Beta-elements and divided congruences Jens Hornbostel and Niko Naumann Abstract The f-invariant is an injective homomorphism from the 2-line of the Adams-N* *ovikov spec- tral sequence to a group which is closely related to divided congruences of* * elliptic modular forms. We compute the f-invariant for two infinite families of fi-elements * *and explain the relation of the arithmetic of divided congruences with the Kervaire invaria* *nt one problem. 1. Introduction One of the most successful tools for studying the stable homotopy groups of sph* *eres is the Adams- Novikov spectral sequence (ANSS) Es,t2= Exts,tMU*MU(MU *, MU *) ) ssst-s(S0). The corresponding filtration on sss*:= sss*(S0) defines a succession of invaria* *nts of framed bordism, each being defined whenever all of its predecessors vanish, the first one of wh* *ich is simply the degree d : F0,*=F1,*= sss0-! E0,02= Z which is an isomorphism. The next invariant, defined for all n > 0, is the e-in* *variant e : sssn= F1,n+1-! E1,n+12 Q=Z, c.f. [Sw , Chapter 19]. Though defined purely homotopy-theoretic here, the e-in* *variant is well-known to encode subtle geometric information. For its relation to index theory via th* *e j-invariant see [APS , Theorem 4.14]. The e-invariant vanishes for all even n = 2k > 2, thus giving ri* *se to the f-invariant f : sss2k= F2,2k+2-! E2,2k+22. The understanding of this invariant is fragmentary at the moment. In particular* *, there is no index- theoretic interpretation of it comparable to the one available for the e-invari* *ant. As a first step towards understanding the f-invariant, G. Laures [L1] showed ho* *w elliptic homology can be used to consider the f-invariant _______________________________________________________________________________* *__ 2000 Mathematics Subject Classification 55Q10, 55T15, 55N34, 11F33 Keywords: beta elements, f-invariant, divided congruences Jens Hornbostel and Niko Naumann f : sss2k-! E2,2k+22,! D__k+1 Q=Z as taking values in a group which is closely related to divided congruences of * *modular forms. Note that this is similar to the role taken by complex K-theory in the study of the * *e-invariant. Strictly speaking, at this point we had better switched from MU to BP. In fact, we will * *always work locally at a fixed prime p in the following. This surprising connection of stable homotopy theory with something as genuinel* *y arithmetic as divided congruences certainly motivates to ask for a thorough understanding of * *how these are related by the f-invariant. The main purpose of this paper is to make this relation explicit. We also include a fairly self-contained review of G. Laures' above version of t* *he f-invariant to help the reader who might be interested in making his own computations. We now revie* *w the individual sections in more detail. In section 2, we first remind the reader of the fi-elements which generate the * *2-line of the ANSS (with a little exception at the prime 2). We then construct, for suitable Hopf * *algebroids, a complex which is quasi-isomorphic to the cobar complex and which will facilitate later * *computations. Finally, we show how to use elliptic homology to obtain the f-invariant as above. In section 3, we recall some fundamental results of N. Katz on the arithmetic o* *f divided congruences and point out an interesting relation between BP-theory and the mod p Igusa tow* *er (Theorem 5). Next, we give some specific computations of modular forms and divided congruenc* *es for 1(3) which n* *+2 we will need to study the f-invariant of the Kervaire elements fi2n,2n2 Ext2,2 * *[BP ] at the prime p = 2. In section 4, we first compute the f-invariants of the infinite family of fi-el* *ements fit for t > 1 not divisible by p (Theorem 10). Then we explain how to approach the problem of com* *puting the Chern numbers determining the fi2n,2n(see [L2, Corollary 4.2.5] for the case of fi1 a* *t the prime 3). We do this by explicit computations in BP-theory for dimension 2 and 6 (Theorem 18). * *The computations get very complicated in higher dimensions. In order to use divided congruences,* * we then compute the f-invariants of the family fis2n,2nfor n > 0 and s > 1 odd (Theorem 19). We* * hope that clever use of divided congruences will enable us to compute the Chern numbers determin* *ing the fi2n,2nfor 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 possible to project to the element fi2n,2nusing divid* *ed congruences - and consequently rephrase the Kervaire invariant one problem - was communicated to * *us by G. Laures. 2.The construction of the f-invariant We remind the reader of the construction of fi-elements in section 2.1. In sect* *ion 2.2, we construct a complex which is quasi-isomorphic to the cobar complex and in which we will c* *ompute represen- tatives for some of the fi-elements. This is used in section 2.3 where we expla* *in how to express the f-invariant of elements in Ext2,2k[BP ] in terms of divided congruences. 2 Beta-elements and divided congruences 2.1 Beta-elements in stable homotopy We review some facts on Brown-Peterson homology BP at the prime p and fi-elemen* *ts. See [MRW ] and [R ] for more details. The Brown-Peterson spectrum BP has coefficient ring * *BP* = Z(p)[v1, v2, ...] with vi in dimension 2(pi- 1). The universal p-typical formal group law is defi* *ned over this ring. The couple (A, ) := (BP *, BP*BP ) becomes a Hopf algebroid in a standard way * *and we have BP *BP = BP* Z(p)[t1, t2, ...] such that the left unit jL of the Hopf algebroid* * (A, ) is the standard inclusion. The right unit jR is determined over Q by the formula in [R , Theore* *m A.2.1.27]. Choosing the Hazewinkel generators [R , A.2.2.1] for the vi, a short computation yields * *jR(v1) = v1+ pt1 and jR(v2) = v2+ v1tp1- vp1t1 mod p. We have the chromatic resolution of BP* as a left BP*BP -comodule BP*! M0 ! M1 ! ... which gives rise to the chromatic spectral sequence Exts,*BP*BP(BP *, Mt) ) Exts+t,*BP*BP(BP *, BP*). This allows the construction of elements in Ext*,*BP*BP(BP *, BP*), the so call* *ed Greek-letter elements. Strictly speaking, these elements arise from comodule sequences 0 ! Nn ! Mn ! N* *n+1 ! 0, but for our computations we will need the related comodule sequences (1) and (2* *) below. See [MRW , Lemma 3.7 and Remark 3.8] for the relationship between them. We abbrevi* *ate Hn(.) := Extn,*BP*BP(BP *, .) in the following. To construct the fi-elements [MRW , p. 476/477], choose integers t, s, r > 1 s* *uch that (pr, vs1, xt0n) BP *is an invariant ideal where t = pnt0, (p, t0) = 1 and xn is a homogeneous p* *olynomial in v1, v2 and v3 considered as an element of v-12BP*=(pr, vs1) (see [R , p. 202] or [MRW * * , p. 476]), for example x0 = v2. Consider the two short exact sequences of BP*BP -comodules r (1) 0 ! BP* p!BP *! BP*=(pr) ! 0 vs1 r r s (2) 0 ! 