )... γ n. We also denote by Γ the associated cochain comlex. Following [R, Definition A ], we define the reduced cobar comlex (usually denoted as C Γ (A, A)) as being the subcomlex Γ Γ with Γ n := Γ An 0 for n 1 where Γ := ker(ɛ) and Γ := A. This is a subcomlex because (γ) γ 1 1 γ Γ A Γ for any γ Γ. We now assume that (A, Γ) is a flat Hof algebroid such that i) A (and hence Γ) is torsion free. ii) The ma A 2 := (A ) 2 φ Γ := Γ, a b a.η R (b) is an isomorhism (η L is suressed from notation). iii) The -algebra A is augmented by some τ : A. Remark 2. The above assumtions i)-iii) are fulfilled by the flat Hof algebroid (BP, BP BP): The roof of ii) follows from the fact that over any -algebra any two (-tyical) formal grou laws are isomorhic via a unique strict isomorhism. If BP E is a non-zero Landweber exact algebra [HS], then (E, E E) also fulfils all the above assumtions: E is a Z () -algebra on which is not a zero divisor, hence E is torsion free. Conditions ii), iii) and the flatness of of (E, E E) are inherited from (BP, BP BP) because E E E BP BP BP BP E, c.f. [Na, Proosition 10] for the flatness. We define the cosimlicial abelian grou D by D n := A (n+1) (n 0) with cofaces i : D n D n+1 to be given by i (a 0... a n ) := a a n with the 1 in the (i + 1) st osition (i = 0,..., n + 1) and codegeneracies σ i : D n D n 1 (i = 0,...,, n 1) defined by σ i (a 0... a n ) := a 0... a i a i+1... a n. By ii) above we have for any n 0 an isomorhism (3) φ n : D n = A (n+1) (A A ) A n φ n Γ A n = Γ n which mas a 0... a n a 0... a n 2 a n 1.η R (a n ) and one checks that φ is an isomorhism of cosimlicial grous and hence of cochain comlexes. We have H (D ) = H 0 (D ) =, a contracting homotoy being given by (4) H n : D n = A (n+1) Define a subcomlex Σ D by D n 1 = A n ; a 0... a n τ(a 0 )a 1... a n. 4

6 Beta-elements and divided congruences Σ n := n i (D n 1 ) = i=1 n i=1 A i A (n i) D n = A (n+1), for n 1 and Σ 0 := 0. One checks that the comosition ι n : Γ n Γ n Γ n (φn ) 1 D n D n /Σ n is injective for all n 0. This is obvious for n = 0, and for n 1 it follows from ( n 1 ker(σi )) ( n i=1 im ( i )) = 0, which in turn is an easy consequence of the cosimlicial identities: One shows by descending induction on 1 j n that ( n 1 ker(σi )) ( n i=j im ( i )) = 0. Observe that D n /Σ n is isomorhic to the grou labelled E G n in [L1,. 404] for A = BP. We define a cochain comlex by the exactness of (5) 0 Γ ι D /Σ π 0, hence n A (n+1) /Σ n + im (Γ An ). From the definitions of the differential of D and Σ n one obtains (B n n denoting the boundaries) n /B n A (n+1) / Σ n + im (Γ An ), where Σ n := Σ n + A n = n+1 i=1 A (i 1) A (n+1 i). The alternative descrition of Ext we are aiming for is the following. Proosition 3. For any n 1, the connecting homomorhism δ of (5) is an isomorhism H n ( ) δ H n+1 (Γ ) = Ext n+1. Proof. One readily sees that the contracting homotoy (4) of D resects the subcomlex Σ in ositive dimensions, hence the middle term of (5) is acyclic in these dimensions. To exlicitly comute δ, it is useful to note that the differential of D /Σ has the simle form D n /Σ n d D n+1 /Σ n+1, [a 0... a n ] [1 a 0... a n ], as is immediate from the definitions. To comute δ 1, we consider the zig-zag Ext n+1 Z(Γ n+1 ) ιn+1 D n+1 /Σ n+1 Hn+1 D n /Σ n π n n n /B n δ 1 5 H n ( ).

7 Jens Hornbostel and Niko Naumann One checks that the dotted arrow exists and is the inverse of δ. Now let be a rime, r 1 an integer, and let δ : Ext n Γ (A, A/r ) Ext n+1 be the connecting homomorhism associated to the short exact sequence of Γ-comodules 0 A r A A/ r 0. Consider the diagram (6) δ : Ext n Γ (A, A/r ) Z(Γ n A A/ r ) {z Γ n dz r Γ n+1 } α ν n /B n Ext n+1 δ H n ( ). Here, α(z) := [y] for any y Γ n+1 satisfying dz = r y and ν(z) := π n ( r ι n (z)) mod B n. The uer horizontal line is δ by definition and one checks that ν factors through the dotted arrow and makes the diagram commutative. Hence, when dislaying an element of im (δ ) in H ( ) rather than in the usual cobar comlex, one does not have to comute the cobar differential imlicit in α but only the (easier) ma ν. Finally, assume that everything in sight is graded where the grading on D n,γ n, etc. is by total degree. For a fixed k Z, consider the commutative diagram n /B n = A (n+1) / Σ n + im (Γ An ) H n ( ) A (n+1),k j / Σ n,(k) + im (Γ An ) k H n,k ( ) δ Ext n+1 Ext n+1,k. Here Σ n,(k) := Σ n A (n+1),k. One checks that j is well defined and injective, and H n,k ( ) is defined to be the ull back of H n ( ) along j. The commutative diagram induces an isomorhism H n,k ( ) Ext n+1,k as indicated. For examle, for (A, Γ) = (BP, BP BP) and n = 1 we obtain an inclusion (7) Ext 2,k (BP BP ) (k) BP k BP + (BP + BP ) (k) which is imortant for us since the f-invariant is defined in terms of the grou on the right hand side. To effectively comute reresentatives of β-elements in the comlex one roceeds as follows. Let t, s, r 1 be integers as in section 2.1 and δ, δ the coboundary mas introduced there. Fix k Z and x H 0,k (A/( r, v1 s )), that is x C 0,k Γ (A/(r, v1)) s = (A/( r, v1)) s k is an invariant element (C Γ indicates the reduced cobar comlex). As δ is the connecting homomorhism determined by the short exact sequence of comlexes obtained by alying C Γ to (2), we comute δ(x) as follows: Lift x to y (A/( r )) k and comute the cobar differential 6