2s(p-1)BP*=(pr) ! BP *=(p ) ! BP*=(p , v1) ! 0. Using the induced boundary maps ffi : H0(BP *=(pr, vs1)) -! H1(BP *=(pr)) and ffi0: H1(BP *=(pr)) -! H2(BP *) we define fit,s,r:= ffi0ffi(xt0n). It is known [MRW , Theorem 2.6], [Sh] for which indices (t, s, r) the elements* * fit,s,rare non-zero in H2(BP*). In this case the order of fit,s,ris pr. By construction, we have fit,s* *,r2 H2,2t(p2-1)-2s(p-1)(BP *). n+2 Example 1. For p = 2 and n > 1, the element fi2n,2n:= fi2n,2n,12 Ext2,2BP*BP(BP* * *, BP*) is called the Kervaire element. It is mapped via the Thom reduction [R , Theorem 5.4.6] to th* *e element h2n+12 n+2 Ext2,2HZ=2*HZ=2(HZ=2*, HZ=2*). The latter element survives to a non-zero elemen* *t of ss2n+2-2(S0) in the Adams spectral sequence if and only if (as W. Browder [B , Theorem 7.1] has* * shown) there exists 3 Jens Hornbostel and Niko Naumann a framed manifold of dimension 2n+2 - 2 with non-vanishing Kervaire invariant. * *Whether or not this is the case is unknown for n > 5, for n 6 4 see [BJM ], [KM ]. 2.2 The rationalised cobar complex The standard way of displaying elements in Extn := Extn(A, A) of a flat Hopf al* *gebroid (A, ) is by means of the cobar complex. In this section we shall give another descriptio* *n of this Ext group as a subquotient of (A Q) n needed to compute f-invariants (Proposition 3). T* *he results of this section are a more algebraic version of [L1, Section 3.1]. Let (A, ) be a Hopf algebroid with structure maps jL, jR, ffl and . This dete* *rmines a cosimplicial abelian group .as follows: Set n := An with cofaces @i : n -! n+1; @0(fl1* * . . .fln) := 1 fl1 . . .fln; @i(fl1 . . .fln) := fl1 . . . (fli) . . .fln (1 6 i 6 n) and @n* *+1(fl1 . . .fln) := fl1 . . .fln 1 for n > 1 and @0 := jR, @1 := jL for n = 0 and codegeneracies* * oei: n+1 -! n, oei(fl0 . . .fln) := fl0 . . .ffl(fli) . . .fln. We also denote by .the as* *sociated cochain complex. Following [R , Definition A.1.2.11], we define the reduced cobar complex (usual* *ly denoted as C.(A, A)) __. __n __ An __ * * __0 as being the subcomplex .with := for n_> 1_where := ker(ffl)_and* * := A. This is a subcomplex because (fl) - fl 1 - 1 fl 2 A for any fl 2 . We now assume that (A, ) is a flat Hopf algebroid such that i) A (and hence ) is torsion free. ii) The map AQ2 := (A Q) 2 -OE! Q := Q, a b 7! a.jR(b) is an isomorphism (jL * *is suppressed from notation). iii) The Q-algebra AQ is augmented by some o : AQ -! Q. Remark 2. The above assumptions i)-iii) are fulfilled by the flat Hopf algebroi* *d (BP *, BP*BP ): The proof of ii) follows from the fact that over any Q-algebra any two (p-typic* *al) formal group laws are isomorphic via a unique strict isomorphism. If BP* -! E* is a non-zero* * Landweber exact algebra [HS ], then (E*, E*E) also fulfils all the above assumptions: E* is a Z* *(p)-algebra on which p is not a zero divisor, hence E* is torsion free. Conditions ii), iii) and the f* *latness of of (E*, E*E) are inherited from (BP *, BP*BP ) because E*E ' E* BP* BP*BP BP* E* , c.f. [Na , * *Proposition 10] for the flatness. We define the cosimplicial abelian group D.by Dn := AQ(n+1)(n > 0) with cofaces* * @i : Dn -! Dn+1 to be given by @i(a0 . . .an) := a0 . . .1 . . .an with the 1 in the* * (i + 1)st position (i = 0, . .,.n + 1) and codegeneracies oei : Dn -! Dn-1 (i = 0, . .,.,* * n - 1) defined by oei(a0 . . .an) := a0 . . .aiai+1 . . .an. By ii) above we have for any n > * *0 an isomorphism n A n (3) OEn : Dn = AQ(n+1)' (AQ Q AQ) AQn OE-! Q Q = n Q which maps a0 . . .an 7! a0 . . .an-2 an-1.jR(an) and one checks that OE.is * *an isomorphism of cosimplicial groups and hence of cochain complexes. We have H.(D.) = H0(D.) = Q, a contracting homotopy being given by (4) Hn : Dn = AQ(n+1)-! Dn-1 = AQn ; a0 . . .an 7! o(a0)a1 . . .an. Define a subcomplex . D.by 4 Beta-elements and divided congruences Xn Xn n := @i(Dn-1) = AQi Q AQ(n-i) Dn = AQ(n+1), i=1 i=1 for n > 1 and 0 := 0. One checks that the composition __n n n (OEn)-1n n n 'n : ,! ,! Q -! D -! D = isPinjective for all n > 0. This is obvious for n = 0, and for n > 1 it follows* * from (\n-1i=0ker(oei)) \ ( ni=1im(@i)) = 0, which in turn is an easy consequence of thePcosimplicial id* *entities: One shows by descending induction on 1 6 j 6 n that (\n-1i=0ker(oei))\( ni=jim(@i)) = 0.* * Observe that Dn= n is isomorphic to the group labelled E* Gn*in [L1, p. 404] for A = BP. We define a cochain complex Q.by the exactness of __.'. . . ss. . (5) 0 -! -! D = -! Q -! 0, __ An hence Qn ' AQ(n+1)= n + im( ). From the definitions of the differential of D.and n one obtains (BQn Qn deno* *ting the boundaries) __ An Qn=BQn ' AQ(n+1)=" n+ im( ), P n+1 (i-1) (n+1-i) where " n:= n + Q AQn = i=1AQ Q AQ . The alternative description of Extwe are aiming for is the following. Proposition 3. For any n > 1, the connecting homomorphism ffi of (5) is an isom* *orphism __. n+1 Hn(Q.) -ffi!Hn+1( ) = Ext . Proof.One readily sees that the contracting homotopy (4) of D. respects the sub* *complex . in __ positive dimensions, hence the middle term of (5) is acyclic in these dimension* *s. |__| To explicitly compute ffi, it is useful to note that the differential of D.= .h* *as the simple form Dn= n -d!Dn+1= n+1 , [a0 . . .an] 7! [1 a0 . . .an], as is immediate from the definitions. To compute ffi-1, we consider the zig-zag oooo_ _'n+1//_ _Hn+1//_ __n+1n+1n+1nnssn////_n////_nn Extn+1__________________________________Z(DD =)=Q Q =BQOO ____________ffi-1______________________________| _______________ | ___________________________O| _________--_____________* *______Hn(Q.).? 5 Jens Hornbostel and Niko Naumann One checks that the dotted arrow exists and is the inverse of ffi. Now let p be a prime, r > 1 an integer, and let ffi0: Extn(A, A=pr) -! Extn+1be* * the connecting r homomorphism associated to the short exact sequence of -comodules 0 -! A .p-!A* * -! A=pr -! 0. Consider the diagram __n __n __n+1ff (6) ffi0: Extn(A, A=pr)oZ(ooo_A A=pr)oooo{z_2 |dz 2 pr _}___//Extn+1OO ________ | || ________________'ffi| fflffl| ___((_____| Qn=BQn oo_______?Hn(Q.).`_ __n+1 Here, ff(z) := [y] for any y 2 satisfying dz = pry and (z) := ssn(p-r'n(z* *)) mod BQn. The upper horizontal line is ffi0 by definition and one checks that factors throu* *gh the dotted arrow and makes the diagram commutative. Hence, when displaying an element of im(ffi0* *) in H.(Q.) rather than in the usual cobar complex, one does not have to compute the cobar differe* *ntial implicit in ff but only the (easier) map . Finally, assume that everything in sight is graded where the grading on Dn, n, * *etc. is by total degree. For a fixed k 2 Z, consider the commutative diagram __ An n `. ffi Qn=BQn ~=AQ(n+1)=" n+ im( )oo___H?(Q ) __'__//_Extn+1 OO OO| OO| j|| || || ?O| __ An ` ?O| ' O|? AQ(n+1),k=" n,(k)+ im( o)ko__?Hn,k(Q.)___//__Extn+1,k. Here " n,(k):= " n\ AQ(n+1),k. One checks that j is well defined and injective,* * and Hn,k(Q.) is defined to be the pull back of Hn(Q.) along j. The commutative diagram induces * *an isomorphism Hn,k(Q.) -'!Extn+1,kas indicated. For example, for (A, ) = (BP *, BP*BP ) and n = 1 we obtain an inclusion (BP Q BPQ)(k) (7) Ext2,k _____________________________ BP kBP + (BP Q Q + Q BPQ)(k) which is important for us since the f-invariant is defined in terms of the grou* *p on the right hand side. To effectively compute representatives of fi-elements in the complex Q.one proc* *eeds as follows. Let t, s, r > 1 be integers as in section 2.1 and ffi, ffi0the coboundary maps intr* *oduced there. Fix k 2 Z and x 2 H0,k(A=(pr, vs1)), that is x 2 C0,k(A=(pr, vs1)) = (A=(pr, vs1))k is an invariant element (C indicates the reduced cobar complex). As ffi is the* * connecting homomor- phism determined by the short exact sequence of complexes obtained by applying * *C to (2), we compute ffi(x) as follows: Lift x to y 2 (A=(pr))k and compute the cobar differ* *ential 6 Beta-elements and divided congruences __k __ r k 1,k r d = jR - jL : C0,k(A=pr) = (A=pr)k -! (A=pr A ) = ( =p ) = C (A=p ) __ __k obtaining d(y) 2 ( =pr)k = =pr. Note that this computation requires knowledge* * of jR(y) mod pr. Now d(y) will be divisible by vs1, hence __ r k d(y) = vs1z in( =p ) __ 1,k-2s(p-1) * * __k-2s(p-1) with z 2 ( =pr)k-2s(p-1)= C (A=pr) representing ffi(x). Lift z to some w* * 2 . To proceed, we use diagram (6): w lies in the last but one group of the top row, h* *ence we compute (w) 2 H2(Q.) which is our representative for ffi0(ffi(x)) 2 Ext2. This require* *s to compute OE-1. Ob- serve for example that OE-1(t1) = 1_v1-v1_1p. 2.3 Elliptic homology theories and divided congruences: the f-invariant We refer the reader to [L1] or [HBJ ] for the notion of elliptic homology with * *respect to the congruence subgroup 1(N). In this section, E denotes the spectrum associated to the homol* *ogy theory with coefficient ring E* = M*(Z(p), 1(N)), see section 3.2 for the notation. Finall* *y, ff : BP -! E denotes the orientation. By the naturality of the constructions in section 2.2 we have a commutative dia* *gram for any k > 0 " ff 2,k Ext2,k[BPO]_____________//"E`xt[E]" ` | | (7)| (7')| fflffl| fflffl| _____(BPQ_BPQ)(k)____ff_ff//__(EQ_EQ)(k)__ BPkBP+(BPQ Q+Q BPQ)(k) EkE+(EQ Q+Q EQ)(k). The injectivity of ff holds for any Landweber exact theory E of height at least* * two, [L2, Proof of 4.3.2]. To proceed, however, we will use a more subtle property of E, namely* * the topological q-expansion principle. We put Xk Mk D__k:= {f = fi2 EQ,2i|there areg0 2 Q, gk 2 EQ,2ksuch that(f + g0+ gk)(q* *) 2 Z(p)[[q]]} i=0 i=0 where f(q) denotes the q-expansion of f at the cusp infinity, and for = 1(N)* * we set Z := Z[_1_N, iN ] if N > 1 and ZSL2(Z):= Z[1_6] as in [L1]. We then define (EQ EQ)(2k) X X 0 (8) '2 : __________________________-! D__ Q=Z , fi gj 7! -q (fi)gj, E2kE + (EQ Q + Q EQ)(2k) k i+j=k i+j=k where q0(f) is the constant term of the q-expansion of f at the cusp infinity. * *The composition '2O ' is injective [L1, Proposition 3.9] and hence so is the f-invariant f : Ext2,2k[BP ] ,! D__k Q=Z . 7 Jens Hornbostel and Niko Naumann We remark that our grading of E* is the topological one, i.e. elements of dimen* *sion 2k correspond to modular forms of weight k. G. Laures describes Ext2using the canonical Adams resolution [R , Definition 2.* *2.10] instead of the cobar resolution. Proposition 4. For k > 0 even, the above definition of the f-invariant Ext2,k-!* * D__k=2 Q=Z coincides with the one given in [L1]. (k) Proof.We have to show that the map Ext2,k,! _____(BPQ_BPQ)_______BPkBP+(BPQwQ+Q* *eBPQ)(k)constructed in section 2.2,(7) coincides with the one of [L1]. We know [R , Lemma A.1.2.9 (b)] that, u* *p to chain homo- topy, there is a unique map from the unreduced cobar resolution A A.to the* * canonical Adams resolution BP*(BP ^ B~P^.) where B~P! S0 j!BP !d B~Pis an exact triangle in th* *e stable homo- * * ss*(idBP^2^d^n) topy category, see also [Br, Lemma 3.7] Now one checks that A(n+1)~=ss*(BP ^n* *+2) ! ss*(BP ^2^ ( B~P)^n) is a map of chain complexes where the isomorphism follows * *from [R , Lemma 2.2.7] and induction. So the claim reduces to the fact that the triangle ss*(idBP^2^d) Z2,k( A.)_____________________________//Z2,k(ss*(BP ^2^ ( B~P)^.)) RR iii RRRR iiiiii RRRR iiiii RR))R ttiii _____(BPQ_BPQ)(k)____ BPkBP+(BPQ Q+Q BPQ)(k) commutes. This follows using that our maps o, H2 and ss*(d ^ d) correspond to r* *, ae and the isomor- phism D1= "1~=G2 = ss*(( B~P)^2) Q in [L1, section 3.1] (where we define o by* * mapping_all vito zero). * * |__| Note the degree shift and the factor 2 for the f-invariant sss2k-! D__k+1 Q=Z * *(both are missing in [L1, p. 411]). 3. Arithmetic computations In section 3.1, we review results of N. Katz on divided congruences and establi* *sh a relation between BP -theory and the mod p Igusa tower (Theorem 5). In section 3.2 we give explic* *it computations for elliptic homology of level 3 and the corresponding divided congruences. 3.1 Divided congruences We review parts 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 policy in al* *gebraic topology we wish to consider 1(N)-structures instead, which are representable only for N >* * 5. More seriously, one has to check that the relevant part of [K1 ] works for this different modul* *i problem. This we did, but we will not explain the details here and only remark that both the geo* *metric irreducibility of the moduli spaces and the irreducibility of the Igusa tower are valid for 1* *(N). Furthermore, we will use these results for p > 5 and N = 1 and for p = 2 and N = 3. These cases* * can be handled by using auxiliary rigid level structures and taking invariants under suitable fin* *ite groups_as in [K1 ]. Fix a level N > 5, a prime p not dividing N and a primitive N-th root of unity * *i 2 Fp. We 8 Beta-elements and divided congruences put k := Fp(i), W := W (k) (Witt vectors) and also denote by i 2 W the Teichm"u* *ller lift of i. Finally, K denotes the field of fractions of W and for any Z(p)-algebra R we de* *note by Mk(R, 1(N)) the R-module of holomorphic modular forms for 1(N) of weight k and defined ove* *r R , see e.g. [K3 ] or [L1]. The 1(N) is omitted from the notation if it is clear from the c* *ontext. We fix a lift Ep-12 Mp-1(W ) of the Hasse invariant. The existence of such a lift puts furthe* *r restrictions on p and N which are satisfied in our applications. We define the ring of divided congruences D by M*(W ) D := {f 2 M*(K)|f(q) 2 W [[q]]} M*(K), where f(q) is the q-expansion of f at the cusp infinity. For n > 0 we also defi* *ne _ n ! M Dn := D \ Mi(K) D and i=0 Mn D__n:= Dn + K + Mn(K) Mi(K) i=0 which is consistent with the definition of the previous section. The group D__k* *considered