8 Beta-elements and divided congruences d = η R η L : C 0,k Γ (A/r ) = (A/ r ) k (A/ r A Γ) k = (Γ/ r ) k = C 1,k Γ (A/r ) obtaining d(y) (Γ/ r ) k = Γ k / r. Note that this comutation requires knowledge of η R (y) mod r. Now d(y) will be divisible by v s 1, hence d(y) = v s 1z in (Γ/ r ) k with z (Γ/ r ) k 2s( 1) = C 1,k 2s( 1) Γ (A/ r ) reresenting δ(x). Lift z to some w Γ k 2s( 1). To roceed, we use diagram (6): w lies in the last but one grou of the to row, hence we comute ν(w) H 2 ( ) which is our reresentative for δ (δ(x)) Ext 2. This requires to comute φ 1. Observe for examle that φ 1 (t 1 ) = 1 v 1 v Ellitic homology theories and divided congruences: the f-invariant We refer the reader to [L1] or [HBJ] for the notion of ellitic homology with resect to the congruence subgrou Γ 1 (N). In this section, E denotes the sectrum associated to the homology theory with coefficient ring E = M (Z (), Γ 1 (N)), see section 3.2 for the notation. Finally, α : BP E denotes the orientation. By the naturality of the constructions in section 2.2 we have a commutative diagram for any k 0 Ext 2,k [BP] α Ext 2,k [E] (7) (7) ι (BP BP ) (k) α α (E E ) (k). BP k BP+(BP + BP ) (k) E k E+(E + E ) (k) The injectivity of α holds for any Landweber exact theory E of height at least two, [L2, Proof of 4.3.2]. To roceed, however, we will use a more subtle roerty of E, namely the toological q-exansion rincile. We ut D k := {f = k f i k E,2i there are g 0, g k E,2k such that (f + g 0 + g k )(q) Z Γ () [[q]]} where f(q) denotes the q-exansion of f at the cus infinity, and for Γ = Γ 1 (N) we set Z Γ := Z[ 1 N, ζ N] if N > 1 and Z SL2(Z) := Z[ 1 6 ] as in [L1]. We then define (8) ι 2 : (E E ) (2k) E 2k E + (E + E ) (2k) D k /Z, i+j=k f i g j i+j=k q 0 (f i )g j, where q 0 (f) is the constant term of the q-exansion of f at the cus infinity. The comosition ι 2 ι is injective [L1, Proosition 3.9] and hence so is the f-invariant f : Ext 2,2k [BP] D k /Z. 7

9 Jens Hornbostel and Niko Naumann We remark that our grading of E is the toological one, i.e. elements of dimension 2k corresond to modular forms of weight k. G. Laures describes Ext 2 using the canonical Adams resolution [R, Definition ] instead of the cobar resolution. Proosition 4. For k > 0 even, the above definition of the f-invariant Ext 2,k D k/2 /Z coincides with the one given in [L1]. Proof. We have to show that the ma Ext 2,k (BP BP ) (k) BP k BP+(BP + BP ) (k) we constructed in section 2.2,(7) coincides with the one of [L1]. We know [R, Lemma A (b)] that, u to chain homotoy, there is a unique ma from the unreduced cobar resolution Γ A Γ A to the canonical Adams resolution BP (BP ΣBP ) where BP S 0 η BP d ΣBP is an exact triangle in the stable homotoy category, see also [Br, Lemma 3.7] Now one checks that Γ A(n+1) = π (BP n+2 ) π (id BP 2 d n ) π (BP 2 (ΣBP) n ) is a ma of chain comlexes where the isomorhism follows from [R, Lemma 2.2.7] and induction. So the claim reduces to the fact that the triangle Z 2,k (Γ A ) π (id BP 2 d) (BP BP ) (k) BP k BP+(BP + BP ) (k) Z 2,k (π (BP 2 (Σ BP) )) commutes. This follows using that our mas τ, H 2 and π (d d) corresond to r, ρ and the isomorhism D 1 / Σ 1 = G 2 = π ((ΣBP) 2 ) in [L1, section 3.1] (where we define τ by maing all v i to zero). Note the degree shift and the factor 2 for the f-invariant π2k s D /Z (both are missing in k+1 [L1,. 411]). 3. Arithmetic comutations In section 3.1, we review results of N. Katz on divided congruences and establish a relation between BP-theory and the mod Igusa tower (Theorem 5). In section 3.2 we give exlicit comutations for ellitic homology of level 3 and the corresonding divided congruences. 3.1 Divided congruences We review arts of [K1]. Some technical remarks are in order: In loc. cit. N. Katz works with level-n structures of fixed determinant for N 3. To confirm with general olicy in algebraic toology we wish to consider Γ 1 (N)-structures instead, which are reresentable only for N 5. More seriously, one has to check that the relevant art of [K1] works for this different moduli roblem. This we did, but we will not exlain the details here and only remark that both the geometric irreducibility of the moduli saces and the irreducibility of the Igusa tower are valid for Γ 1 (N). Furthermore, we will use these results for 5 and N = 1 and for = 2 and N = 3. These cases can be handled by using auxiliary rigid level structures and taking invariants under suitable finite grous as in [K1]. Fix a level N 5, a rime not dividing N and a rimitive N-th root of unity ζ F. We 8