in [L1] differs from the D__kabove because the ring of holomorphic modular forms has be* *en localised in [L1]. This difference is not serious because the f-invariant factors through holomorp* *hic modular forms [L1, Proposition 3.13]. The ring D carries a uniformly continuous Z*p-action (the diamond operators) de* *fined by X X [ff]( fi) := ffifi, i i where ff 2 Z*pand fi2 Mi(K). We put 0 := Z*pand n := 1+pnZp for n > 1 and V1,* *n:= (D=pD) n for n > 0. Then V1,0 V1,1 . . .D=pD is an ind-'etale Z*p-Galois extension, the mod p Igusa tower. So, V1,0 V1,1is * *a (Z=p)*-Galois extension and for all n > 2 V1,n-1 V1,nis an 'etale Z=p-extension and hence an* * Artin-Schreier ex- tension. An immediate computation with diamond operators shows that the composi* *tion M*(W ) ! D ! D=pD factors through V1,1 D=pD. It is a result of P. Swinnerton-Dyer that * *this induces an isomorphism (9) M*(W )=(p, Ep-1- 1) -'!V1,1, see [K1 , Corollary 2.2.8]. By Artin-Schreier theory, given n > 2 and x 2 D=pD satisfying [ff](x) = x for a* *ll ff 2 n and [1 + pn-1](x) = x + 1 one has V1,n= V1,n-1[x] and the minimal polynomial of x o* *ver V1,n-1is T p- T - a for some a 2 V1,n-1. 9 Jens Hornbostel and Niko Naumann At this point we can establish a first relation between BP-theory and divided c* *ongruences. Consider the ring extensions M*(W ) D W [[q]]. We have a formal group F over M*(W ) induced by the universal elliptic curve. T* *he base change of F to W [[q]] is the formal completion of a Tate elliptic curve and is thus isom* *orphic to dGm. Implicit in [K1 ] is the fact that D is the minimal extension of M*(W ) over which F bec* *omes isomorphic to dGm, i.e. D is obtained from M*(W ) by adjoining the coefficients of an isom* *orphism F ' dGm defined over W [[q]]. This is what underlies N. Katz' construction [K1 , Sectio* *n 5] of a sequence of elements dn 2 D which modulo p constitute a sequence of Artin-Schreier generato* *rs for the mod p Igusa tower. Since the elements tn 2 BP *BP are the coefficients of the universal isomorphis* *m of a p-typical formal group law, one may expect a relation between the tn and the dn. To formu* *late this, denote by ff : BP* -! M*(W ) the classifying map of F and consider the composition (3) 2ff ff 2 -q0 id OE : BP*BP BP*BP Q ' BP Q -! M*(K) -! M*(K). Note that the map '2 in (8) composed with the orientation ff is a quotient of O* *E. The topological_q-expansion principle guarantees that Tn := OE(tn) 2 D for n > * *1 and we can thus define Tn := (Tn mod pD) 2 D=pD. __ __ __ * * __ Theorem_5. For any n > 1 we have [1 + pk](T n) = Tn for k > n and [1 + pn](T n)* * = Tn+ 1. Hence Tn is an Artin-Schreier generator for the extension V1,n V1,n+1. P Proof.Let ! = ( n>1antn-1)dt be the expansion along infinity of a normalised (* *i.e. a1 = 1) invariant differentialPon the universal elliptic curve. Then an 2 Mn-1(W ) and * *the logarithm of the p-typification is n>0apn_pntpn 2 M*(K)[[t]], i.e. the classifying map ff * *: BP* -! M*(W ), when tensored with Q, sends ln 2 BP Q,2(pn-1)to apn_pn2 Mpn-1(K), see [R , Theorem A* *.2.1.27] for the definition of the ln (=~n in the notation of loc. cit.). Defining d0 := 1 and dn (n > 1) recursively by Xn dpi a n (10) _n-i_i= _p_n, i=0 p p _ N. Katz shows in [K1 , Corollary 5.7] that the dn:= (dn mod p) 2 D=pD behave un* *der the diamond __ P n pi operators as claimed for the Tn. In BP *BP Q we have jR(ln) = i=0litn-i and * *we apply OE to this relation to obtain apn_ Xn q0(api) pi = ______Tn-i pn i=0 pi which, using (10), implies 10 Beta-elements and divided congruences Xn dpi Xn q0(a i) i (11) _n-i_i= ____p_iTnp-i. i=0 p i=0 p We now proceed by induction on n > 1. For n = 1 we have d1 + 1=p = q0(a1)T1 + q* *0(ap)=p. Also, q0(a1) = 1 since a1 = 1 and q0(ap) 2 1 + pW because_ap_reduces mod p to the Has* *se invariant which has q-expansion equal to 1. We obtain T1 = d1+ ff for some ff 2 k. As ff * *is invariant under all diamond operators, our claim for n = 1 is obvious. Assume that n > 2. From (11) and a1 = 1 we obtain Xn dpi Xn q0(a i) i Tn = _n-i_i- ____p_iTnp-i. i=0 p i=1 p For k > n we know that the terms involving di are invariant mod p under [1 + pk* *] whereas the remaining terms are likewise invariant by the induction hypothesis. Finally, we have Xn dpi Xn q0(a i) * * i [1 + pn]Tn = [1 + pn]dn + [1 + pn]( _n-i_i) - [1 + pn]( ____p_iT* *np-i). i=1 p i=1 p Here we have_[1 +_pn]dn dn + 1 (pD) and the remaining terms are invariant. Th* *us, indeed,_ [1 + pn](T n) = Tn+ 1. * * |__| 3.2 Modular forms For a prime p > 5, the following is well known [L1, Appendix]: M*(Z(p), 1(1)) = Z(p)[E4, E6], where E4 and E6 are the Eisenstein series of level one of the indicated weight.* * For the discriminant , the ring of meromorphic modular forms is given by Z(p)[E4, E6, -1] and the * *usual orientation BP*- ! Z(p)[E4, E6, -1] is Landweber exact of height 2 and factors through Z(p)[E4, E6]. A similar resu* *lt holds for p > 3 and M*(Z(p), 1(2)) = Z(p)[ffi, ffl]. The purpose of this section is to give analogous results for 1(3) and p = 2, c* *.f. [St] for related results. Consider the elliptic curve E : y2 + a1xy + a3y = x3 defined over R := Z[1=3][a1, a3, -1] where = a33(a31- 27a3) is the discrimin* *ant of the given Weierstrass equation. Note that, unlike in level one, is not irreducible as a* * polynomial in a1 and 11 Jens Hornbostel and Niko Naumann a3 and we put f := a3, g := a31- 27a3, hence = f3g. The section P := (0, 0) 2 E(R) is of exact order 3 in every geometric fibre as * *follows from [Si1, III,2.3] and ! := dx=(2y + a1x + a3) is an invariant differential on E. The following may be compared with [St, Lemma 11]: Proposition 6. The above tuple (E=R, !, P ) is the universal example of an elli* *ptic curve over a Z[1=3]-scheme together with a point of order 3 and a non-zero invariant differe* *ntial. Proof.We have to show that whenever T is a Z[1=3]-scheme and E0=T is an ellipti* *c curve with non-zero invariant differential !0and P 02 E0(T ) of exact order 3, there is a * *unique map OE : T -! Spec(R) such that OE*(E, P, !) = (E0, P 0, !0). We show the uniqueness of OE fi* *rst. This amounts to seeing that the only change of coordinates x = u2x0+ r , y = u3y0+ u2sx0+ t with r, s, t 2 R and u 2 R* (see [Si1, III Table 1.2]) preserving (E, P, !) is * *the identity, i.e. r = s = t = 0 and u = 1. From x0(P ) = y0(P ) = 0 we obtain r = t = 0. Next, a4 = a04implies -sa3 = 0, h* *ence s = 0 because = f3g and thus also f = a3 is a unit in R. Finally, !0= u! forces u = 1. Given the uniqueness of OE in general, its existence is a local problem on T an* *d we can assume that T = Spec(S) is affine and E0=T is given by a Weierstrass equation with coeffici* *ents a0i2 S. Moving P 0to (0, 0) gives a06= 0. We claim that a032 S*: This can be checked on geomet* *ric fibres where it follows from [Si1, III,2.3] and the fact that (0, 0) has order 3 (if a03vani* *shed on some geometric fibre the point (0, 0) would have order 2 in that fibre). Using this, one finds* * a transformation such that (dy)P0 = 0 in E0=T,P0, hence a02= a04= 0. We thus have some _ : T -! Spec* *(R) such that _*(E, P ) = (E0, P 0) and _*(!) = u!0for some u 2 S*. Adjusting _ using u, i.e.* * multiplying_the a0i by u-i, we obtain the desired OE. * * |__| We conclude that the ring of meromorphic modular forms is given as Mmer*(Z(2), 1(3)) = Z(2)[a1, a3, -1] with aiof weight i, and likewise for any other prime different from 3 in place * *of 2. As usual, t = -x=y is a local parameter at infinity for E=R which is normalised* * for ! and hence determines a 2-typical formal group law over Mmer*(Z(2), 1(3)). Using [Si1, p.* * 113] one checks that the corresponding classifying map ff : BP* -! Mmer*(Z(2), 1(3)) satisfies ff(v1) = a1and ff(v2) = a3for the Hazewinkel generators vi. Thus ff m* *akes Mmer*(Z(2), 1(3)) a Landweber exact BP algebra of height 2. Using the orders of f and g at the two cusps 0 and 1 of X1(3), one can check th* *at the ring of holomorphic modular forms is given by (12) M*(Z(2), 1(3)) = Z(2)[a1, a3]. 12 Beta-elements and divided congruences We stick to the notations of section 3.1 for p = 2 and N = 3. For example, i de* *notes a primitive cube- root of unity and W = W (F2(i)) = W (F4) = Z2[i] is the unique unramified quadr* *atic extension of Z2. To study divided congruences we will need to know the q-expansions of a1 and a3* *. Given a Dirichlet character O, we consider it as a function on Z as usual and define for k > 0 an* *d n > 1 X oeOk(n) := O(d)dk. 16d|n In the following, O will always denote the unique non-trivial character mod 3 O : (Z=3Z)* -! C*. Proposition 7. The q-expansions of a1 and a3 at the cusp infinity are given as * *follows. X O a1(q) = (1 + 2i)(1 + 6 oe0(n)qn) and n>1 1 X O n a3(q) = (1 + 2i)(-_ + oe2(n)q ) inW [[q]]. 9 n>1 Proof.From (12) we know that rkM1(Z(2)) = 1 and rkM3(Z(2)) = 2. Using [K2 , sec* *tion 2.1.1] we see that X O (13) 6G1,O(q) = 1 + 6 oe0(n)qn 2 M1(Z(2)) and n>1 1 X O n G3,O(q) = -_ + oe2(n)q 2 M3(Z(2)). 9 n>1 We have evaluated L(0, O) = 1=3 and L(-2, O) = -2=9 using [Ne, Theorem VII.2.9]* * and [Wa , formula following Proposition 4.1 and Exercise 4.2(b)]. It is easy to see that G3,O(0) = 0, i.e. G3,Ovanishes at the cusp 0. Below, we * *explain how to compute the following values of a1 and a3 at the cusps zero and infinity. (14) a1(1) = 1 + 2i 1 (15) a3(1) = -_ (1 + 2i) 9 (16) a3(0) = 0. Using these values and the dimensions of the spaces of modular forms of weight * *1 and 3, we conclude that a1 = 6(1 + 2i)G1,Oand a3 = (1 + 2i)G3,O, hence that a1 and a3 have desired* * q-expansions by (13). We are using the fact that the map M3(Z(2)) C = M3(C) -! C2, f 7! (f(1)* *, f(0)) is an isomorphism, as follows from the theory of Eisenstein series. To establish (14) and (15) one has to evaluate a1 and a3 at the tuple (T (q), !* *can, P ) consist- ing of the Tate curve T (q)=Z((q)), its canonical invariant differential !can a* *nd a specific sec- tion P 2 T (q)(Z[i]((q)))[3]. To do so, one may use J. Tate's uniformisation [* *Si2, p. 426] to 13 Jens Hornbostel and Niko Naumann write T (q)=(Z[[q]]=(q3)) in Weierstrass form, the point P having coordinates (* *X(q, i), Y (q, i)). One then uses Weierstrass transformations to bring (T (q), !can, P )=Z[[q]]=(q3) to* * the standard form of Proposition 6. The coefficients a1 and a3 of the Weierstrass equation thus obta* *ined are by defi- nition a1(1) and a3(1). The computation for (16) is similar, the point P has to* * be replaced_by Q = (X(q, q1=3), Y (q, q1=3)). * * |__| Remark 8. In E. Hecke's notation [He], we have a1 = 9i_ssG1(o, 0, 1, 3) and a3 * *= 27i_4ss3G3(o, 0, 1, 3). Note that a1(q) 1 mod 2, hence a1 2 M1(Z(2), 1(3)) is a lift of the Hasse in* *variant for p = 2. From section 3.1 we know that (9) V1,0= V1,1' M*(W, 1(3))=(2, a1- 1) = k[a3] 0(a1)-a1 __ (k := W=2W = F4) and that, for T := q______22 D, T := (T mod 2) 2 D=2D is an Ar* *tin-Schreier __2 __ generator for V1,1 V1,2, in particular T + T 2 V1,1' k[a3] and for later use * *we will need the following more precise result. __2 __ Proposition 9. T + T = 1 + a3. Proof.Recall that the q-expansionPmap V1,1 D=pDP,! k[[q]] is injective [K1 , (* *1.4.6) for m = 1]. In k[[q]] we have T = n>1oeO0(n)qn and T 2= n>1oeO0(n)q2n, hence X O O T 2+ T = (oe0(n=2) + oe0(n))qn, n>1 where we understand that oeO0(n=2) = 0 for n odd. To complete the proof, one ne* *eds to check that for all n > 1 one has oeO0(n=2) + oeO0(n) oeO2(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 compute the f-invariants of two infinite families of fi-ele* *ments including the Kervaire elements fi2n,2nand explain the relation of our results with the Kerva* *ire invariant one problem. 4.1 f(fit) for t not divisible by p Fix a prime p and the level N as N = 1 for p > 5, N = 5 for p = 3 and N = 3 for* * p = 2. We keep the notations of section 3.1 for this choice2of p and N. Given an integer t > 1 not* * divisible by p, recall that fit= ffi0ffi(vt2) 2 Ext2,2t(p -1)-2(p-1)[BP ] has its f-invariant in D__n * * Q=Z, n := t(p2- 1) - (p - 1). When trying to express f(fit) in terms of divided congruences, we encounter wha* *t is in fact the major obstacle at the moment for using the arithmetic of divided congruences in* * homotopy theory: The group D__n Q=Z is not directly related to D. Instead, we have D__n= D+K +Mn* *(K) by definition and there is a canonical surjection 14 Beta-elements and divided congruences _ n ! _ ! M Mn ss : Dn Q=Z ' Mi(K) =Dn -! D__n Q=Z ' Mi(K) =D__n i=0 i=0 which is split because its kernel is divisible, hence W -injective. In particul* *ar, ss remains surjective when restricted to p-torsion (17) ss : Dn Q=Z[p] -! D__n Q=Z[p], note that f(fit) 2 D__n Q=Z[p]. The group Dn Q=Z[p] is related to the ring of* * divided congruences as follows: (18) _ : Dn Q=Z[p] -'!Dn=pDn ,! D=pD, where the first arrow is multiplication by p and the injectivity of the last ma* *p is immediate. What we will do is to compute some element in D=pD in the image of _ which under ss * *projects to f(fit). At the low risk of confusion we will continue to label such an element, which i* *s in general not unique, as f(fit). Recall that we have fixed an elliptic orientation ff : BP* -* *! M*(W ) and denote 0(ff(v1)) T := ff(v1)-q___p2 D=pD. We also put b := ((q0(ff(v2)) mod p) 2 k. Theorem 10. For an integer t > 1 not divisible by the fixed prime p, we have f(fit) = bt- (T p- T + b)t2 V1,0 D=pD. Proof.Note first that from section 3.1 we know that T is an Artin-Schreier gene* *rator for V1,1 V1,2, hence T p- T 2 V1,1. A short computation with diamond operators, which we leave* * to the reader, shows that in fact T p- T 2 V1,0, hence also bt- (T p- T + b)t2 V1,0. We introduce a := q0(OE(v1)) and compute as explained at the end of section 2.2* * using the notations introduced there. From jRv2 v2+ v1tp1- vp1t1 mod p we obtain Xt `t' jRvt2 vt2+ vt-i2vi1ti1(tp-11- vp-11)imod p, i=1 i hence Xt `t' w = vt-i2vi-11ti1(tp-11- vp-11)i i=1 i and ` ' ` ' i 1X t t t-i i-1 1 v1- v1 1 (w) = _ (v2 v1 1) _____________ . p i=1 i p _` ' !i 1___v1-_v1__1_p-1 p-1 (BP Q BPQ)(2n) - v1 1 2 _______________________________. p BP 2nBP + (BP Q Q + Q BPQ)(2n) As in section 2.3 we apply '2O (ff ff) to this expression to obtain, denoting* * ff(v1) 2 Mp-1(W ) as v1 for simplicity, 15 Jens Hornbostel and Niko Naumann ` ' ` 'i_` ' p-1 !i 1 Xt t t-ii-1 v1- a v1- a p-1 -_ b a _____ _____ - a = p i=1 i p p " _ ` ' _` ' ! !t# 1 t v1- a v1- a p-1 p-1 -__ -b + a _____ _____ - a + b = pa p p __ ` ' _` ' ! !t ! n -1_ v1- a v1- a p-1 p-1 t M a _____ _____ - a + b - b 2 Mi(K). pa p p i=0 This is a representative for f(fit) in Dn Q=Z[p] to which we have to apply th* *e map _ from (18) to obtain an element in D=pD. For this, note that a 1 mod p because v1 reduce* *s to the Hasse invariant mod p. This allows us to put a = 1 in the above expression (but not t* *o replace v1-a_pby v1-1_ 2 p ; this would require the congruence a 1 mod p , which does not hold in ge* *neral). We then obtain indeed -((T (T p-1- 1) + b)t- bt) = bt- (T p- T + b)t. * * __ * *|__| Remark 11. Assume that p > 5 in the situation of Theorem 10. In general, the el* *liptic orientation will not map v1 to the Eisenstein series Ep-1 of weight p - 1 and level one. Bu* *t ff(v1) and Ep-1 can only differ by a modular form divisible by p and we may thus change the ori* *entation to force ff(v1) = Ep-1. Assuming this, we see that f(fi1,1,1) = Ep-1-1_p2- 1_p(Ep-1-1_p)* *p, as first computed by G. Laures [L1, p. 414] (where the second summand is missing). Remark 12. The injectivity of the f-invariant together with the known structure* * of Ext2[BP ] pro- vides some non-trivial information about the arithmetic of divided congruences * *as follows. Fix some x 2 Ext2,k[BP ] of order pr. Then f(x) 2 D__k Q=Z will be of order pr, hence a* * representative of f(x) in Dk Q=Z will be of order ps for some s > r. Thus the f-invariant relat* *es the order of a fi-element to the (non-)existence of a certain divided congruence. Let us assume that r = 1 as is the case for all fit,s,rconsidered in this artic* *le. Then our results show that our representatives in D Q=Z have order p and the non-trivial addit* *ional information on divided congruences is then that they do not lie in the kernel of D Q=Z -!* * D__ Q=Z. To give an example, assume that we are in the situation of Remark 11. The arith* *metic of dividedLcon- gruences shows that F := Ep-1-1_p2- 1_p(Ep-1-1_p)p 2 Dp(p-1) Q=Z is of order p,* * i.e. F 2 p(p-1)i=0Mi(K) has a q-expansion with denominator exactly p. The additional information is the* *n that for any ff 2 K and f 2 Mp(p-1)(K) the q-expansion of F + ff + f will still have exact denomina* *tor p. Example 13. Fix p = 5 and set g2 := _1_12E4 and g3 := -1_216E6 as in [K1 ]. The* * comparison of the logarithm of the universal p-typical formal group law [R ] and the correspondin* *g coefficients of the logarithm of the elliptic curve (E, !) [K1 , (5.0.3)] (p-typification does not * *change these coefficients) ap2-ap+1p shows that the orientation a maps v1 to ap and v2 to _______p, the ai denoting * *the normalised * * 6 (multiplied with -1=2) aiof [K1 , p. 351]. One deduces that v1 maps to -8g2 and* * v2 maps to a25-a5_5. A computation with Maple shows that a25= 129761280g32g23+ 32440320g43+ 3784704g* *62(and also that the correct value for the unnormalised a11 is -2520g2g3 and not -512g2g3).* * It follows that q0(v1) = -2_3and q0(v2) = -4900_310, so Theorem 10 may be rephrased in terms of* * the Eisenstein series g2 and g3. 16 Beta-elements and divided congruences 4.2 Projecting to the Kervaire element In this section, we compute f(fis2n,2n) for n > 0 and s > 1 odd at the prime p * *= 2. Using this, we are able to determine a single coefficient in the f-invariant of a (U, fr)2- ma* *nifold of dimension 2n the non-vanishing of which is necessary and sufficient for the corner of X to b* *e 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 cert* *ain fi-elements, due to W. Browder [B ]. Fix some n > 3. We have a homomorphism K : sss2n-2-! Z=2 which sends the class of a stably framed manifold to its Kervaire invariant. Co* *nsider on the other hand the composition n 2,2n 2 K0: sss2n-2-! sss2n-2[2] -! E2,21[HZ=2] ,! E2 [HZ=2] = Z=2 . hn-1. Here, the first map is the projection to the 2-primary part, the second is the * *projection onto F 2=F 3 in the (classical) Adams spectral sequence at p = 2, the third is an edge homom* *orphism and the final equality is due to J. Adams, [R , 3.4.1, c)]. Proposition 14. K = K0. Proof.For any y 2 sss2n-2[2] we have K(y) = 1 if and only if y has Adams filtra* *tion 2. This_is_ implicit in [B ], c.f. [BJM2 , p. 144]. * * |__| We can easily obtain a similar homotopy theoretic description of K using BP ins* *tead of HZ=2. n Proposition 15. Let n > 2. Then Ext2,2is a direct sum of cyclic groups of order* * 2. It is generated by the element ff1.ff2n-1-1and the elements fis2i,2iwith s odd and i > 0 such t* *hat (3s-1)2i+1= 2n and the case (s, i) = (1, 0) has to be omitted. Proof.This follows from [R , Corollary 5.4.5]. Observe that the __fftin loc. ci* *t equals fft_as t is odd, see [Sh, Theorem 1.5] or [R , Theorem 5.2.6]. * * |__| n Remark 16. The Lemma shows that the number of generators of Ext2,2is [n=2] + 1 * *for n > 3. The low dimensional cases are as follows. Ext2,4: ff21 Ext2,8: ff1ff3, fi2,2 Ext2,16: ff1ff7, fi4,4, fi3,1 Ext2,32: ff1ff15, fi8,8, fi6,2 Ext2,64: ff1ff31, fi16,16, fi12,4, fi11,1 Ext2,128: ff1ff63, fi32,32, fi24,8, fi22,2 Ext2,256: ff1ff127, fi64,64, fi48,16, fi44,4, fi43,1. 17 Jens Hornbostel and Niko Naumann Now we consider the composition n 2,2n K00: sss2n-2-! sss2n-2[2] -! E2,21[BP ] ,! E2 [BP ] -! Z=2 . fi2n-2* *,2n-2 which is defined in analogy with K0, the final map being the projection to the * *Z=2-summand generated by fi2n-2,2n-2. Proposition 17. K00= K. Proof.We have the Thom reduction : Ext*[BP ] -! Ext*[HZ=2] which satisfies (* *fi2n-2,2n-2) = n h2n-1and is zero on all other generators of Ext2,2[BP ], see [R , 5.4.6, a)] an* *d Proposition_15. The result then follows from Proposition 14. * * |__| Let X be a (U, fr)2-manifold of dimension 2n. From the above, we see that the c* *orner of X is a Kervaire manifold if and only if the f-invariant of X contains fi2n,2nas a su* *mmand. Thus, one certainly wants a more geometric description of the f-invariant (or just its pr* *ojection to fi2n-2,2n-2). In principle, it is possible to obtain such a description in terms of Chern num* *bers of X, simply because they determine the (U, fr)2-bordism class of X [L2], but the necessary * *computations become quite complicated already in low dimensions. At the end of this section, we wil* *l explain how divided congruences might simplify such computations. 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 decom* *position of its stable tangent bundle T X = T X(0) T X(1)and we have Chern classes c(j)i2 H2i(X, Z) a* *ccordingly (i > 0, j = 0, 1). Theorem 18. a) Let X be a (U, fr)2-manifold of dimension 4 and put q :=< c(0)1c* *(1)1, [X] >2 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 put q :=< c(0)1(c(1)31+ c(1)* *1c(1)2+ c(1)3) + (c(0)2+ c(0)21)(c(1)2+ c(1)21), [X] >2 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. P i Proof.If i>0lix2 is the logarithm of the universal 2-typical formal group law* *, then (see [L2, Example 4.2.4]) exp(x) = x - l1x2+ 2l21x3- (5l31+ l2)x4 (mod x5) and thus x 2 2 3 3 4 Q(x) := ______= 1 + l1x - l1x + (2l1 + l2)x (mod x ). exp(x) Q For indeterminates xi of dimension 2 we set := iQ(xi). Denoting by ci the i* *-th elementary symmetric function in the xione gets (using the definition of the Hazewinkel ge* *nerators) v1 (2)= __c1 2 v21 2 (4)= __(3c2- c1) and 4 v31 3 v2 3 (6)= __(4c1- 13c1c2+ 16c3) + __(c1- 3c1c2+ 3c3). 8 2 18 Beta-elements and divided congruences 2,4 To prove part a) one has that BPQ =(BP Q Q + Q BPQ) is a one-dimensional Q* *-vector space generated by v1 v1. Moreover, one checks that the image of BP 4BP is generate* *d (over Z(2)) by v1_v1_ v1_v1_ 2 2 and that 4 is a representative of ff1. To see the latter, observe that * *ff1 is represented by t1, hence [R , A.1.2.15] ff21is represented in the cobar complex by t1 t1 = (1* * t1)(t1 1), and then one applies the description of ffi-1 given in section 2.2. Using the notations * *introduced in [L2], one computes that v1 v1 2,4 (4) KBP<2>(T X)(4)= c(0)1c(1)1____ inBP Q =(BP Q Q + Q BPQ) , 4 hence the image of the corner of X in Ext2,4is represented by q_2. v1_v1_2= q .* * ff21. The final assertion follows because the only non-trivial element of sss2has Adams-Novikov filtratio* *n precisely 2. For part b), we know that Ext2,8is generated by ff1__ff3and fi2,2. As fi2,2is a* * permanent cycle in the ANSS whereas ff1__ff3is not we know that the image of X in Ext2,8is a multi* *ple of fi2,2. One computes that in the notation of section 2.2 ffi(v22) is represented by z = t41* *+v21t21in C1(A=2). Hence fi2,2= ffi0ffi(v22) is represented in BP Q2,8=(BP Q Q + Q BPQ)(8)by -1_8(v1 * * v31) + 5_16(v21 v21) - 3_ 3 3 2 2 3 4 8(v1 v1). Computing enough of the image of v1t1, v1t1, v1t12and2t1 under BP8BP * *,! BP8BP Q ' BP Q2,8-! BP Q2,8=(BP Q Q + Q BPQ )(8), one sees that v1_v1_82 im (BP 8BP) * *and fi2,2is 2 v2 represented by v1__1_16. We provide the following argument for general n, for t* *he proof here wenneed the casenn = 3. Observe that by the computations of the previous sections, the * *image of Ext2,2in BP Q2,2=(BP Q,2n Q+Q BP Q,2n+BP 2nBP) is given by representatives consisting of* * summands of the form vi1vk2 vj1with rational coefficients. Moreover, these elements map und* *er the isomorphism OE of section 2.2 to polynomials in v1, v2 and t1. In other words, the f-invariant* * in bidegree (2, 2n) factors through the subgroup generated by elements vi1vk2 vj1modulo elements of the fo* *rm OE-1(vi1vj2tk1), 1 vj1and vi1vj2 1. Denote this quotient by B2n. A computation using the resu* *lts of the previous section shows that __c_22n-1v2n-21 v2n-21is not zero in B2n for c an odd integ* *er. More precisely, no elementnin the relations defining the quotient B2ncontains such a summand. Amon* *g the elements in Ext2,2, the image of fi2n-2,2n-2in B2ncontains such a summand and the other gen* *erators exhibited n in Proposition 15 do not. Thus we have a well-defined map Ext2,2! Z=2 given by * *mapping an element to 1 if it admits a representative in B2n having a summand __c_22n-1v2n* *-21 v2n-21with c odd. This map is a projection to fi2n-2,2n-2. We would like to consider the summands* * of KBP<2>(T X)(8) (2) (6)+ (4) (4)+ (6) (2)mod (BP Q Q + Q BPQ)(8) as elements in B23. Once this is achieved, we have to consider only the summand* *s involving v21 v21. The only summand which is not already given by a representative in B23is c(0)1(* *c(1)31- 3c(1)1c(1)2+ 3c(1)3)v1_v2_4in (2) (6). One computes (for p = 2 and the Hazewinkel generat* *ors as before) that 3 v1 v2 v2 v1 v3 1 * * v v v2 v2 OE-1(t2) = 1_v2_2- v2_1_1+ 1_v1_4- ____18+ _1__4-3_1__8. Hence we have OE-1(t1t* *2) = -_1__24- _1__1_16+..., 2 v2 * * (0) (1)3 so we have to look at the coefficients of v1_v2_4and v1__1_16which by the above* * equal c1 (c1 - 3c(1)1c(1)2+ 3c(1)3) and (3c(0)2- c(0)21)(3c(1)2- c(1)21). We also have that c(* *0)1(c(1)31- 3c(1)1c(1)2+ 3c(1)3) equals c(0)1(c(1)31+ c(1)1c(1)2+ c(1)3) and (3c(0)2- c(0)21)(3c(1)2- c(1)21) eq* *uals (c(0)2+ c(0)21)(c(1)2+_c(1)21) modulo 2 when evaluated on [X]. Now the assertion follows as in part a). * * |__| Of course, it is possible to do similar but more complicated computations for 2* *n-dimensional (U, fr)2-manifolds in case n > 4. We always have a projection to Z=2 looking at* * the power of 2 in the denominator of the coefficient of the summand v2n-21 v2n-21. The comp* *utation then re- duces to compute those (2i)which contribute to this summand. The diligent read* *er may thus 19 Jens Hornbostel and Niko Naumann prove statements of the following form: The element h2n-1survives (equivalently* *: there is a framed manifold in dimension 2n - 2 having Kervaire invariant 1) if and only if < Fn(c* *(0)i, c(1)j), [X] > is odd for a certain explicit polynomial Fn. The main problem in the computation o* *f Fn is to find representatives in B2n (that is sums of vi1vk2 vj1) for elements arising in th* *e (2n-i) (i). For n = 3 this was done using t1t2. In the case n > 3, it will be necessary to find* * suitable elements in BP 2nBP which will involve tifor larger i and the computation of OE-1 of these * *elements. The rest of this section is devoted to the computation of the f-invariant in di* *mensionn2n+at1the prime 2. More generally, we compute the f-invariant of fis2n,2n2 Ext2,(3s-1)2fo* *r all n > 0 and s > 1 odd. We use the notations of section 3.1 for p = 2 and N = 3 and those of section 3.* *2 and write T := a1-1_22 D=2D which is an Artin-Schreier generator for the extension V1,0= * *V1,1' k[a3] V1,2. Theorem 19. The image of the f-invariant in V1,2is given by f(ff1ff2n+1-1) = T forn > 0, f(fis) = 1 + as3fors > 3 odd , f(fis2,2) = 1 + a2s3fors > 1 odd and n 3.2n-2s f(fis2n,2n) = (a23+ a3 ) fors > 1 odd andn > 2. Proof.For the first line, recall that mod 2 we have fft:= fft,1:= ffi(vt1). One* * computes that in the cobar complex ff1fftis represented by t1 1_2[(2t1+ v1)t- vt1] , use the descri* *ption of the product in the cobar complex of [R , A.1.2.15]. Using that OE is a ring isomorphism and th* *e description of ffi-1, see section 2.2, one further computes that ff1fft is represented by -1_4v1 vt* *1in the usual quotient of BPQ2, c.f. (7). The second line is a special case of Theorem 10 (use Proposition 9) and implies* * the third line because x1 x20= v22mod v21, recall the invariant sequences (2, v2n1, xn) from section* * 2.1. The only case requiring a longer computation is f(fi4,4): In the notation of section 2.1 we have r = 1, s = t = 4 and x2 = v42- v31v322 H0,24(BP =(2, v41)). This value of x2 follows from the definition in [MRW , p. 476] or [R , Theorem* * 5.2.13])_after cancelling all possible multiples of v41. One computes that, in the notation of section 2.* *2, z 2 ( =2)16is given as z = t81+ v41t41+ v22t1(t1+ v1) + v1v2t21(t21+ v21) + v21t31(t1+ * *v1)3 = v1v22t1+ v22t21+ v31v2t21+ +v51t31+ v1v2t41+ v31t51+ v21t61+ * *t81. __ One then computes that f(fi4,4) = (w), where w 2 is a lift of z as in sectio* *n 2.2, is as claimed, using the relation in Proposition 9 and that q0(a1) = q0(a3) = 1 mod 2 (see Pro* *position 7). Now the value for f(fis4,4) for any s follows immediatelyPfrom the derivation p* *roperty of the con- necting homomorphism ffi, namely ffi(xn) = ffi(x)( n-1i=0jR(x)ijL(x)n-1-i). Al* *ternatively, f(fis4,4) may be computed directly for any odd s using that (q0 id)jL(x2) = 0 mod 2, jR * *and OE-1 are ring homomorphisms and Proposition 9. We obtain f(fis2n,2n) = f(fis4,4)n-2 for all s* * and n_>_3 because xn = x2n-1for all n > 3. * * |__| Fix some n > 3. To explain the relevance of the above computation for the probl* *em of projecting the f-invariant to fi2n-2,2n-2we contemplate the following diagram. 20 Beta-elements and divided congruences _________ss0________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *__________________________________________@ ____________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *______________________________________ ___"f_______________________________&&________________* *_________ (19) Ext2,2n[BPO]__//_iCpD__2n-1OOQ=Z[2] Z=2" ` CCC (17)OO| || CCC || | CCC O_"(18)//_ | 'CCCCCD2n-1OOQ=Z[2] D=2DOO || CCC || (section3.||1)|| C!!C?O| " ?O| fflffl| V"1,2O________//V1,2ss_//k = F4. By the results in section 3.1, V1,2is k-free on the set {ai3T j|i > 0, j = 0, 1* *} and ss is defined to be the projection to the coefficient of a2n-23. The map ss0is defined to be the projec* *tion to the generator fi2n-2,2n-2, c.f. Proposition 15. Theorem 19 determines representatives in V"1,* *2:= D2n-1 Q=Z[2]\V1,2 n for all generators of Ext2,2[BP ] and thus defines the map '. We know that (19)* * commutes when ss and ss0are omitted. Theorem 20. The diagram (19) is commutative. * * n Proof.By inspection of Proposition 15 and Theorem 19 , the only generator of Ex* *t2,2[BP ] whose__ f-invariant contains a2n-23is the Kervaire element fi2n-2,2n-2. * * |__| Corollary 21. Let n > 3 and X a (U, fr)2-manifold of dimension 2n. Then the cor* *ner of X has Kervaire invariant one if and only if the f-invariant of X admits a represe* *ntative in V"1,2which contains the summand a2n-23. Note that the coefficient of a2n-23in the f-invariant of X can rather easily be* * expressed in terms of Chern numbers of X. The very reason that this does not give us the Chern number* *s determining the Kervaire elements is the indeterminacy in the above constructions caused by* * the projection D2n-1 Q=Z[2] -! D__2n-1 Q=Z[2]. References APS M. Atiyah, V. Patodi, I. Singer, Spectral asymmetry and Riemannian geometr* *y II, Math. Proc. Cam- bridge Philos. Soc. 78 (1975), no. 3, 405-432. 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, 533-550. BJM2 M. Barratt, J. Jones, M. Mahowald, The Kervaire invariant and the Hopf in* *variant, in: Algebraic topology (Seattle, Wash., 1985), 135-173, Lecture Notes in Math., 1286, Sp* *ringer, Berlin, 1987. B W. Browder, The Kervaire invariant of framed manifolds and its generalizat* *ion, Ann. Math. 90 (1969), 157-186. Br R. Bruner, The homotopy theory of H1 ring spectra, in: H1 Ring Spectra and* * their Applications, Lecture Notes in Math., 1176, Springer, Berlin, 1986. He E. Hecke, Theorie der Eisensteinschen Reihe h"oherer Stufe und ihre Anwend* *ung auf Funktionentheorie und Arithmetik, Abh. Math. Seminar der Hamb. Univ. 5 (1927), 199-224 oder * *Kapitel 24 aus E. Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Gttingen 1959. HBJ F. Hirzebruch, T. Berger, R. Jung, Manifolds and modular forms, Aspects of* * Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig, 1992. 21 Beta-elements and divided congruences HS M. Hovey, N. Strickland, Comodules and Landweber exact homology theories, * *Adv. Math. 192 (2005), 427-456. K1 N. Katz, Higher congruences between modular forms, Ann. Math. 101 (1975), * *332-367. K2 N. Katz, The Eisenstein measure and p-adic interpolation, Amer. J. Math. 9* *9 (1977), no. 2, 238-311. K3 N. Katz, p-adic properties of modular schemes and modular forms, in: Modul* *ar functions of one variable III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 69-* *190. Lecture Notes in Mathematics, 350, Springer, Berlin, 1973. KM S. Kochman, M. Mahowald, On the computation of stable stems, Comtemp. Math* *. 181 (1995), 299- 316. L1 G. Laures, The topological q-expansion principle, Topology 38 (1999), 387-* *425. L2 G. Laures, On cobordism of manifolds with corners, Trans. AMS 352 (2000), * *5667-5688. MRW H. Miller, D. Ravenel and W. Wilson, Periodic phenomena in the Adams-Novi* *kov spectral sequence, Ann. Math. 106 (1977), 469-516. Na N. Naumann, Comodule categories and the geometry of the stack of formal gr* *oups, math.AT/0503308. Ne J. Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wisse* *nschaften, 322, Springer-Verlag, Berlin, 1999. R D. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure * *and Applied Mathemat- ics, 121, Academic Press, Inc., Orlando, FL, 1986. Sh K. Shimomura, Novikov's Ext2at the prime 2, Hiroshima Math. J. 11 (1981), * *499-513. Si1 J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathema* *tics, 106, Springer-Verlag, New York, 1986. Si2 J. Silverman, Advanced topics in the arithmetic of elliptic curves, Gradua* *te Texts in Mathematics, 151, Springer-Verlag, New York, 1994. St N. Strickland, Notes on level three structures on elliptic curves, preprin* *t (2000), available at http://www.shef.ac.uk/personal/n/nps/papers/. Sw R. Switzer, Algebraic topology_homotopy and homology, Reprint of the 1975 * *original, Classics in Mathematics, Springer-Verlag, Berlin, 2002. Wa L. Washington, Introduction to cyclotomic fields, Second edition, Graduate* * Texts in Mathematics, 83, Springer-Verlag, New York, 1997. Jens Hornbostel and Niko Naumann jens.hornbostel@mathematik.uni-regensburg.de niko.naumann@mathematik.uni-regensburg.de NWF I- Mathematik, Universit"at Regensburg, 93040 Regensburg 22