10 Beta-elements and divided congruences ut k := F (ζ), W := W (k) (Witt vectors) and also denote by ζ W the Teichmüller lift of ζ. Finally, K denotes the field of fractions of W and for any Z () -algebra R we denote by M k (R, Γ 1 (N)) the R-module of holomorhic modular forms for Γ 1 (N) of weight k and defined over R, see e.g. [K3] or [L1]. The Γ 1 (N) is omitted from the notation if it is clear from the context. We fix a lift E 1 M 1 (W ) of the Hasse invariant. The existence of such a lift uts further restrictions on and N which are satisfied in our alications. We define the ring of divided congruences D by M (W ) D := {f M (K) f(q) W [[q]]} M (K), where f(q) is the q-exansion of f at the cus infinity. For n 0 we also define ( n ) D n := D M i (K) D and D n := D n + K + M n (K) n M i (K) which is consistent with the definition of the revious section. The grou D k considered in [L1] differs from the D k above because the ring of holomorhic modular forms has been localised in [L1]. This difference is not serious because the f-invariant factors through holomorhic modular forms [L1, Proosition 3.13]. The ring D carries a uniformly continuous Z -action (the diamond oerators) defined by [α]( i f i ) := i α i f i, where α Z and f i M i (K). We ut Γ 0 := Z and Γ n := 1+ n Z for n 1 and V 1,n := (D/D) Γn for n 0. Then V 1,0 V 1,1... D/D is an ind-étale Z -Galois extension, the mod Igusa tower. So, V 1,0 V 1,1 is a (Z/) -Galois extension and for all n 2 V 1,n 1 V 1,n is an étale Z/-extension and hence an Artin-Schreier extension. An immediate comutation with diamond oerators shows that the comosition M (W ) D D/D factors through V 1,1 D/D. It is a result of P. Swinnerton-Dyer that this induces an isomorhism (9) M (W )/(, E 1 1) V 1,1, see [K1, Corollary 2.2.8]. By Artin-Schreier theory, given n 2 and x D/D satisfying [α](x) = x for all α Γ n and [1 + n 1 ](x) = x + 1 one has V 1,n = V 1,n 1 [x] and the minimal olynomial of x over V 1,n 1 is T T a for some a V 1,n 1. 9

11 Jens Hornbostel and Niko Naumann At this oint we can establish a first relation between BP-theory and divided congruences. Consider the ring extensions M (W ) D W [[q]]. We have a formal grou F over M (W ) induced by the universal ellitic curve. The base change of F to W [[q]] is the formal comletion of a Tate ellitic curve and is thus isomorhic to Ĝm. Imlicit in [K1] is the fact that D is the minimal extension of M (W ) over which F becomes isomorhic to Ĝm, i.e. D is obtained from M (W ) by adjoining the coefficients of an isomorhism F Ĝm defined over W [[q]]. This is what underlies N. Katz construction [K1, Section 5] of a sequence of elements d n D which modulo constitute a sequence of Artin-Schreier generators for the mod Igusa tower. Since the elements t n BP BP are the coefficients of the universal isomorhism of a -tyical formal grou law, one may exect a relation between the t n and the d n. To formulate this, denote by α : BP M (W ) the classifying ma of F and consider the comosition φ : BP BP BP BP (3) BP 2 α α M (K) 2 q0 id M (K). Note that the ma ι 2 in (8) comosed with the orientation α is a quotient of φ. The toological q-exansion rincile guarantees that T n := φ(t n ) D for n 1 and we can thus define T n := (T n mod D) D/D. Theorem 5. For any n 1 we have [1 + k ](T n ) = T n for k > n and [1 + n ](T n ) = T n + 1. Hence T n is an Artin-Schreier generator for the extension V 1,n V 1,n+1. Proof. Let ω = ( n1 a nt n 1 )dt be the exansion along infinity of a normalised (i.e. a 1 = 1) invariant differential on the universal ellitic curve. Then a n M n 1 (W ) and the logarithm of the -tyification is a n n0 t n n M (K)[[t]], i.e. the classifying ma α : BP M (W ), when tensored with, sends l n BP,2( n 1) to a n M n n 1(K), see [R, Theorem A ] for the definition of the l n (=λ n in the notation of loc. cit.). Defining d 0 := 1 and d n (n 1) recursively by (10) n d i n i i = a n n, N. Katz shows in [K1, Corollary 5.7] that the d n := (d n mod ) D/D behave under the diamond oerators as claimed for the T n. In BP BP we have η R (l n ) = n l it i n i and we aly φ to this relation to obtain which, using (10), imlies a n n = n q 0 (a i) i T i n i 10

12 Beta-elements and divided congruences (11) n d i n i i = n q 0 (a i) i T i n i. We now roceed by induction on n 1. For n = 1 we have d 1 + 1/ = q 0 (a 1 )T 1 + q 0 (a )/. Also, q 0 (a 1 ) = 1 since a 1 = 1 and q 0 (a ) 1 + W because a reduces mod to the Hasse invariant which has q-exansion equal to 1. We obtain T 1 = d 1 + α for some α k. As α is invariant under all diamond oerators, our claim for n = 1 is obvious. Assume that n 2. From (11) and a 1 = 1 we obtain T n = n d i n i i n i=1 q 0 (a i) i T i n i. For k > n we know that the terms involving d i are invariant mod under [1 + k ] whereas the remaining terms are likewise invariant by the induction hyothesis. Finally, we have [1 + n ]T n = [1 + n ]d n + [1 + n ]( n i=1 d i n i i ) [1 + n ]( n i=1 q 0 (a i) i T i n i ). Here we have [1 + n ]d n [1 + n ](T n ) = T n + 1. d n + 1 (D) and the remaining terms are invariant. Thus, indeed, 3.2 Modular forms For a rime 5, the following is well known [L1, Aendix]: M (Z (), Γ 1 (1)) = Z () [E 4, E 6 ], where E 4 and E 6 are the Eisenstein series of level one of the indicated weight. For the discriminant, the ring of meromorhic modular forms is given by Z () [E 4, E 6, 1 ] and the usual orientation BP Z () [E 4, E 6, 1 ] is Landweber exact of height 2 and factors through Z () [E 4, E 6 ]. A similar result holds for 3 and M (Z (), Γ 1 (2)) = Z () [δ, ɛ]. The urose of this section is to give analogous results for Γ 1 (3) and = 2, c.f. [St] for related results. Consider the ellitic curve E : y 2 + a 1 xy + a 3 y = x 3 defined over R := Z[1/3][a 1, a 3, 1 ] where = a 3 3 (a3 1 27a 3) is the discriminant of the given Weierstrass equation. Note that, unlike in level one, is not irreducible as a olynomial in a 1 and 11

13 Jens Hornbostel and Niko Naumann a 3 and we ut f := a 3, g := a a 3, hence = f 3 g. The section P := (0, 0) E(R) is of exact order 3 in every geometric fibre as follows from [Si1, III,2.3] and ω := dx/(2y + a 1 x + a 3 ) is an invariant differential on E. The following may be comared with [St, Lemma 11]: Proosition 6. The above tule (E/R, ω, P ) is the universal examle of an ellitic curve over a Z[1/3]-scheme together with a oint of order 3 and a non-zero invariant differential. Proof. We have to show that whenever T is a Z[1/3]-scheme and E /T is an ellitic curve with non-zero invariant differential ω and P E (T ) of exact order 3, there is a unique ma φ : T Sec (R) such that φ (E, P, ω) = (E, P, ω ). We show the uniqueness of φ first. This amounts to seeing that the only change of coordinates x = u 2 x + r, y = u 3 y + u 2 sx + t with r, s, t R and u R (see [Si1, III Table 1.2]) reserving (E, P, ω) is the identity, i.e. r = s = t = 0 and u = 1. From x (P ) = y (P ) = 0 we obtain r = t = 0. Next, a 4 = a 4 imlies sa 3 = 0, hence s = 0 because = f 3 g and thus also f = a 3 is a unit in R. Finally, ω = uω forces u = 1. Given the uniqueness of φ in general, its existence is a local roblem on T and we can assume that T = Sec (S) is affine and E /T is given by a Weierstrass equation with coefficients a i S. Moving P to (0, 0) gives a 6 = 0. We claim that a 3 S : This can be checked on geometric fibres where it follows from [Si1, III,2.3] and the fact that (0, 0) has order 3 (if a 3 vanished on some geometric fibre the oint (0, 0) would have order 2 in that fibre). Using this, one finds a transformation such that (dy) P = 0 in Ω E /T,P, hence a 2 = a 4 = 0. We thus have some ψ : T Sec (R) such that ψ (E, P ) = (E, P ) and ψ (ω) = uω for some u S. Adjusting ψ using u, i.e. multilying the a i by u i, we obtain the desired φ. We conclude that the ring of meromorhic modular forms is given as M mer (Z (2), Γ 1 (3)) = Z (2) [a 1, a 3, 1 ] with a i of weight i, and likewise for any other rime different from 3 in lace of 2. As usual, t = x/y is a local arameter at infinity for E/R which is normalised for ω and hence determines a 2-tyical formal grou law over M mer (Z (2), Γ 1 (3)). Using [Si1,. 113] one checks that the corresonding classifying ma α : BP M mer (Z (2), Γ 1 (3)) satisfies α(v 1 ) = a 1 and α(v 2 ) = a 3 for the Hazewinkel generators v i. Thus α makes M mer (Z (2), Γ 1 (3)) a Landweber exact BP algebra of height 2. Using the orders of f and g at the two cuss 0 and of X 1 (3), one can check that the ring of holomorhic modular forms is given by (12) M (Z (2), Γ 1 (3)) = Z (2) [a 1, a 3 ]. 12

14 Beta-elements and divided congruences We stick to the notations of section 3.1 for = 2 and N = 3. For examle, ζ denotes a rimitive cuberoot of unity and W = W (F 2 (ζ)) = W (F 4 ) = Z 2 [ζ] is the unique unramified quadratic extension of Z 2. To study divided congruences we will need to know the q-exansions of a 1 and a 3. Given a Dirichlet character χ, we consider it as a function on Z as usual and define for k 0 and n 1 σ χ k (n) := χ(d)d k. 1d n In the following, χ will always denote the unique non-trivial character mod 3 χ : (Z/3Z) C. Proosition 7. The q-exansions of a 1 and a 3 at the cus infinity are given as follows. a 1 (q) = (1 + 2ζ)(1 + 6 n1 σ χ 0 (n)qn ) and a 3 (q) = (1 + 2ζ)( n1 σ χ 2 (n)qn ) in W [[q]]. Proof. From (12) we know that rk M 1 (Z (2) ) = 1 and rk M 3 (Z (2) ) = 2. Using [K2, section 2.1.1] we see that (13) 6G 1,χ (q) = n1 σ χ 0 (n)qn M 1 (Z (2) ) and G 3,χ (q) = n1 σ χ 2 (n)qn M 3 (Z (2) ). We have evaluated L(0, χ) = 1/3 and L( 2, χ) = 2/9 using [Ne, Theorem VII.2.9] and [Wa, formula following Proosition 4.1 and Exercise 4.2(b)]. It is easy to see that G 3,χ (0) = 0, i.e. G 3,χ vanishes at the cus 0. Below, we exlain how to comute the following values of a 1 and a 3 at the cuss zero and infinity. (14) a 1 ( ) = 1 + 2ζ (15) a 3 ( ) = 1 (1 + 2ζ) 9 (16) a 3 (0) = 0. Using these values and the dimensions of the saces of modular forms of weight 1 and 3, we conclude that a 1 = 6(1 + 2ζ)G 1,χ and a 3 = (1 + 2ζ)G 3,χ, hence that a 1 and a 3 have desired q-exansions by (13). We are using the fact that the ma M 3 (Z (2) ) C = M 3 (C) C 2, f (f( ), f(0)) is an isomorhism, as follows from the theory of Eisenstein series. To establish (14) and (15) one has to evaluate a 1 and a 3 at the tule (T (q), ω can, P ) consisting of the Tate curve T (q)/z((q)), its canonical invariant differential ω can and a secific section P T (q)(z[ζ]((q)))[3]. To do so, one may use J. Tate s uniformisation [Si2,. 426] to 13

15 Jens Hornbostel and Niko Naumann write T (q)/(z[[q]]/(q 3 )) in Weierstrass form, the oint P having coordinates (X(q, ζ), Y (q, ζ)). One then uses Weierstrass transformations to bring (T (q), ω can, P )/Z[[q]]/(q 3 ) to the standard form of Proosition 6. The coefficients a 1 and a 3 of the Weierstrass equation thus obtained are by definition a 1 ( ) and a 3 ( ). The comutation for (16) is similar, the oint P has to be relaced by = (X(q, q 1/3 ), Y (q, q 1/3 )). Remark 8. In E. Hecke s notation [He], we have a 1 = 9i π G 1(τ, 0, 1, 3) and a 3 = 27i 4π 3 G 3 (τ, 0, 1, 3). Note that a 1 (q) 1 mod 2, hence a 1 M 1 (Z (2), Γ 1 (3)) is a lift of the Hasse invariant for = 2. From section 3.1 we know that V 1,0 = V 1,1 (9) M (W, Γ 1 (3))/(2, a 1 1) = k[a 3 ] (k := W/2W = F 4 ) and that, for T := q0 (a 1 ) a 1 2 D, T := (T mod 2) D/2D is an Artin-Schreier generator for V 1,1 V 1,2, in articular T 2 + T V 1,1 k[a 3 ] and for later use we will need the following more recise result. Proosition 9. T 2 + T = 1 + a 3. Proof. Recall that the q-exansion ma V 1,1 D/D k[[q]] is injective [K1, (1.4.6) for m = 1]. In k[[q]] we have T = n1 σχ 0 (n)qn and T 2 = n1 σχ 0 (n)q2n, hence T 2 + T = n1(σ χ 0 (n/2) + σχ 0 (n))qn, where we understand that σ χ 0 (n/2) = 0 for n odd. To comlete the roof, one needs to check that for all n 1 one has σ χ 0 (n/2) + σχ 0 (n) σχ 2 (n) mod 2, and we leave this exercise in elementary number theory to the reader. 4. f-invariants and Kervaire invariant one In this section, we comute the f-invariants of two infinite families of β-elements including the Kervaire elements β 2 n,2 n and exlain the relation of our results with the Kervaire invariant one roblem. 4.1 f(β t ) for t not divisible by Fix a rime and the level N as N = 1 for 5, N = 5 for = 3 and N = 3 for = 2. We kee the notations of section 3.1 for this choice of and N. Given an integer t 1 not divisible by, recall that β t = δ δ(v t 2 ) Ext2,2t(2 1) 2( 1) [BP] has its f-invariant in D n /Z, n := t( 2 1) ( 1). When trying to exress f(β t ) in terms of divided congruences, we encounter what is in fact the major obstacle at the moment for using the arithmetic of divided congruences in homotoy theory: The grou D n /Z is not directly related to D. Instead, we have D n = D+K +M n (K) by definition and there is a canonical surjection 14

16 Beta-elements and divided congruences ( n ) ( n ) π : D n /Z M i (K) /D n D n /Z M i (K) /D n which is slit because its kernel is divisible, hence W -injective. In articular, π remains surjective when restricted to -torsion (17) π : D n /Z[] D n /Z[], note that f(β t ) D n /Z[]. The grou D n /Z[] is related to the ring of divided congruences as follows: (18) ψ : D n /Z[] D n /D n D/D, where the first arrow is multilication by and the injectivity of the last ma is immediate. What we will do is to comute some element in D/D in the image of ψ which under π rojects to f(β t ). At the low risk of confusion we will continue to label such an element, which is in general not unique, as f(β t ). Recall that we have fixed an ellitic orientation α : BP M (W ) and denote T := α(v 1) q 0 (α(v 1 )) D/D. We also ut b := ((q 0 (α(v 2 )) mod ) k. Theorem 10. For an integer t 1 not divisible by the fixed rime, we have f(β t ) = b t (T T + b) t V 1,0 D/D. Proof. Note first that from section 3.1 we know that T is an Artin-Schreier generator for V 1,1 V 1,2, hence T T V 1,1. A short comutation with diamond oerators, which we leave to the reader, shows that in fact T T V 1,0, hence also b t (T T + b) t V 1,0. We introduce a := q 0 (φ(v 1 )) and comute as exlained at the end of section 2.2 using the notations introduced there. From η R v 2 v 2 + v 1 t 1 v 1 t 1 mod we obtain t ( ) t η R v2 t v2 t + v2 t i v i i 1t i 1(t 1 1 v 1 1 ) i mod, hence and ν(w) = 1 ( (1 v1 v 1 1 w = i=1 i=1 t i=1 ( ) t v2 t i v1 i 1 t i i 1(t 1 1 v 1 1 ) i t ( ) ( ) t 1 (v2 t i v1 i 1 v1 v 1 1 i 1) i ) 1 v 1 1 1) i (BP BP ) (2n) BP 2n BP + (BP + BP ) (2n). As in section 2.3 we aly ι 2 (α α) to this exression to obtain, denoting α(v 1 ) M 1 (W ) as v 1 for simlicity, 15

17 Jens Hornbostel and Niko Naumann 1 t ( ) ( ) ( t b t i a i 1 v1 a i (v1 ) ) a 1 i a 1 = i i=1 [ ( 1 ( ) ( ( ) ) ) b t v1 a v1 a 1 t ] + a a 1 + b = a (( ( ) ( ( ) ) ) 1 v1 a v1 a 1 t ) n a a 1 + b b t M i (K). a This is a reresentative for f(β t ) in D n /Z[] to which we have to aly the ma ψ from (18) to obtain an element in D/D. For this, note that a 1 mod because v 1 reduces to the Hasse invariant mod. This allows us to ut a = 1 in the above exression (but not to relace v 1 a by v 1 1 ; this would require the congruence a 1 mod 2, which does not hold in general). We then obtain indeed ((T (T 1 1) + b) t b t ) = b t (T T + b) t. Remark 11. Assume that 5 in the situation of Theorem 10. In general, the ellitic orientation will not ma v 1 to the Eisenstein series E 1 of weight 1 and level one. But α(v 1 ) and E 1 can only differ by a modular form divisible by and we may thus change the orientation to force α(v 1 ) = E 1. Assuming this, we see that f(β 1,1,1 ) = E Laures [L1,. 414] (where the second summand is missing). 1 ( E 1 1 ), as first comuted by G. ) D ( 1) /Z is of order, i.e. F ( 1) M i (K) Remark 12. The injectivity of the f-invariant together with the known structure of Ext 2 [BP] rovides some non-trivial information about the arithmetic of divided congruences as follows. Fix some x Ext 2,k [BP] of order r. Then f(x) D k /Z will be of order r, hence a reresentative of f(x) in D k /Z will be of order s for some s r. Thus the f-invariant relates the order of a β-element to the (non-)existence of a certain divided congruence. Let us assume that r = 1 as is the case for all β t,s,r considered in this article. Then our results show that our reresentatives in D /Z have order and the non-trivial additional information on divided congruences is then that they do not lie in the kernel of D /Z D /Z. To give an examle, assume that we are in the situation of Remark 11. The arithmetic of divided congruences shows that F := E ( E 1 1 has a q-exansion with denominator exactly. The additional information is then that for any α K and f M ( 1) (K) the q-exansion of F + α + f will still have exact denominator. Examle 13. Fix = 5 and set g 2 := 1 12 E 4 and g 3 := E 6 as in [K1]. The comarison of the logarithm of the universal -tyical formal grou law [R] and the corresonding coefficients of the logarithm of the ellitic curve (E, ω) [K1, (5.0.3)] (-tyification does not change these coefficients) shows that the orientation a mas v 1 to a and v 2 to a 2 a +1, the a i denoting the normalised (multilied with 1/2) a i of [K1,. 351]. One deduces that v 1 mas to 8g 2 and v 2 mas to a 25 a A comutation with Male shows that a 25 = g2 3g g g6 2 (and also that the correct value for the unnormalised a 11 is 2520g 2 g 3 and not 512g 2 g 3 ). It follows that q 0 (v 1 ) = 2 3 and q0 (v 2 ) = 4900, so Theorem 10 may be rehrased in terms of the Eisenstein series 3 10 g 2 and g 3. 16

18 4.2 Projecting to the Kervaire element Beta-elements and divided congruences In this section, we comute f(β s2 n,2n) for n 0 and s 1 odd at the rime = 2. Using this, we are able to determine a single coefficient in the f-invariant of a (U, fr) 2 - manifold of dimension 2 n the non-vanishing of which is necessary and sufficient for the corner of X to be a Kervaire manifold, that is having Kervaire invariant one. See [L2] for the notion of cobordism of manifolds with corners. We begin by recalling the well-known relation of the Kervaire invariant to certain β-elements, due to W. Browder [B]. Fix some n 3. We have a homomorhism K : π s 2 n 2 Z/2 which sends the class of a stably framed manifold to its Kervaire invariant. Consider on the other hand the comosition K : π s 2 n 2 π s 2 n 2[2] E 2,2n [HZ/2] E 2,2n 2 [HZ/2] = Z/2 h 2 n 1. Here, the first ma is the rojection to the 2-rimary art, the second is the rojection onto F 2 /F 3 in the (classical) Adams sectral sequence at = 2, the third is an edge homomorhism and the final equality is due to J. Adams, [R, 3.4.1, c)]. Proosition 14. K = K. Proof. For any y π2 s n 2 [2] we have K(y) = 1 if and only if y has Adams filtration 2. This is imlicit in [B], c.f. [BJM2,. 144]. We can easily obtain a similar homotoy theoretic descrition of K using BP instead of HZ/2. Proosition 15. Let n 2. Then Ext 2,2n is a direct sum of cyclic grous of order 2. It is generated by the element α 1 α 2 n 1 1 and the elements β s2 i,2 i with s odd and i 0 such that (3s 1)2i+1 = 2 n and the case (s, i) = (1, 0) has to be omitted. Proof. This follows from [R, Corollary 5.4.5]. Observe that the α t in loc. cit equals α t as t is odd, see [Sh, Theorem 1.5] or [R, Theorem 5.2.6]. Remark 16. The Lemma shows that the number of generators of Ext 2,2n is [n/2] + 1 for n 3. The low dimensional cases are as follows. Ext 2,4 : α 2 1 Ext 2,8 : α 1 α 3, β 2,2 Ext 2,16 : α 1 α 7, β 4,4, β 3,1 Ext 2,32 : α 1 α 15, β 8,8, β 6,2 Ext 2,64 : α 1 α 31, β 16,16, β 12,4, β 11,1 Ext 2,128 : α 1 α 63, β 32,32, β 24,8, β 22,2 Ext 2,256 : α 1 α 127, β 64,64, β 48,16, β 44,4, β 43,1. 17

19 Now we consider the comosition Jens Hornbostel and Niko Naumann K : π s 2 n 2 π s 2 n 2[2] E 2,2n [BP] E 2,2n 2 [BP] Z/2 β 2 n 2,2 n 2 which is defined in analogy with K, the final ma being the rojection to the Z/2-summand generated by β 2 n 2,2 n 2. Proosition 17. K = K. Proof. We have the Thom reduction Φ : Ext [BP] Ext [HZ/2] which satisfies Φ(β 2 n 2,2 n 2) = h 2 n 1 and is zero on all other generators of Ext2,2n [BP], see [R, 5.4.6, a)] and Proosition 15. The result then follows from Proosition 14. Let X be a (U, fr) 2 -manifold of dimension 2 n. From the above, we see that the corner of X is a Kervaire manifold if and only if the f-invariant of X contains β 2 n,2 n as a summand. Thus, one certainly wants a more geometric descrition of the f-invariant (or just its rojection to β 2 n 2,2 n 2). In rincile, it is ossible to obtain such a descrition in terms of Chern numbers of X, simly because they determine the (U, fr) 2 -bordism class of X [L2], but the necessary comutations become quite comlicated already in low dimensions. At the end of this section, we will exlain how divided congruences might simlify such comutations. We then would like to generalise Theorem 18 below to higher dimensions. Recall [L2, section 4.1] that if X is a (U, fr) 2 -manifold then there is a decomosition of its stable tangent bundle T X = T X (0) T X (1) and we have Chern classes c (j) i H 2i (X, Z) accordingly (i 0, j = 0, 1). Theorem 18. a) Let X be a (U, fr) 2 -manifold of dimension 4 and ut q :=< c (0) 1 c(1) 1, [X] > Z. Then q is odd if and only if the corner of X has Kervaire invariant 1. If q is even, then the corner of X is the boundary of a framed manifold. b) Let X be a (U, fr) 2 -manifold of dimension 8 and ut q :=< c (0) 1 (c(1)3 1 + c (1) 1 c(1) 2 + c (1) 3 ) + (c(0) 2 + c (0)2 1 )(c (1) 2 + c (1)2 1 ), [X] > Z. Then q is odd if and only if the corner of X has Kervaire invariant 1. If q is even, then the corner of X is the boundary of a framed manifold. Proof. If i0 l ix 2i is the logarithm of the universal 2-tyical formal grou law, then (see [L2, Examle 4.2.4]) ex(x) = x l 1 x 2 + 2l 2 1x 3 (5l l 2 )x 4 (mod x 5 ) and thus (x) := x ex(x) = 1 + l 1x l1x (2l1 3 + l 2 )x 3 (mod x 4 ). For indeterminates x i of dimension 2 we set Π := i (x i). Denoting by c i the i-th elementary symmetric function in the x i one gets (using the definition of the Hazewinkel generators) Π (2) = v 1 2 c 1 Π (4) = v2 1 4 (3c 2 c 2 1) and Π (6) = v3 1 8 (4c3 1 13c 1 c c 3 ) + v 2 2 (c3 1 3c 1 c 2 + 3c 3 ). 18

20 Beta-elements and divided congruences To rove art a) one has that BP 2,4 /(BP + BP ) is a one-dimensional -vector sace generated by v 1 v 1. Moreover, one checks that the image of BP 4 BP is generated (over Z (2) ) by v 1 v 1 2 and that v 1 v 1 4 is a reresentative of α1 2. To see the latter, observe that α 1 is reresented by t 1, hence [R, A ] α1 2 is reresented in the cobar comlex by t 1 t 1 = (1 t 1 )(t 1 1), and then one alies the descrition of δ 1 given in section 2.2. Using the notations introduced in [L2], one comutes that K BP <2>(T X) (4) = c (0) 1 c(1) 1 v 1 v 1 4 in BP 2,4 /(BP + BP ) (4), hence the image of the corner of X in Ext 2,4 is reresented by q 2 v1 v 1 2 = q α1 2. The final assertion follows because the only non-trivial element of π2 s has Adams-Novikov filtration recisely 2. For art b), we know that Ext 2,8 is generated by α 1 α 3 and β 2,2. As β 2,2 is a ermanent cycle in the ANSS whereas α 1 α 3 is not we know that the image of X in Ext 2,8 is a multile of β 2,2. One comutes that in the notation of section 2.2 δ(v2 2) is reresented by z = t4 1 + v2 1 t2 1 in C1 (A/2). Hence β 2,2 = δ δ(v2 2 ) is reresented in BP 2,8 /(BP + BP ) (8) by 1 8 (v 1 v1 3) (v2 1 v2 1 ) 3 8 (v3 1 v 1). Comuting enough of the image of v1 3t 1, v1 2t2 1, v3 1 t 1 and t 4 1 under BP 8BP BP 8 BP BP 2,8 BP 2,8 /(BP + BP ) (8), one sees that v2 1 v2 1 8 im (BP 8 BP) and β 2,2 is reresented by v2 1 v We rovide the following argument for general n, for the roof here we need the case n = 3. Observe that by the comutations of the revious sections, the image of Ext 2,2n in /(BP,2 n + BP,2 n +BP 2 nbp) is given by reresentatives consisting of summands of BP 2,2n the form v1 i vk 2 vj 1 with rational coefficients. Moreover, these elements ma under the isomorhism φ of section 2.2 to olynomials in v 1, v 2 and t 1. In other words, the f-invariant in bidegree (2, 2 n ) factors through the subgrou generated by elements v1 i vk 2 vj 1 modulo elements of the form φ 1 (v1 i vj 2 tk 1 ), 1 v j 1 and vi 1 vj 2 1. Denote this quotient by B 2n. A comutation using the results of the revious c section shows that v 2n 2 2 2n 1 1 v1 2n 2 is not zero in B 2 n for c an odd integer. More recisely, no element in the relations defining the quotient B 2 n contains such a summand. Among the elements in Ext 2,2n, the image of β 2 n 2,2 n 2 in B 2n contains such a summand and the other generators exhibited in Proosition 15 do not. Thus we have a well-defined ma Ext 2,2n Z/2 given by maing an c element to 1 if it admits a reresentative in B 2 n having a summand v 2n 2 2 2n 1 1 v1 2n 2 with c odd. This ma is a rojection to β 2 n 2,2n 2. We would like to consider the summands of K BP <2>(T X) (8) Π (2) Π (6) + Π (4) Π (4) + Π (6) Π (2) mod (BP + BP ) (8) as elements in B 2 3. Once this is achieved, we have to consider only the summands involving v1 2 v2 1. The only summand which is not already given by a reresentative in B 2 3 is c (0) 1 (c(1)3 1 3c (1) 1 c(1) 2 + 3c (1) 3 ) v 1 v 2 4 in Π (2) Π (6). One comutes (for = 2 and the Hazewinkel generators as before) that φ 1 (t 2 ) = 1 v 2 2 v v3 1 4 v 1 v v2 1 v v Hence we have φ 1 (t 1 t 2 ) = v 1 v 2 4 v2 1 v , so we have to look at the coefficients of v 1 v 2 4 and v2 1 v which by the above equal c (0) 1 (c(1)3 1 3c (1) 1 c(1) 2 + 3c (1) 3 ) and (3c(0) 2 c (0)2 1 )(3c (1) 2 c (1)2 1 ). We also have that c (0) 1 (c(1)3 1 3c (1) 1 c(1) 2 + 3c (1) 3 ) equals c (0) 1 (c(1)3 1 + c (1) 1 c(1) 2 + c (1) 3 ) and (3c(0) 2 c (0)2 1 )(3c (1) 2 c (1)2 1 ) equals (c (0) 2 + c (0)2 1 )(c (1) 2 + c (1)2 1 ) modulo 2 when evaluated on [X]. Now the assertion follows as in art a). Of course, it is ossible to do similar but more comlicated comutations for 2 n -dimensional (U, fr) 2 -manifolds in case n 4. We always have a rojection to Z/2 looking at the ower of 2 in the denominator of the coefficient of the summand v1 2n 2 v1 2n 2. The comutation then reduces to comute those Π (2i) which contribute to this summand. The diligent reader may thus 19

21 Jens Hornbostel and Niko Naumann rove statements of the following form: The element h 2 n 1 survives (equivalently: there is a framed manifold in dimension 2 n 2 having Kervaire invariant 1) if and only if < F n (c (0) i, c (1) j ), [X] > is odd for a certain exlicit olynomial F n. The main roblem in the comutation of F n is to find reresentatives in B 2 n (that is sums of v1 i vk 2 vj 1 ) for elements arising in the Π(2n i) Π (i). For n = 3 this was done using t 1 t 2. In the case n > 3, it will be necessary to find suitable elements in BP 2 nbp which will involve t i for larger i and the comutation of φ 1 of these elements. The rest of this section is devoted to the comutation of the f-invariant in dimension 2 n at the rime 2. More generally, we comute the f-invariant of β s2 n,2 n Ext2,(3s 1)2n+1 for all n 0 and s 1 odd. We use the notations of section 3.1 for = 2 and N = 3 and those of section 3.2 and write T := a D/2D which is an Artin-Schreier generator for the extension V 1,0 = V 1,1 k[a 3 ] V 1,2. Theorem 19. The image of the f-invariant in V 1,2 is given by f(α 1 α 2 n+1 1) = T for n 0, f(β s ) = 1 + a s 3 for s 3 odd, f(β s2,2 ) = 1 + a 2s 3 for s 1 odd and f(β s2 n,2n) = (a2n 3 + a3 3 2n 2 ) s for s 1 odd and n 2. Proof. For the first line, recall that mod 2 we have α t := α t,1 := δ(v1 t ). One comutes that in the cobar comlex α 1 α t is reresented by t [(2t 1 + v 1 ) t v1 t ], use the descrition of the roduct in the cobar comlex of [R, A ]. Using that φ is a ring isomorhism and the descrition of δ 1, see section 2.2, one further comutes that α 1 α t is reresented by 1 4 v 1 v1 t in the usual quotient of BP 2, c.f. (7). The second line is a secial case of Theorem 10 (use Proosition 9) and imlies the third line because x 1 x 2 0 = v2 2 mod v2 1, recall the invariant sequences (2, v2n 1, x n) from section 2.1. The only case requiring a longer comutation is f(β 4,4 ): In the notation of section 2.1 we have r = 1, s = t = 4 and x 2 = v 4 2 v 3 1v 3 2 H 0,24 (BP/(2, v 4 1)). This value of x 2 follows from the definition in [MRW,. 476] or [R, Theorem ]) after cancelling all ossible multiles of v 4 1. One comutes that, in the notation of section 2.2, z (Γ/2)16 is given as z = t v 4 1t v 2 2t 1 (t 1 + v 1 ) + v 1 v 2 t 2 1(t v 2 1) + v 2 1t 3 1(t 1 + v 1 ) 3 = v 1 v 2 2t 1 + v 2 2t v 3 1v 2 t v 5 1t v 1 v 2 t v 3 1t v 2 1t t 8 1. One then comutes that f(β 4,4 ) = ν(w), where w Γ is a lift of z as in section 2.2, is as claimed, using the relation in Proosition 9 and that q 0 (a 1 ) = q 0 (a 3 ) = 1 mod 2 (see Proosition 7). Now the value for f(β s4,4 ) for any s follows immediately from the derivation roerty of the connecting homomorhism δ, namely δ(x n ) = δ(x)( n 1 η R(x) i η L (x) n 1 i ). Alternatively, f(β s4,4 ) may be comuted directly for any odd s using that (q 0 id)η L (x 2 ) = 0 mod 2, η R and φ 1 are ring homomorhisms and Proosition 9. We obtain f(β s2 n,2 n) = f(β s4,4) n 2 for all s and n 3 because x n = x 2 n 1 for all n 3. Fix some n 3. To exlain the relevance of the above comutation for the roblem of rojecting the f-invariant to β 2 n 2,2 n 2 we contemlate the following diagram. 20

22 Beta-elements and divided congruences (19) Ext 2,2n [BP] f D 2 n 1 /Z[2] Z/2 (17) π ι (18) D 2 n 1 /Z[2] D/2D V 1,2 V 1,2 (section 3.1) π k = F 4. By the results in section 3.1, V 1,2 is k-free on the set {a i 3 T j i 0, j = 0, 1} and π is defined to be the rojection to the coefficient of a 2n 2 3. The ma π is defined to be the rojection to the generator β 2 n 2,2n 2, c.f. Proosition 15. Theorem 19 determines reresentatives in V 1,2 := D 2 n 1 /Z[2] V 1,2 for all generators of Ext 2,2n [BP] and thus defines the ma ι. We know that (19) commutes when π and π are omitted. Theorem 20. The diagram (19) is commutative. Proof. By insection of Proosition 15 and Theorem 19, the only generator of Ext 2,2n [BP] whose f-invariant contains a 2n 2 3 is the Kervaire element β 2 n 2,2 n 2. Corollary 21. Let n 3 and X a (U, fr) 2 -manifold of dimension 2 n. Then the corner of X has Kervaire invariant one if and only if the f-invariant of X admits a reresentative in V 1,2 which contains the summand a 2n 2 3. Note that the coefficient of a 2n 2 3 in the f-invariant of X can rather easily be exressed in terms of Chern numbers of X. The very reason that this does not give us the Chern numbers determining the Kervaire elements is the indeterminacy in the above constructions caused by the rojection D 2 n 1 /Z[2] D 2 n 1 /Z[2]. References APS M. Atiyah, V. Patodi, I. Singer, Sectral asymmetry and Riemannian geometry II, Math. Proc. Cambridge Philos. Soc. 78 (1975), no. 3, BJM M. Barratt, J. Jones, M. Mahowald, Relations amongst Toda brackets and the Kervaire invariant in dimension 62, J. London Math. Soc. (2) 30 (1984), no. 3, BJM2 M. Barratt, J. Jones, M. Mahowald, The Kervaire invariant and the Hof invariant, in: Algebraic toology (Seattle, Wash., 1985), , Lecture Notes in Math., 1286, Sringer, Berlin, B W. Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. Math. 90 (1969), Br He HBJ R. Bruner, The homotoy theory of H ring sectra, in: H Ring Sectra and their Alications, Lecture Notes in Math., 1176, Sringer, Berlin, E. Hecke, Theorie der Eisensteinschen Reihe höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik, Abh. Math. Seminar der Hamb. Univ. 5 (1927), oder Kaitel 24 aus E. Hecke, Mathematische Werke, Vandenhoeck & Rurecht, Gttingen F. Hirzebruch, T. Berger, R. Jung, Manifolds and modular forms, Asects of Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig,