Realisations of Kummer-Chern-Eisenstein-motives
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(2) 24. A. Caspar. Recall that S/Q represents the coarse moduli space of polarised abelian surfaces with real multiplication by O F . To get a fine moduli space (i.e. a universal family of abelian surfaces), one has to introduce a level N -structure. Otherwise one is just left with a moduli stack S. For this we consider the congruence subgroup K N := {g ∈ K 0 |g ≡ id mod N }. Now according to [21], the scheme S K N /Q represents for N ≥ 3 the fine moduli space of polarised abelian surfaces with real multiplication by O F and level N -structure. There are different ways to compactify our surface S. First, there is the (singular) Baily–Borel compactification S → S. Basically, this is S = S ∪ (PSL2 (O F )\P1F ), and here in our case (with class number 1 and full level) it is just given by adding a singular point ∞ at infinity (a cusp). This cusp singularity can be resolved in a canonical way. S → S. For the complex surface S(C) this is due to Hirzebruch (as in [17]), but it can also be done over Q according to Rapoport ([21] and [10]). This resS−S olution. S is the smooth toroidal compactification ([1]). The boundary. S∞ :=. 1. is a polygon with rational components S∞,i P . We get our motive, which is an extension of a Tate motive by a Dirichlet motive, as a piece of a long exact sequence. Our approach is a little bit different from Harder’s general proposal in [16], 1.4, where he considers a smooth closed subscheme Y ⊂ X of a smooth projective scheme X/Q. For such a pair one gets the long exact sequence ... → Hci (X − Y, Z) → H i (X − Y, Z) → H i (N˙ Y ) → ..., where N˙ Y denotes the punctured normal bundle of Y . Here in our situation the closed subscheme Y =. S∞ ⊂. S = X is not smooth. It is a divisor with normal crossings. Let j : S →. S be the open embedding of S/Q into the toroidal compactification. S/Q. It induces an exact triangle j! Q → R j∗ Q → R j∗ Q/j! Q → j! Q[1] of complexes of sheaves on. S, i.e. a sequence in the derived category of. S. This gives rise to the long exact sequence ... → H i (. S, j! Q) → H i (. S, R j∗ Q) → H i (. S, R j∗ Q/j! Q) → ... . By definition we have H i (. S, j! Q) = Hci (S, Q) and H i (. S, R j∗ Q) = H i (S, Q) and i S, R j∗ Q/j! Q) (Sect. 1.2). Our understandour duty is to compute the motives H (. ing of these motives is a touch old fashioned (see [14], 1, [16], 1). We discuss the realisations of motives in more detail below (Sect. 1.4). For example there are the Tate motives Q(−n) = H 2n (Pn , Q). In Sect. 1.2 we show the above sequence gives rise to an exact sequence of motives 0 → Q(0) → Hc2 (S, Q) → H 2 (S, Q) → Q(−2) → 0. With the interior cohomology H!2 (S, Q) := Im(Hc2 (S, Q) → H 2 (S, Q)) we get an extension 0 → H!2 (S, Q) → H 2 (S, Q) → Q(−2) → 0. Now the bottom H!2 (S, Q) is a semi-simple module for the Hecke algebra and we can find a direct 2 (S, Q(1)) = Q(0)⊕Q(0)χ , where Q(0)χ is the Dirichlet motive summand HCH D D.
(3) Realisations of Kummer–Chern–Eisenstein-motives. 25. for the quadratic character χ D in the sense of Deligne ([7], 6). It is generated by 2 the first Chern class c1 (L 1 ⊗ L −1 2 ) ∈ H! (S, Q), where L 1 ⊗ L 2 is the line bundle of Hilbert modular forms of weight (2, 2) on S. Hence after twisting and splitting the summand Q(0) one is faced with 2 0 → Q(0)χ D → HCHE (S, Q(1))(−) → Q(−1) → 0, 2 (S, Q(1))(−)] ∈ Ext1 i.e. an element [HCHE MMQ (Q(−1), Q(0)χ D ), which we call a Kummer–Chern–Eisenstein motive. We have various realisations of our motive. For each prime l there is an l2 (S, Q(1))(−)] ∈ Ext1MGAL (Ql (−1), Ql (0)χ D ), which is adic realisation [HCHE,l a mixed Gal(Q/Q)-module. We show that this extension group is isomorphic to the subgroup of norm one elements in l-adic completion of F ∗ tensorised with Ql . 1 −1 Set. ε := ε− 2 ζ F (−1) .. Theorem. For each prime number l the l-adic realisation of our Kummer–Chern– 2 (S, Q(1))(−)] is. ε. Eisenstein motive [HCHE,l The corresponding l-adic Galois representation is inducedbe the Kummer field ∞ √ )α −1 (σ ) extension F l. ε, i.e. Gal(Q/Q) σ → χ D (σ ) τ ε (σ−1 , ε, ζl ∞ attached to. where α is the cyclotomic character and τ ε (σ ) is defined by σ. 0. ε) √ l∞. ε. (l. √ ∞. α. =. (σ ) τ ε (σ ) ζl ∞ .. There is the Hodge-de Rham extension, which is a mixed Hodge-de Rham struc2 ture [HCHE,∞ (S, Q(1))(−)] ∈ Ext1MHd RQ (Q(−1), Q(0)χ D ) and we can identify this group with R. 2 (S, Q(1))(−)] Theorem. The Hodge-de Rham realisation of our motive [HCHE,∞ is log. ε.. There is a general construction of a so called Kummer motive K . ε attached to. ε (cf. Sect. 1.4), and these two theorems tell us that the realisations of our motive are exactly those of the Kummer motive K . ε, 2 2 (S, Q(1))(−)], [HCHE,l (S, Q(1))(−)] = (log. ε,. ε) = (K . ε∞ , K . εl ), [HCHE,∞ 2 (S, Q(1))(−)] = K . but we do not know, whether actually [HCHE ε. Moreover, one observes (cf. Sect. 4.1) that such a Kummer motive K . ε is actually a Kummer-1-motive M ε in the sense of [5] and we appoint a candidate for M ε . Consider −1. L= L1 ⊗ L 2 , where. L i is the unique prolongation of L i to the compact surface. S, such that its first Chern class restricted to the boundary is trivial. Thus, if we denote −1 by u : Pic(. S) → Pic(. S∞ ) the restriction map, we get u( L1 ⊗ L 2 ) ∈ Pic0 (. S∞ ). u 0 0. We know that Pic ( S∞ ) Gm and get a 1-motive [Z(χ D ) · L → Pic ( S∞ )], where L is the submodule generated by. L. Z(χ D ) ·. L → Pic0 (. S∞ )]∨ ) ⊗ Q of the dual Theorem. Consider the motive T ([Z(χ D ) ·. u 0 ∨. Kummer-1-motive [Z(χ D ) · L → Pic ( S∞ )] . Then u. u 2 (S, Q(1))(−)] Tl ([Z(χ D ) ·. L → Pic0 (. S∞ )]∨ ) ⊗ Ql . [HCHE,l 2 In particular, we have [HCHE,l (S, Q(1))(−)] Tl (M ε ) ⊗ Ql ..
(4) 26. A. Caspar. There is a third motive attached to the surface S, the Hodge-one-motive η S . It is induced by the Hodge structure given by the second cohomology of S. Using [3] and the previous theorem we obtain. Theorem. The Kummer-1-motive attached to ε−2 is isomorphic to a submotive of the Hodge-one-motive η S . In particular, the dual of the Kummer–Chern–Eisenstein 2 (S, Q(1))(−)]∨ is isomorphic to a submotive of the realisation motive [HCHE T (η S ) ⊗ Q of the Hodge-one-motive. 1. The Kummer–Chern–Eisenstein motive Let F be a real quadratic number field with conventions and notations fixed in the introduction. Let S be the Hilbert modular surface of full level as discussed in the introduction. 1.1. The line bundles of modular forms Since the first Chern class of the line bundle of Hilbert modular forms on S is the one of the main ingredients for our construction, we investigate it more precisely. Let us start with the complex situation (i.e. with the Betti realisation). By [8], II.7, we know that the line bundle of modular forms is a product L 1 ⊗ L 2 , where each factor corresponds to the factor of automorphy (cz 1 + d)2 , resp. (c z 2 + d )2 . The sections of L 1 ⊗ L 2 on S(C) are Hilbert modular forms f (z 1 , z 2 ) of weight (2, 2), and we know that L 1 ⊗ L 2 =
(5) 2S(C) . We have the Chern class map c1 : Pic(S(C)) → H 2 (S(C), Q(1)), where Q(1) = 2πi Q. The map c1 is inexp duced by the exponential sequence 0 → Z(1) → O S(C) → O∗S(C) → 0 of sheaves on S(C), where Pic(S(C)) = H 1 (S(C), O∗S(C) ). Now we want to extend the line S(C). This is given by bundles L 1 , L 2 on S(C) to the compact surface. Lemma 1.1. Let L i ∈ Pic(S(C)), i = 1, 2 be the line bundles on S(C) as above. S(C)) with trivial Chern class on Then there is a unique line bundle L i in Pic(. 0 (. ∈ Pic S (C)), such that its restriction to the open the boundary, L i |. ∞ S∞ (C) part S(C) is L i . In particular, with the interior cohomology H!2 (S(C), Q(1)) := Im Hc2 (S(C), Q(1)) → H 2 (S(C), Q(1)) we have c1 (L i ) ∈ H!2 (S(C), Q(1)). L 2 compare e.g. [8], IV.2. The Proof. For the construction of the extensions L 1, 2. S∞ (C)) denotes the tensor product is L 1 ⊗ L 2 =
(6) (log S∞ (C)), where
(7) 2 (log. sheaf of differentials on. S(C), which may have poles of at most simple order along the boundary. S∞ (C). To proof the uniqueness, we observe that there is, according to [10], Lemma 2.2, the following commutative triangle Hc2 (S(C), Q(1)) RRR RRR RRR RRR R( 2 2 /. H (S(C), Q(1)), H ( S(C), Q(1)).
(8) Realisations of Kummer–Chern–Eisenstein-motives. 27. i.e. H!2 (S(C), Q(1)) = Im H 2 (. S(C), Q(1)) → H 2 (S(C), Q(1)) . Now the Chern class c1 ( L i ) ∈ H 2 (. S(C), Q(1)) of the above line bundle is a preimage of c1 (L i ), 0. that has trivial Chern class on the boundary, i.e. L i |. S∞ (C) ∈ Pic ( S∞ (C)). We have to see that this extension is indeed unique. We prove that the extension cannot be modified by a divisor, that is supported on the boundary. S∞ (C). For this we use S∞ (C) is the fact that the intersection matrix (Si · S j )i j of the boundary divisor. negative definite. In particular, the self-intersection number Si2 is at least −2, see for example [8], II.3. This means that a divisor, which is only supported on. S∞ (C) has non zero degree. And hence we cannot modify a line bundle with trivial Chern class on. S∞ (C), for example our above L i , by a boundary divisor. Compare also [10], Hilfssatz 2.3. L 2 over. S(C) are now meromorphic Hilbert modular forms The sections of L1 ⊗ L 2 is trivial on the polygon at infinity, since of weight (2, 2). The product L1 ⊗ there is, up to a constant, a non-vanishing section, see [8], III. Lemma 3.2. But the S∞ (C) is not trivial. We see this in restriction of each factor L i to. L 2 be the line bundle of meromorphic Hilbert modular Lemma 1.2. Let L1 ⊗ forms of weight (2, 2) on. S(C). Let ε = ε02 ∈ O∗F be as above. Then the restric0 ∗. tion L i |. S∞ (C) ∈ Pic ( S∞ (C)) C of each factor L i to the boundary S∞ (C) is −1. L 1 |. S∞ (C) = ε and L 2 |. S∞ (C) = ε . Proof. Restrict L i to the boundary. S∞ (C) and use the explicit glueing of the components. S∞,i (C) (cf. [17], 2.3) to see what happenend to a section of L i |. S∞ (C) by going around the polygon. S∞ (C). Moreover, we use that fact that the units identify. S∞,i+n (C), see loc. cit. or [1], I.5. S∞,i (C) and. Note that the exponent ±1 of ε has been fixed by the orientation of the boundary. To discuss the other realisations, we must look at the moduli interpretation. For the fine moduli space there is the universal family A/S K N of abelian surfaces, as above, with the zero section s : S K N → A. Let
(9) 1A/SK be the sheaf of relaN. tive differentials. Then we have the Lie algebra Lie(A) := s ∗
(10) 1A/SK , which is a N locally free O F ⊗ O SK N -module of rank one (here we denote by O SK N the structure sheaf of S K N ). And we define, as in [21], 6, a line bundle ω SK N on S K N by ω SK N := NmO F ⊗O SK /O SK Lie(A)∨ , where NmO F ⊗O SK /O SK is the norm N. N. N. N. map from O F ⊗ O SK N to O SK N and Lie(A)∨ the dual Lie algebra. For its global sections we have. Lemma 1.3. Let L 1 ⊗ L 2 be the line bundle of Hilbert modular forms of weight (2, 2) on S K N (C). Then its global sections are global sections of ω⊗2 S K N (C) on S K N (C), i.e. the line bundle L 1 ⊗ L 2 comes from a line bundle on S K N . Proof. This is [6], Lemme 6.12. Note that this is true for any level N , so even for the stack. Still we have to show that each factor L i is defined over F, i.e. a line bundle on S K N × F. For this we base change our scheme S K N over Q to F and observe.
(11) 28. A. Caspar. that O F ⊗ F decomposes into F ⊕ F along the action of Gal(F/Q) = {id, }. So this implies after base change a decomposition of the above O F ⊗ O SK N module Lie(A). But then according to this splitting and the above lemma, we get ω SK N ×F = L 1 ⊕ L 2 , and therefore ω⊗2 S K N ×F = L 1 ⊗ L 2 . For the l-adic Chern class of these line bundles we start with the Kummer sequence of sheaves on S K N × Q, that is 0 → µl n → Gm → Gm → 0. This gives rise to the Chern class map c1 : Pic(S K N × Q) → Het´2 (S K N × Q, Ql (1)), where Pic(S K N × Q) = Het´1 (S K N × Q, Gm,SK N ) = Het´1 (S K N × Q, O∗SK ). N. Remark 1.4. To get rid of the level N -structure, we proceed as in the proof of [21], Lemme 6.12, i.e. we look at the (K N /K 3N )-invariants in Pic(S K 3N × Q), resp. Het´2 (S K 3N × Q, Ql (1)). Finally, we use our considerations in the complex case to get Corollary 1.5. Let L 1 ⊗ L 2 be the line bundle of Hilbert modular forms of weight S × Q) (2, 2) on S × Q. Then there is a unique line bundle L i , i = 1, 2, in Pic(. 0 (. ∈ Pic S × Q), such that its with trivial Chern class on the boundary, L i |. ∞ S∞ ×Q restriction to the open part S × Q is L i . In particular, we have 2 (S × Q, Q (1)). c1 (L i ) ∈ Het,! l ´. Proof. We have discussed this for the complex surface S(C). By the comparison 2 (S × Q, Q (1)). theorems this is also true for the algebraic classes c1 (L i ) ∈ Het,! l ´ By Lemma 1.1 we get this extension in the complex context and in Lemma 1.2 we S × F). showed that the restriction class is defined over F and in particular L i ∈ Pic(. . 1.2. The construction of the motive Recall from the introduction the long exact sequence ... → H i (. S, j! Q) → H i (. S, R j∗ Q) → H i (. S, R j∗ Q/j! Q) → ... . The next step to get our motive is Lemma 1.6. From this long exact sequence one can break the exact sequence 0 → H 1 (. S, R j∗ Q/j! Q) → Hc2 (S, Q) → H 2 (S, Q) → H 2 (. S, R j∗ Q/j! Q) → 0. Proof. By Poincaré duality we just have to prove the vanishing of H 1 (S, Q). This is stated for example in [10], Satz 1.9 (compare also [9]), but without any proof. In the Betti realisation the vanishing can be seen by the use of group cohomology for = PSL2 (O F ). One has H 1 (S(C), Q) = H 1 (, Q) = HomGroups (, Q) HomAb (/[, ], Q). The quotient ab = /[, ] is finite (compare [23], Théorème 3), and so H 1 (S(C), Q) must be trivial. By the comparison theorems this holds then also for the other cohomology theories. .
(12) Realisations of Kummer–Chern–Eisenstein-motives. 29. The complex R j∗ Q/j! Q lives on the boundary. S∞ and is isomorphic to the S∞ →. S denotes the closed embedding. We compute complex i ∗ R j∗ Q, where i :. its cohomology in the following lemma, which is just a special case of a theorem by Pink (see e.g. [14], 2.2.10). Proposition 1.7. Let j : S →. S be the open embedding of the Hilbert modular surS, R j∗ Q/j! Q) Q(0) face S into its toroidal compactification. S. Then we have H 1 (. 2. and H ( S, R j∗ Q/j! Q) Q(−2). Proof. We denote by. S∞,i,i+1 :=. S∞,i ∩. S∞,i+1 the intersection of the two comS∞,i+1 with i = 0, ..., n − 1. Now we distinguish the two cases. ponents. S∞,i and. First, the point P is smooth, i.e. P ∈ /. S∞,i,i+1 for all i. Then we get for the fibre over P, ⎧ ⎪ q=0 ⎨Q(0), q H (Gm , Q) = Q(−1), q=1 ⎪ ⎩ 0, q = 2. If the point P = Pi,i+1 is not smooth, i.e. Pi,i+1 ∈. S∞,i,i+1 , we get for the fibre over Pi,i+1 , ⎧ ⎪ q=0 ⎨Q(0), H q (Gm × Gm , Q) = Q(−1) ⊕ Q(−1), q=1 ⎪ ⎩ Q(−2), q = 2. This leads us to. ⎧ ⎪ Q(0). ⎪ S , ⎪ ⎨ ∞ Q(−1). S∞,i , R q j∗ Q/j! Q = i ⎪ ⎪ ⎪ ⎩ Q(−2). S∞,i,i+1 ,. q=0 q=1 q = 2,. i. Now we put the cohomology classes H p (. S, R q j∗ Q/j! Q) into a diagram, which pq pq S, R q j∗ Q/j! Q) describes the E 2 -term of the spectral sequence, i.e. E 2 = H p (. looks like Q(−2) 0 Pi,i+1 RRR RRR 02 RRdR RRR RRR RR) Q(−1) Q(−2) 0. S∞,i S∞,i RRR RRR 01 RRRd RRR RRR RR( Q(−1) Q(0) Q(0). S∞,i.
(13) 30. A. Caspar. 1 All the other entries are zero. Sinceour polygon is a closed chain of P ’s, we 02 Q(−2) →. conclude that the differential d : S∞,i Q(−2) must have rank. n − 1. To see this, we observe that. Pi,i+1 d 02 is. given by. (P0,1 , . . . , Pn−1,0 ) → (. S∞,0 −. S∞,1 ,. S∞,1 −. S∞,2 , . . . ,. S∞,n−1 −. S∞,0 ), ⎛ ⎜ i.e. can be represented by an (n ×n)-matrix like this ⎜ ⎝. ⎞. 1 −1 1 −1. .. .. . .. −1. −1 1 −1 1. ⎟ ⎟ , which ⎠. is of rank n − 1. Therefore the kernel of d 02 is one copy of Q(−2). This gives 2. of d 21 is Ker(d 21 ) = us H ( S, R j∗ Q/j! Q) Q(−2). Furthermore, the kernel 02. S∞,i Q(−2). The quotient of this by the image of d gives one copy of Q(−2). S, R j∗ Q/j! Q). The second differential map d 01 :. And this is H 3 (. S∞,i Q(−1) → 1. S∞,i Q(−1) is an isomorphism, hence we are left with H ( S, R j∗ Q/j! Q) Q(0). With the interior cohomology H!2 (S, Q) = Im(Hc2 (S, Q) → H 2 (S, Q)) we have an immediate consequence Corollary 1.8. For a Hilbert modular surface S we get a short exact sequence 0 → H!2 (S, Q) → H 2 (S, Q) → Q(−2) → 0. Now we decompose the bottom of the extension H!2 (S, Q) even further. For this we twist the above sequence with Q(1) and get 0 → H!2 (S, Q(1)) → H 2 (S, Q(1)) → Q(−1) → 0. By [10], 1.8, (see also [8], XI.2) we get, according to the semi-simple action of the Hecke algebra (see loc. cit. for the definition), a decomposition H!2 (S, Q(1)) = 2 (S, Q(1)) ⊕ H 2 (S, Q(1)), and furthermore, this Hecke action commutes Hcusp CH 2 (S, Q(1)) collects all the contriwith the action of Galois. The first summand Hcusp butions coming from the cuspidal representations of weight two. (In [10], 1.8, this 2 (S, Q(1)) consists is denoted by Coh0 , the “interesting” part.) The second one HCH of those coming from the one dimensional representations, i.e. the Größencharacters. (In loc. cit. this is denoted by Cohe , the “trivial” part.) Note that in our case of h + = 1 and K f = K 0 there is only a contribution by the trivial character. More2 (S, Q(1)) is spanned by our two Chern classes c (L ) and over, this summand HCH 1 1 c1 (L 2 ) of Sect. 1.1 (compare e.g. [8], XI.2). Now we identify the Dirichlet character χ D with the associated Galois charχD Res / / Gal(F/Q) {±1}, with the Galois Gal(Q(ζ D )/Q) acter Gal(Q/Q) group Gal(Q(ζ D )/Q) (Z/DZ)∗ of the cyclotomic field Q(ζ D )/Q. Then define Q(0)χ D to be the Dirichlet motive for the quadratic character χ D in the sense of Deligne ([7], 6). We explain this latter notion in more detail below. And we get.
(14) Realisations of Kummer–Chern–Eisenstein-motives. 31. Corollary 1.9. We have an isomorphism 2 HCH (S, Q(1)) Res F/Q (Q(0)) = Q(0) ⊕ Q(0)χ D .. Proof. See e.g. [10], Proposition 2.10, or [8], XI.2 Proposition 2.7.. . If we use the above decomposition, our sequence becomes 2 2 0 → HCH (S, Q(1)) → HCHE (S, Q(1)) → Q(−1) → 0, 2 (S, Q(1)) := H 2 (S, Q(1))/H 2 (S, Q(1)). So we identify the bottom where HCHE cusp 2 HCH (S, Q(1)) with Q(0) ⊕ Q(0)χ D , and define. Definition. (Kummer–Chern–Eisenstein motive): We call the extension 2 0 → Q(0) ⊕ Q(0)χ D → HCHE (S, Q(1)) → Q(−1) → 0. a Kummer–Chern–Eisenstein motive. Now we still have the action by the involution ∈ Gal(F/Q). This gives a further decomposition into (±1)-eigenspaces. Regarding this our sequence becomes after splitting the (+1)-eigenspace Q(0), 2 (S, Q(1))(−) → Q(−1) → 0, 0 → Q(0)χ D → HCHE 2 (S, Q(1))(−) := H 2 (S, Q(1))/Q(0). So we get an element with HCHE CHE 2 [HCHE (S, Q(1))(−)] ∈ Ext1MMQ (Q(−1), Q(0)χ D ),. which we call again a Kummer–Chern–Eisenstein motive, and which is now defined over Q. The name bases on the idea that it is an extension of Q(−1) by Q(0)χ D 2 (S, Q(1)) is spanned by the (so it should be Kummer), and that the extension HCHE Chern classes and the section of the restriction map. And the latter one is given by the Eisenstein section. Now we have the various realisations of our motive. There are the l-adic 2 (S, Q(1))(−)] ∈ Ext1MGAL (Ql (−1), Ql (0)χ D ), which are realisations [HCHE,l mixed Galois modules (see e.g. [14], I). And we have the Hodge-de Rham reali2 (S, Q(1))(−)] ∈ Ext1MHd RQ (Q(−1), Q(0)χ D ), which is a mixed sation [HCHE,∞ Hodge-de Rham structure (loc. cit.). 2 (S, Q(1)). The Remark 1.10. We want to fix a generator of the summands of HCH first summand Q(0) is generated by c1 (L 1 ⊗ L 2 ) = c1 (L 1 ) + c1 (L 2 ). For the sec−1 ond one Q(0)χ D , we have the choice between c1 (L 1 ⊗ L −1 2 ) and c1 (L 1 ⊗ L 2 ). These two differ by the action of ∈ Gal(F/Q). Here we choose c1 (L 1 ⊗ L −1 2 ) for a generator of Q(0)χ D . Note that we do not have a canonical choice of the generator, because both classes generate the submodule Q(0)χ D . So we are left with this (±1)-ambiguity. But our results respects this in the sense that if we flip the generator, we get the conjugate result..
(15) 32. A. Caspar. 1.3. The dual Kummer–Chern–Eisenstein motive 2 (S, Q(1))(−)]. In order In Chapter 2 we compute the l-adic realisations of [HCHE 2 to do so, we look at the dual motive [HCHE (S, Q(1))(−)]∨ .. Lemma 1.11. The Kummer–Chern–Eisenstein motive 2 0 → Q(0)χ D → HCHE (S, Q(1))(−) → Q(−1) → 0 2 (S, Q(1))(−)∨ → Q(0)χ becomes by dualising 0 → Q(1) → HCHE D → 0, 1 S, R j∗ Q/j! Q) ⊗ Q(1) the motive as in Proposition where the bottom Q(1) is H (. S, R j∗ Q/j! Q) is canonical dual to H 2 (. S, R j∗ Q/j! Q). 1.7. In particular H 1 (. Proof. Recall that by Lemma 1.6 the sequence 0 → H 1 (. S, R j∗ Q/j! Q) → Hc2 (S, Q) → H 2 (S, Q) → H 2 (. S, R j∗ Q/j! Q) → 0 is the starting point of the construction of the motive. And the outer terms can be identified as 0 → Q(0) → Hc2 (S, Q) → H 2 (S, Q) → Q(−2) → 0, see Proposition 1.7. To get the dual motive, we consider the Poincaré duality pairing Hc2 (S, Q) × H 2 (S, Q) → Q(−2). Hence Hc2 (S, Q(1)) and H 2 (S, Q(1)) are dual to each other, i.e. the dual of the map Hc2 (S, Q(1)) → H 2 (S, Q(1)) is the map itself, and therefore the dual of the kernel is the cokernel and vice versa. Now this means that if we tensorise the above sequence with Q(1), its dual sequence is just the same sequence 0 → Q(1) → Hc2 (S, Q(1)) → H 2 (S, Q(1)) → Q(−1) → 0. So the dual sequence for 0 → H!2 (S, Q(1)) → H 2 (S, Q(1)) → Q(−1) → 0 ∨ is 0 → Q(1) → Hc2 (S, Q(1)) → H!2 (S, Q(1)) → 0. But the cokernel of Q(1) → Hc2 (S, Q(1)) is H!2 (S, Q(1)) the interior cohomology itself, this means ∨ H!2 (S, Q(1)) is self-dual, that is H!2 (S, Q(1)) = H!2 (S, Q(1)). So take the 2 (S, Q(1))(−) → Kummer–Chern–Eisenstein motive 0 → Q(0)χ D → HCHE Q(−1) → 0. Then we get, according to the previous discussion, by dualising 2 (S, Q(1))(−)∨ → Q(0)χ −1 → 0. As χ 2 = the extension 0 → Q(1) → HCHE D D 2 (S, Q(1))(−)∨ → Q(0)χ 1, we end up with 0 → Q(1) → HCHE D → 0 in Ext1MMQ (Q(0)χ D , Q(1)). 2 (S, Q(1))(−)∨ sits in H 2 (S, Q(1)), because Remark 1.12. The dual motive HCHE c 2 our motive [HCHE (S, Q(1))(−)] is by definition a quotient of H 2 (S, Q(1)) and H 2 (S, Q(1)) is the dual of Hc2 (S, Q(1)). We have chosen c1 (L) = c1 (L 1 ⊗ L −1 2 )∈ H!2 (S, Q(1)) for the generator of Q(0)χ D . In the dualising process we have to flip the generator and the dual motive of Q(0)χ D is therefore generated by c1 (L −1 1 ⊗L 2 ). Moreover, by Siegel’s theorem (see e.g. [8], IV.1 and Chapter 3) the cup product of c1 (L 1 ⊗ L −1 2 ) with itself is −4ζ F (−1). And therefore to normalise the generator of the dual motive we have to multiply it with −1/4ζ F (−1).. In the following we give an alternative construction of the dual extension. Start again with the open embedding j of S and the closed embedding i of. S∞ into the toroidal compactification. S. This gives a short exact sequence 0 → j! j ∗ Q → S. By definition we have the identifications Q → i ∗ i ∗ Q → 0 of sheaves on.
(16) Realisations of Kummer–Chern–Eisenstein-motives. 33. H i (. S, j! j ∗ Q) = Hci (S, Q) and H i (. S, i ∗ i ∗ Q) = H i (. S∞ , Q). Therefore, the sequence of sheaves induces a long exact sequence in the cohomology. Since the compact surface. S is simply connected (cf. [8], IV.6—more general the first cohomology with nontrivial coefficients vanishes by [10], Proposition 5.3), we get f1. S∞ , Q) → Hc2 (S, Q) → H 2 (. S, Q) → H 2 (. S∞ , Q) → 0. 0 → H 1 (. We twist this with Q(1) and by the above remark we know that the dual motive 2 (S, Q(1))(−)∨ sits in H 2 (S, Q(1)). Moreover, we have HCHE c Lemma 1.13. Let 0 → H 1 (. S∞ , Q(1)) → Hc2 (S, Q(1)) → Im( f 1 ) → 0 be the short exact sequence that is induced by the above sequence. Then the dual of our 2 (S, Q(1))(−)]∨ sits inside this extension, Kummer–Chern–Eisenstein motive [HCHE that is 0. / H 1 (. S∞ , Q(1)). / H 2 (S, Q(1)) c O. / Im( f 1 ) O. /0. 0. / Q(1). ? / H 2 (S, Q(1))(−)∨ CHE. ? / Q(0)χ D. / 0.. Proof. Consider the square Hc2 (S, Q(1)) f1. Hc2 (S, Q(1)) f2. / H 2 (S, Q(1)). H 2 (. S, Q(1)) By [10], Lemma 2.2 and Hilfssatz 2.3 (see also Proposition 5.3 of loc. cit.) we have that Im( f 2 ) = H!2 (S, Q(1)) Im( f 1 ) and moreover H!2 (S, Q(1)) Im H 2 (. S, Q(1)) → H 2 (S, Q(1)) . So the two kernels H 1 (. S∞ , Q(1)) = Ker( f 1 ) 1. and H ( S, R j∗ Q/j! Q) ⊗ Q(1) = Ker( f 2 ) have to be isomorphic, and we get the bottom of our extension. Moreover, we have, by Lemma 1.1, the unique extension L) ∈ H 2 (. S, Q(1)) of c1 (L) ∈ H!2 (S, Q(1)) and c1 (L) generates Q(0)χ D . Since c1 (. Im( f 1 ) Im( f 2 ), we can consider Q(0)χ D inside Im( f 1 ) 1.4. Realisations of mixed motives In this section, we touch the general theory of motives a little further. The main references are [14], 1, [16], 1, and [7]. (But consult additionally [19], 1.) According to the very general conjectures (see e.g. [18], 3) one expects for mixed Tate motives over a number field k that Ext1MMk (Q(−1), Q(0)) = k ∗ ⊗ Q. But one expects even more. Each of such an extension should come from a so called Kummer motive K a, a ∈ k ∗ . In other words the Kummer motives should exhaust Ext1MMk (Q(−1), Q(0)). Recall briefly the construction of a Kummer motive K a attached to a ∈ k ∗ . For details see e.g. [19], 3.1. Start with X = P1k − {∞, 0} = Gm and the divisor D = {1, a}, a = 1, and consider the relative cohomology sequence 0 → H 0 (Gm , Q) → H 0 (D, Q) → H 1 (X, D, Q) → H 1 (Gm , Q) → 0,.
(17) 34. A. Caspar. which becomes 0 → Q(0) → Q(0) ⊕ Q(0) → K a → Q(−1) → 0. And this gives us K a ∈ Ext1MMk (Q(−1), Q(0)). The realisations are (K a∞ , K al ) =. ( ι:k→C log |ι(a)|, a). Now one observes (cf. Sect. 4.1) that such a Kummer motive K a is actually a Kummer-1-motive Ma in the sense of [5]. Here we meet a slightly 2 (S, Q(1))(−)] is an extension of the Tate different situation, as our motive [HCHE motive Q(−1) by the Dirichlet motive Q(0)χ D . This means it is defined over Q, but its extension class is actually in the real quadratic field F. If we just look at the 2 (S, Q(1))(−)], the situation becomes more transparent. realisations of [HCHE The l-adic realisations 2 (S, Q(1))(−)] is For each prime l the l-adic realisation of our mixed motive [HCHE 1 an extension of Gal(Q/Q)-modules, i.e. an element in ExtMGAL (Ql (−1), Ql (0)χ D ). Let us describe this group. ∗,(l) := Lemma 1.14. nLet ∈ Gal(F/Q) be the nontrivial element. Consider F l ∗ ∗ ∗ lim F / (F ) the l-adic completion of F and define the subgroup of norm-one← − n − ∗,(l) ∗,(l) | · f = f −1 }. Then there is a canonical elements F := { f ∈ F − ∗,(l) ⊗Zl Ql . isomorphism Ext1MGAL (Ql (−1), Ql (0)χ D ) = F. Proof. By definition we have Ext1MGAL (Ql (−1), Ql (0)χ D ) = lim H 1 (Gal(Q/Q), µl n ⊗ χ D ) ⊗Zl Ql . ← − n Recall that µl n ⊗ χ D denotes the Galois module, given by the product of the cyclotomic character α and the quadratic character χ D , i.e. the tensor product of Galois representations. Furthermore, we identify Z/lZ with the roots of unity µl . Now by the Hochschild–Serre spectral sequence one has the exact sequence 0 → H 1 Gal(F/Q), (µl n ⊗ χ D )Gal(Q/F) → H 1 Gal(Q/Q), µl n ⊗ χ D Gal(F/Q) → H 1 Gal(Q/F), µl n ⊗ χ D → H 2 Gal(F/Q), (µl n ⊗ χ D )Gal(Q/F) → . One firstly notes that (µl n ⊗ χ D )Gal(Q/F) is trivial, i.e. H 1 (Gal(F/Q), (µl n ⊗ χ D )Gal(Q/F) ) = 0 = H 2 (Gal(F/Q), (µl n ⊗ χ D )Gal(Q/F) ). Hence the spectral sequence gives us H 1 (Gal(Q/Q), µl n ⊗ χ D ) H 1 (Gal(Q/F), µl n ⊗ χ D )Gal(F/Q) . Now Gal(Q/F) acts only on µl n , and therefore we get a (twisted) Kummer isomor∗,(l) ⊗ χ , where phism for H 1 (Gal(Q/F), µ n ⊗ χ D ). This gives in the limit F D l.
(18) Realisations of Kummer–Chern–Eisenstein-motives. 35. ∗,(l) by conjugation, i.e. by , and on χ by the multiplication Gal(F/Q) acts on F D by −1. This means that, eventually, for the invariant part Gal(F/Q) ∗,(l) ⊗ χ ∗,(l) | · f = f −1 }. { f ∈ F F D . So the l-adic realisations of the Kummer–Chern–Eisenstein motive is determined by ∗,(l) ⊗ Q , which is then independent of l. Moreover, the l-adic extena scalar in F l Zl. sion classes give rise to two dimensional Galois representations χ D (σ ) ∗ , where α is the cyclotomic Gal(Q/Q) → GL2 (Ql ), with σ → 0 α −1 (σ ) (Tate) character and where ∗ denotes the extension class. And by Kummer theory this element star ∗ is given by a Kummer field extension. If one suspects such an l-adic representation to come from a Kummer motive K a, with a ∈ F ∗ of norm one, then the star ∗ is determined by the Kummer-one-cocycle Hence, ∗ = τa (σ )α −1 (σ ), with. √ ∞ σ ·( l a) √ l∞ a. √ ∞ σ ·( l a) . √ l∞ a. = ζlτ∞a (σ ) . Indeed, we prove that our. 2 (S, Q(1))(−)] induces such a Galois representation (Theorem 2.5). [HCHE,l. The Hodge-de Rham realisation To describe Ext1MHd RQ (Q(−1), Q(0)χ D ) we follow basically the exposition in [19], 1.5 (see also [16], 1.5.2, and [14], 4.3.2). We are in the following situation: We have an element [M]∞ = [M] B−d R ∈ Ext1MHd RQ (Q(−1), Q(0)χ D ), i.e. two exact sequences in the commutative diagram. 0. / Q(0)χ D,B ⊗ C √ ·( D)−1 . . 0. / Q(0)χ D,d R ⊗ C. u / MB ⊗ C. sB. / Q(−1) B ⊗ C. /0. ·(2πi)−1 . I∞ . / Q(−1)d R ⊗ C. . / Md R ⊗ C i. / 0,. sd R. that are linked by the comparison isomorphism I∞ : M B ⊗ C Md R ⊗ C. The isomorphism on the right is given by multiplication with (2πi)−1 , and √on the left we ) = D of χ D , see have the multiplication with the inverse of the Gauß-sum G(χ D √ D ∗ explanation below. So we get a representation like 0 2πi , where the star ∗ is our [M]∞ . To get this, we observe that each of the sequences splits in its own category, i.e. we have sections s B and sd R , and the extension class is given by the comparison of s B and sd R . Furthermore, we know for the Hodge filtration F • Md R of Md R that F 1 Md R F 1 Q(−1)d R Q(−1)d R , and this isomorphism gives the section sd R . Now let us describe the rule to get [M]∞ ∈ Ext1MHd RQ (Q(−1), Q(0)χ D ). We start with a generator 1 B of Q(−1) B ⊗ C. This goes via the right isomorphism to −1 in M ⊗ C. (2πi)−1 · 1d R ∈ Q(−1)d R ⊗ C, by sd R we land in Md R ⊗ C and by I∞ B.
(19) 36. A. Caspar. On the other hand we can map 1 B ∈ Q(−1) B ⊗ C by the section s B directly to M B ⊗ C such that the image is in the (−1)-eigenspace of F∞ . Then the difference −1 ((2πi)−1 · 1 ) is in the kernel Q(0)χ s B (1 B ) − I∞ dR D,B ⊗ C. More precisely, it is in iR. If we multiply this with the inverse of the Gauß-sum G(χ D ) of χ D (here it √ is ( D)−1 ), we get exactly our class [M]∞ . We summarise the description of the Hodge-de Rham extension classes in Lemma 1.15. There is an isomorphism Ext1MHd RQ (Q(−1), Q(0)χ D ) iR. We choose here. 1 2πi. as a basis for iR, i.e. we get Ext1MHd RQ (Q(−1), Q(0)χ D ) R.. Proof. For details see e.g. [19], 1.5.. . Note that the image of the realisation functor consists of those elements in R such that acts by −1. Again in the case of a Kummer motive K a oneknows how √ D log a . the realisation class K a∞ looks like (compare [19], 3.1), one gets 0 2πi So in view of the above conjecture, one should have the following identification Ext1MMQ (Q(−1), Q(0)χ D ) (F ∗ )− ⊗ Q, i.e. if we have got a pair (M∞ , Ml ) such that there exists an a ∈ F ∗ of norm one with (M∞ , Ml ) = (log a, a), then this should come from a Kummer motive K a ∈ Ext1MMQ (Q(−1), Q(0)χ D ). In the end of this section I discuss briefly the category of motives, where the constructed Chern–Eisenstein motives are living in. It is the category that is generated by Dirichlet motives and their extensions - cf. [7], 6. One should think of a Dirichlet motive in our special case of a real quadratic number field F/Q and its character χ D as follows: our F/Q is a subfield of Q(ζ D )/Q, where D is the discriminant of F. Hence we get Spec(Q(ζ D )) → Spec(F). The abelian Galois module define Q(0)χ D as the direct Spec(Q(ζ D )) decomposes by the characters, and we summand, corresponding to the projector |(Z/D1 Z)∗ | σ ∈Gal(Q(ζ D )/Q) χ D (σ ) · σ,. −1 where we use that χ D = χ D . (The general definition of Q(0)χ for an arbitrary Dirichlet character χ : (Z/DZ)∗ → E ∗ with values in a field E is given by the above projector with the inverse character χ −1 .) If we restrict this to Spec(F), we get Q(0)χ D as the direct summand, corresponding to the projector 21 (id−), where is the nontrivial element in Gal(F/Q). So we have Spec(F) as a two dimensional motive over Spec(Q), i.e. H 0 (Spec(F)) = Q(0) ⊕ Q(0)χ D . Let us again look at the realisations of Q(0)χ D . There we have (by loc. cit.) the l-adic realisations that are one dimensional Galois modules Ql (0)χ D , i.e. Gal(Q/Q) acts by χ D . For the Hodge-de Rham realisation we have that the Betti realisation Q(0)χ D,B is just Q(0). The Hodge structure of Q(0) ⊗ C is pure of type (0, 0), and the involution F∞ given by the complex conjugation. Furthermore, over C it becomes Q(0)χ D,B ⊗C and isomorphic to the de Rham realisation Q(0)χ D,d R ⊗C. As an appropriate basis of Q(0)χ D,d R we choose the Gauß-sum G(χ D ) of χ D compare [7],√ 6.4. In our case of the quadratic character (with D ≡ 1(mod 4)) this is G(χ D ) = D. But this is a very special case of a Dirichlet motive, in particular its values are yet in Q..
(20) Realisations of Kummer–Chern–Eisenstein-motives. 37. 2. The l-adic realisations of our motive We start with the calculation of the l-adic realisations of our Kummer–Chern– 2 (S, Q(1))(−)]. Eisenstein motive. We determine the action of Gal(Q/Q) on [HCHE,l For each l we get a two dimensional extension 2 0 → Ql (0)χ D → HCHE,l (S, Q(1))(−) → Ql (−1) → 0,. and we know that Gal(Q/Q) acts by the quadratic χ D on the left and by the inverse of the Tate character α on the right term. Hence there is a two dimensional Galois χ D (σ ) ∗ and the star representation Gal(Q/Q) → GL2 (Ql ), with σ → 0 α −1 (σ ). ∗ represents the extension class in Ext1MGAL (Ql (−1), Ql (0)χ D ). Now in accor− ∗,(l) ⊗Zl Ql . In the following sections I dance of Lemma 1.14 this group is F compute this scalar, which belongs to the l-adic realisations of the Kummer–Chern– Eisenstein motive. We know by Kummer theory that this gives rise to Kummer field extensions of F, and the associated Galois representations of our l-adic realisations above come exactly from this field extensions.. 2.1. Detection of the l-adic extension classes The first problem is to find an appropriate recipe that describes such an extension class. In our situation this can be done in the following manner. If we look at the dual motive (compare Lemma 1.11) we get extensions 2 (S, Q(1))(−)∨ → Ql (0)χ D → 0 0 → Ql (1) → HCHE,l. in Ext1MGAL (Ql (0)χ D , Ql (1)). Hence the Galois representation for the dual mod ) ∗ ule looks like σ → α(σ 0 χ D (σ ) . Recall by Sects. 1.2 and 1.3 that the top of the. extension Ql (0)χ D is generated by the first Chern class c1 (L) = c1 (L −1 1 ⊗ L 2) ∈ 2 2 Het,! (S × Q, Ql (1)) ⊂ Het´ (S × Q, Ql (1)), and note that we write again L (instead ´ of L −1 ) to keep the notation easy and we normalise as described in Remark 1.12. 2 (S, Q(1))(−)∨ sits in the cohomology with comFurthermore, the middle HCHE,l 2 (S × Q, Q (1)) - see loc. cit. And the bottom Q (1) comes from pact support Het,c l l ´ 1. the cohomology Het´ ( S × Q, R j∗ Ql /j! Ql ) twisted by Ql (1). Now along the general procedure to get such a group extension class, we must 2 (S ×Q, Q (1)) to c 2 (S ×Q, Q (1)) and find the action lift c1 (L) ∈ Het,! l 1 (L) ∈ Het,c l ´ ´ ofGal(Q/Q) on this lifting c (L). The extension class is then given by the cocycle 1 σ c1 (L) − σ c1 (L), σ ∈ Gal(Q/Q). This is an element in H 2 (S × Q, Ql (1)) et,c ´. that goes to zero in Het´2 (S×Q, Ql (1)), i.e. it is in the kernel, which is Ql (1). We note that the above cocycle σ c (L) − σ c1 (L) is in Ext1 (Ql (0)χ D , Ql (1)).This 1 MGAL.
(21) 38. A. Caspar. is just because ∈ Gal(F/Q) flips the two factors of L = L −1 1 ⊗ L 2 , i.e. · c1 (L) = −c1 (L). We observe that we have the first Chern class of a line bundle, which is defined over F, i.e. it is in the Galois invariant part Het´2 (S × Q,. Ql (1))Gal(Q/F) . This means that the Galois group Gal(Q/F) acts trivially on c1 (L), c1 (L) = c and in particular, σ c1 (L) = c1 (L), resp. σ 1 (L). Recall that (by Sect. 1.1) 2 (S × Q, Q (1)) come uniquely from the first Chern the above c1 (L i )’s in Het,! l ´ classes of the line bundles L i of modular forms on the toroidal compactified surface. S × Q, i.e. c1 (L i ) = c1 ( L i )| S×Q . So in order to lift our class c1 (L) ∈ H 2 (S × Q, Ql (1)) to H 2 (S × Q, Ql (1)), we can lift the class c1 (. L) in et,! ´. et,c ´. 2 (S × Q, Q (1)). The following lemma shows that those Het´2 (. S × Q, Ql (1)) to Het,c l ´ two liftings are equal.. 2 (S × Q, Q (1)) and lift this to c Lemma 2.1. Start with c1 (L) ∈ Het,! l 1 (L) ∈ ´ 2 2. Het,c (S × Q, Ql (1)). Let c1 ( L) ∈ Het´ ( S × Q, Ql (1)) be the Chern class of the ´ extended line bundle. L. Then c L) via H 2 (S × Q, Ql (1)) → 1 (L) maps to c1 (. et,c ´. Het´2 (. S × Q, Ql (1)), i.e. c1 (. L) lifts to c 1 (L).. 2 (S × Q, Q (1)) Proof. Again according to [10], Lemma 2.2, we know that Het,! l ´ is isomorphic to the image of the restriction map Het´2 (. S × Q, Ql (1)) → Het´2 (S × Q, Ql (1)). And moreover, we have that c1 (. L) ∈ H 2 (. S × Q, Ql (1)) is the unique et ´. lift of c1 (L) ∈ Het´2 (S × Q, Ql (1)) - see Corollary 1.5.. . Our next goal is the construction of a diagram, which contains all cohomology groups that play along. For this we turn over to finite coefficient µl n , i.e. we conn c1 / H 2 (. sider the map c1(l ) that is given by Het´1 (. S × Q, Gm ) S × Q, Zl (1)) et ´ QQQ QQQ QQQ c1 mod l n QQQ (l n ) Q( c1 H 2 (. S × Q, µl n ). et ´ (l n ). 2. and we get the class c1 ( L) ∈ Het´ ( S × Q, µl n ), and furthermore we observe that (l n ) L) = lim c1 (. L) ∈ Het´2 (. S ×Q, Zl (1)). We know (Lemma 1.11) that the bottom c1 (. ← − n Ql (1) of our extension comes from Het´1 (. S ×Q, R j∗ Ql /j! Ql )⊗Ql (1), and the l-adic S∞ × Q, µl n ) ⊗ Ql = version of our Lemma 1.13 gives the isomorphism lim Het´1 (. ← − n Het´1 (. S × Q, R j∗ Ql /j! Ql ) ⊗ Ql (1). To construct the diagram, we start with the open. S∞ × Q into the toroidal embedding j of S × Q and the closed embedding i of. S × Q with compactification. S × Q. We have the following diagram of sheaves on. exact rows and columns..
(22) Realisations of Kummer–Chern–Eisenstein-motives. 0. 39. 0. 0. / j! j ∗ Gm,. / j! j ∗ Gm,. / j! j ∗ µl n ,. S S S LLL LLL LLL α L% α1 / Gm,. / Gm,. / µl n ,. S S S LLL LLL L β LLL % ∗G / i ∗ i ∗ µl n ,. / / i i i i ∗ Gm,. ∗ ∗ S m, S S. 0. 0. 0. 0. 0. /0. /0. /0. 0. where in each row sits a Kummer sequence. In the next lemma we take a look at the cohomology of these sheaves. We abbreviate SQ := S × Q, etc. Lemma 2.2. The above diagram of sheaves on. SQ gives rise to the following diagram of cohomology groups with Galois equivariant maps and exact rows and columns. /. 1 (. Het SQ , Gm ) ´. 0. . 1 (. Het S∞,Q , µl n ) ´. /. 1 (S , G ) Het,c m Q ´. /. 0. . 1 (. S∞,Q , µl n ) Het ´. . 2 (S , µ n ) Het,c Q l ´. /. . . 1 (. Het S∞,Q , i ∗ Gm,. S) ´. /. /. 1 (. Het SQ , Gm ) ´. . 2 (S , G ) Het,c m Q ´. /. . . 2 (S , µ n ) Het,c Q l ´. /. 1 (. Het SQ , Gm ) ´. 1 (. Het S∞,Q , i ∗ Gm,. S) ´. . /. 1 (S , G ) Het,c m Q ´. /. . 2 (. Het SQ , µl n ) ´. /. . 1 (. Het S∞,Q , i ∗ Gm,. S) ´. . /. /. 2 (S , G ) Het,c m Q ´. . 2 (. Het SQ , Gm ) ´. . 2 (. Het S∞,Q , µl n ) ´. . 0. i (S × Q, G ) and Proof. By definition we have Het´i (. S × Q, j! j ∗ Gm,. m S ) = Het,c ´ i i ∗. Het´ ( S × Q, j! j µl n ,. (S × Q, µl n ). For µl n one also knows Het´i (. S× S ) = Het,c ´ i (. i (. ∗G Q, i ∗ i ∗ µl n ,. ) = H S ×Q, µ ). But for G we only know H S×Q, i i n ∞ m ∗ l S m,. S) et ´ et ´ i ∗. Het´ ( S∞ ×Q, i Gm,. S ). The compact surface S is simply connected (cf. [8], IV.6), 1. i.e. H ( S × Q, µl n ) vanishes (cf. also [10], Proposition 5.3). et ´.
(23) 40. A. Caspar. The l-adic version of Lemma 1.13 tells us that we can find our dual extension in the above big diagram. Let 2 0 → lim Het´1 (. S∞ × Q, µl n ) ⊗ Ql → lim Het,c (S × Q, µl n ) ⊗ Ql → Im( f 1 ) → 0 ← − − ´ f1 ← n. n. be the short exact sequence, coming from the above diagram in Lemma 2.2. Then 2 (S, Q(1))(−)]∨ sits inside this extension. the l-adic realisation [HCHE,l 2.2. Galois action on the liftings The idea to describe the Galois action is based on the above big diagram in Lemma (l n ) 2.2. It relates the Chern class c1 (. L) ∈ Het´2 (. S × Q, µl n ) of the line bundle 1. S∞ × Q. To L ∈ Het´ ( S × Q, Gm ) and the restriction of L to the boundary. get this restriction. L|. , we compose the above “restriction map” in the diaS∞ ×Q 1 1 1 (i ∗ ) : ∗. gram R : Het´ ( S × Q, Gm ) → Het´ ( S∞ × Q, i Gm,. S ) with the map Het ´ 1 1 ∗ Het´ (. S∞ × Q, i Gm,. (. S∞ × Q, Gm ), which is induced by the morphism S ) → Het ´ ∗ i G → G of sheaves on. S∞ × Q. Now we add to the bottom part of the m, S. m, S∞. diagram in Lemma 2.2 the map Het´1 (i ∗ ), that is SQ , Gm ) Het´1 (. Het´1 (. S∞,Q , µl n ). Het´1 (. S∞,Q , µl n ). / H 1 (. SQ , Gm ) et ´. R. . / H 1 (. S∞,Q , i ∗ Gm,. S) et ´ . κ. 1 (i ∗ ) Het ´. / H 1 (. S∞,Q , Gm ) et ´. κ. R. . / H 1 (. S∞,Q , i ∗ Gm,. S) et ´ . 1 (i ∗ ) Het ´. / H 1 (. S∞,Q , Gm ). et ´. We note that the restriction to the boundary factorises into Het´1 (i ∗ ) ◦ R, i.e. 0. (Het´1 (i ∗ )◦ R)(. L) =. L|. S∞ ×Q . By Sect. 1.1 we know L|. S∞ ×Q ∈ Gm,Q = Pic ( S∞ × −2 Q) ⊂ Het´1 (. S∞ × Q, Gm ) and even more. L|. S∞ ×Q = ε . This restriction class. is the obstruction of the triviality of the extension. In the case it would L|. S∞ ×Q. 1 (. vanish, then the lifting c S × Q, Gm ), 1 (L) would come from an element in Het,c ´ i.e. it would be Galois invariant. To prepare the proof of Theorem 2.5 below, we must investigate some diagram chases in the diagram of Lemma 2.2. This exhibits how the Galois action 2 on [HCHE,l (S, Q(1))(−)]∨ and the above Kummer sequence in the cohomology (l n ) are linked. We start with c1 (. L) ∈ Het´2 (. S × Q, µl n ) in the big diagram. This S × Q, Gm ). If we send. L to Het´1 (. S∞ × comes from the line bundle. L ∈ Het´1 (. 2 1 ∗ ∗. Q, i Gm,. ( S × Q, µl n ) to Het´ ( S∞ × Q, i Gm,. S ), we get a map from Het S ). But ´ this is not precisely correct, as there is some arbitrariness caused by the liftings. To understand this we consider the map Het´2 (α1 ⊕ α2 ) : Het´2 (. S × Q, µl n ) →.
(24) Realisations of Kummer–Chern–Eisenstein-motives. 41. Het´2 (. S × Q, Gm ) ⊕ Het´2 (. S∞ × Q, µl n ), which is induced by the right bottom corner of the big diagram. The kernel Ker(Het´2 (α1 ⊕ α2 )) is the subgroup of those classes that are Chern classes of line bundles and that have furthermore trivial Chern class n on the boundary. S∞ × Q. Note that by construction c1(l ) (. L) ∈ Ker(Het´2 (α1 ⊕ α2 )). S × Q, Gm ) → And secondly, consider the map Het´1 (β) defined by R ◦ κ : Het´1 (. i (−) indicates that these morphisms actuS∞ × Q, i ∗ Gm,. ). The notation H Het´1 (. S et ´ ally come from morphisms of sheaves in the above diagram. We want to construct two maps Ker(Het´2 (α1 ⊕ α2 )) → Coker(Het´1 (β)). Let us write the whole diagram again. /. 0. . /. 1 (S , G ) Het,c m Q ´. /. 0. . /. . λ. /. 1 (S , G ) Het,c m Q ´. . /. 1 (. Het SQ , Gm ) ´. 1 (. Het S∞,Q , µl n ) ´. 2 (S , µ n ) Het,c Q l ´. /. 1 (. Het SQ , Gm ) ´. . . 1 (. S∞,Q , µl n ) Het ´. . 2 (S , µ n ) Het,c Q l ´. /. . w. 1 (. Het S∞,Q , i ∗ Gm,. S) ´. /. . 1 (. Het S∞,Q , i ∗ Gm,. S) ´. /. . . LLL LLL LLL κ L&. 2 (S , G ) Het,c m Q ´. . R. /. κ. e. 1 (β)) Coker(Het ´. j. . 1 (. Het S∞,Q , i ∗ Gm,. S) ´. ?. /. . 2 (S , G ) Het,c m Q ´. . 2. H (α1 ) et ´ 2 (. / Het´2 ( SQ , Gm ) Het SQ , µl n ) ´. eJJJ JJJ JJJ J2 R. λ R. 1 (. Het SQ , Gm ) ´. H 2 (α2 ) et ´. . 2 (. Het S∞,Q , µl n ) ´. 2 (α ⊕ α )) Ker(Het 1 2 ´. . 0 . Since we divide out the image of Het´1 (β) we know that : Ker(Het´2 (α1 ⊕ α2 )) → Coker(Het´1 (β)) is a well-defined map. But in the diagram we have the dotted arrows S∞ × Q, i ∗ Gm,. λ from Ker(Het´2 (α1 ⊕ α2 )) to Het´1 (. S ), where that sits now in the right top corner. If we compose this with the map κ, induced by the Kummer map, we get a morphism κ ◦ λ, which lands in the same quotient Coker(Het´1 (β)) of S∞ × Q, i ∗ Gm,. Het´1 (. S ) as the . The following lemma shows that these are equal. Lemma 2.3. With the above notations, the two maps κ ◦ λ and are equal. Proof. First we have to see that κ ◦ λ is indeed well-defined. For this regard again the above diagram of cohomology groups. We know that (l n ) c1 (. L) ∈ Ker(Het´2 (α1 ⊕ α2 )) ⊂ Het´2 (. S × Q, µl n ), or any other element, comes 2 from a lifting in Het,c (S × Q, µl n ). If we send this lifting via the natural co´ 2 (S × Q, G ), the image cannot vanish. Otherwise boundary morphism to Het,c m ´. 1 (S × Q, G ), and the lifting would be the class, coming from an element in Het,c m ´.
(25) 42. A. Caspar. this would contradict by exactness the non-vanishing of the boundary class of. L in 2 (S×Q, G ) Het´1 (. S∞ ×Q, i ∗ Gm,. ). But we know that we can lift this element in H m S et,c ´ (l n ). 2 (α )). So we are in the right to Het´1 (. S∞ × Q, i ∗ Gm,. ), since c ( L) ∈ Ker(H 1 S 1 et ´ top corner. Apply the Kummer map κ and we are done. We have to take into considerations the non-uniqueness of the liftings. The first step does not effect any2 (S × Q, µ n ) is unique up to H 1 (. S∞ × Q, µl n ) and thing, since the lift to Het,c l ´ et ´ S∞ × Q, i ∗ Gm,. this is the kernel of the Kummer map. The second lift to Het´1 (. S) 1. is unique up to elements in H ( S × Q, Gm ). Here we use that we actually want et ´. to land into Coker(Het´1 (β)), hence this ambiguity is divided out. Now I show that κ ◦ λ and the have to coincide. Since the rows in the diagram are exact, it is sufficient to consider the map Ker(Het´2 (α )) → Coker(Het´1 (β)), where S × Q, Gm ) . Then we have a Ker(H 2 (α )) = Ker H 2 (S × Q, µl n ) → H 2 (. et ´. et,c ´. et ´. “snake diagram” 2 (α )) Ker(Het ´. /. 1 (. Het SQ , Gm ) ´. 1 (. Het SQ , C (α )) ´. /. . H 2 (α ) et ´ 2 (. Het SQ , Gm ) ´. 2 (S , µ n ) Het,c Q l ´. r tt rrr tt H 1 (β) rr H 2 (α ) t t et ´ et ´ t H 2 (v) rr H 1 (u) t r 1 t r y z et ´ et ´ H (β) et ´ 1 1 2 2 ∗. / Het´ ( S∞,Q , i Gm, S ) / Het´ ( SQ , K(β)) / Het´ ( SQ , Gm ) Het ( SQ , Gm ) ´. /. . 1 (β)) Coker(Het ´. which is induced by the following exact diagram of sheaves on. S × Q, 0. / j! j ∗ µl n ,. S. 0. / K(β). α. / Gm,. S. / C(α ). /0. v. u. / Gm,. S. β. / i ∗ i ∗ Gm,. S. / 0.. With the notation C(α ) for the cokernel of α , and K(β) for the kernel of β. Now one just imitate the proof of the snake lemma to get the well-defined map Ker(Het´2 (α )) → Coker(Het´1 (β)). The diagram does not fulfill exactly the assumption of the snake lemma, but here we are in a special case, which makes the things work. The “snake diagram” explains to us the two maps from Ker(Het´2 (α1 ⊕ α2 )) to Coker(Het´1 (β)) via the left bottom corner and the right top corner of the big diagram. First of all, this is due to the fact that there are two morphisms Het´1 (. S ×Q, C(α )) → 2 1 S ×Q, K(β)). One factorisation is given if we start with Het´ (. S ×Q, C(α )) and Het´ (. 2 (S × Q, µ n ) go via the natural coboundary map to Het´2 (S × Q, j! j ∗ µl n ,. l S ) = Het,c ´ and compose this with the morphism induced on the second cohomology.And for.
(26) Realisations of Kummer–Chern–Eisenstein-motives. 43. the other round we start with Het´1 (. S × Q, C(α )) and go via the induced map to 2 (. S × Q, i ∗ i ∗ Gm,. S × Q, K(β)). Het´1 (. S ) and apply the other coboundary map to Het ´ Now we are left to show that we get in the first case and κ ◦ λ in the second one, so eventually by the commutativity they have to be equal. To do so we observe that u and v in the above diagram can be factorise in two ways: for we have 0. / j! j ∗ µl n ,. S. α. / Gm,. S. / C(α ). /0 v. u. 0. / µl n ,. S. 0. / K(β). 0. / j! j ∗ µl n ,. S. α1. / Gm,. S. /0. / i ∗ i ∗ Gm,. S. / 0.. / C(α ). /0. / Gm,. S. / Gm,. S. β. and for λ α. / Gm,. S. v. u. 0. / j! j ∗ Gm,. S. / Gm,. S. 0. / K(β). / Gm,. S. β. / i ∗ i ∗ Gm,. S. /0. / i ∗ i ∗ Gm,. S. / 0.. This is just due to the commutativity of our initial diagram of sheaves.. . We come back to our situation with the (l n ). L) ∈ Ker(Het´2 (α1 ⊕ α2 )) and Corollary 2.4. Let Het´1 (i ∗ ) be as above. Let c1 (. (l n ) L)) = let ε ∈ O∗F be as fixed in the very beginning. Then Het´1 (i ∗ ) (κ ◦ λ)(c1 (. (l n ) Het´1 (i ∗ ) (c1 (. L)) = ε−2 . Proof. First recall the construction of λ (Lemma 2.3), i.e. we have the following piece in the big diagram / H 1 (. S × Q, Gm ) S × Q, Gm ) Het´1 (. et ´ TTTT 1 (β) H TTTTet´ TTTT TTT) 1 1 (. ∗ / Het´ (. S∞ × Q, i Gm,. ) H S × Q, i ∗ Gm,. ∞ S S) κ et ´ 2 (S × Q, G ) Het,c m ´.
(27) 44. A. Caspar. 2 (S × Q, G ) to H 1 (. The lift from Het,c S∞ × Q, i ∗ Gm,. m S ) is well-defined up to an ´ et ´ 1. image coming from H ( S × Q, Gm ), but now we choose this such that the lift goes et ´. (l ) L)). We can do this, since we know that the image of the lift via under κ to (c1 (. (l n ) L)) ∈ κ is also only well-defined up to the image of Het´1 (β). We know that (c1 (. 1 1 ∗. ( S × Q, Gm ). Since Het´ ( S∞ × Q, i Gm,. S ) comes from the line bundle L ∈ Het ´ (l n ) 1 ∗ L)) the restriction map factorises into H (i ) ◦ R, we have that H 1 (i ∗ ) (c (. n. et ´. et ´. 1. ×Q, and we computed this in Lemma equals the restriction ofn L to the boundary S∞ −1 (l ). 1 ∗ −2. |. = L ⊗ L 1.2, i.e. Het´ (i ) (c1 ( L)) = L|. 1 2 S∞ ×Q S∞ ×Q = ε . Let us now turn over to the Galois action and our main theorem. Recall that χ D (σ ) ∗ , our big goal is to determine the Galois representations σ → 0 α −1 (σ ) which come from the l-adic realisations of our Kummer–Chern–Eisenstein motive 2 (S, Q(1))(−)]. The above discussion gives the star ∗, and shows that it comes [HCHE indeed from a Kummer extension of our real quadratic field F. We sum this up in the ε := Theorem 2.5. Let ε ∈ O∗F be as fixed in the very beginning and define. −1. 1. 2 (S, Q(1))(−)] is. ε. ε− 2 ζ F (−1) . Then [HCHE,l The corresponding l-adic Galois representation the Kummer field is induced by ∞ √ l χ D (σ ) τ ε (σ )α −1 (σ ) ∞ attached to. ε, i.e. σ → , where τ (σ ) extension F. ε, ζl −1. is defined by. σ. ∞√ l. ε √ l∞. ε. 0. =. α. (σ ). τ (σ ) ζl ∞ ε .. Proof. By Kummer theory we know that the first assertion follows from the second one. We take off with the cocycle σ c 1 (L) − c 1 (L), where the class c 1 (L) is the 2. lifting of c1 ( L) ∈ H ( S × Q, Ql (1)) (compare Lemma 2.1). That cocycle gives et ´. 2 the extension class of the dual Galois module [HCHE, (S, Ql (1))(−)]∨ . We know et ´ that the dual motive sits in the following sequence (see Lemma 1.13) 2 S∞ × Q, µl n ) ⊗ Ql → lim Het,c (S × Q, µl n ) ⊗ Ql → Im( f 1 ) → 0, 0 → lim Het´1 (. ← − ´ ← − n. n. where this sequence itself is part of the big diagram of Lemma 2.2. Now we apply the diagram chases in this diagram (loc. cit.). For start, we go again back to finite (l n ) coefficients, i.e. consider the class c1 (. L) ∈ Het´2 (. S × Q, µl n ), and we write the action now multiplicatively. Furthermore, we should remind ourselves that L is a certain power of L - see Remark 1.4. For the sake of simplicity, write in the fol −1 (l n ) (l n ) lowing for this power again L. So we get the element σ c1 (L) c1 (L) 2 (S × Q, µ n ), which vanishes under the restriction map to H 2 (. in Het,c S × Q, µl n ), l ´ et ´ 1. i.e. it is in the kernel, and this is H ( S∞ × Q, µl n ). By construction of λ, more et ´.
(28) Realisations of Kummer–Chern–Eisenstein-motives. 45. precisely by the commutativity of S∞ × Q, µl n ) Het´1 (. / H 1 (. S∞ × Q, i ∗ Gm,. S) et ´. 2 (S × Q, µ n ) Het,c l ´. / H 2 (S × Q, Gm ) et,c ´ (obs). we conclude furthermore that up to an l n -torsion element tl n that we have the equality −1 n n −1 (l n ) (l n ) σ c1 (L) = σ λ c1(l ) (. L) λ c1(l ) (. L) . c1 (L) (obs) The ambiguity given by tl n comes from the line bundles in Het´1 (. S × Q, Gm ) that map via R to n −1 n (l ) (l ) L) λ c1 (. L) . σ λ c1 (. (obs). I call this torsion element tl n with (“obs” := “obstruction”). These line bunS × Q, Gm ). Now dles, causing the trouble are also l n - torsion elements in Het´1 (. S × Q, Gm ) is of finite rank. (Note that Pic0 (. S × Q) vanishes.) Thus we can Het´1 (. (obs). only have finitely many tl n ’s in the image. Assume that these line bundles are all of l k -torsion. Otherwise, we choose a suitable power, i.e. take k 0. So let us consider again our situation with l n -coefficients for n ≥ k, −1 n n −1 (obs) (l n ) (l n ) (l ) (l ) σ c1 (L) = σ λ c1 (. L) λ c1 (. L) · tl k . c1 (L) Now if we raise this to l k -th power, the obstruction vanishes, i.e. −1 n−k n−k −1 n−k ) n−k ) (l (l σ c1 (L) (L) = σ λ c1(l ) (. L) λ c1(l ) (. L) . c1 Hence in the limit −1 ∞ ∞ −1 ∞) ∞) (l (l (l ) (l ) σ c1 (L) = σ λ c1 (. L) λ c1 (. L) , c1 (L) i.e. −1 −1 σ c c = σ λ c1 (. L) λ c1 (. L) . 1 (L) 1 (L) n n By Corollary 2.4 we know (κ ◦ λ) c1(l ) (. L) = c1(l ) (. L) and therefore −1 σ λ c1 (. L) λ c1 (. L) =σ. −1 l∞ l∞ c1 (. L) c1 (. L) ,.
(29) 46. A. Caspar. √ ∞ where l − indicates the preimage κ −1 of the Kummer map. We conclude that we have the equality −1 −1 l∞ l∞ . (L) c (L) = σ c ( L) c ( L) σ c 1 1 1 1 S∞ × Q, µl n ⊗ Ql . in lim Het´1. ← − n This group is the bottom of our extension Het´1 (. S × Q, R j∗ Ql /j! Ql ) ⊗ Ql (1) Ql (1). Now we have to determine this element in Ql (1). For this consider again the diagram (Sect. 2.2) / H 1 (. S × Q, Gm ) et ´. S × Q, Gm ) Het´1 (. R. Het´1 (. S∞ × Q, µl n ). / H 1 (. S × Q, i ∗ Gm,. ∞ S) et ´. R. κ. 1 (i ∗ ) Het ´. Het´1 (. S∞ × Q, µl n ). / H 1 (. S × Q, Gm ) ∞ et ´ O ? Gm. / H 1 (. S × Q, i ∗ Gm,. ∞ S) et ´ 1 (i ∗ ) Het ´. κ. / H 1 (. S × Q, Gm ) ∞ et ´ O / G? m. If we apply the above Corollary 2.4, we get −1 l −1 l n n ln ln (l n ). (l n ). 1 ∗ Het´ (i ) σ (c1 ( L)) (c1 ( L)) ε−2 ε−2 , =σ where σ. √ √ √ l n −2 l n −2 −1 l n −2 ε ε ∈ F ε , ζl n is an element in the Kummer. extension of degree l n attached to ε−2 . Moreover, this is exactly the Galois cocycle attached to this field extension. Now the two Kummer maps above have the S∞ × Q, µl n ), and hence we see that in the limit lim same kernel, which is Het´1 (. ← − n −1 ∞ ∞ −1 ∞ √ l l (c1 (. that σ c c = σ l (c1 (. L)) L)) =σ ε−2 1 (L) 1 (L) −1 ∞ √ l ε−2 . All this yields eventually to the Galois representation σ → ∞√ l σ ε−2 τ −2 (σ ) α(σ ) τε−2 (σ )α −1 (σ ) , where l ∞√ = ζl ∞ε . 0 χ (σ ) D. ε−2. 2 (S, Q(1))(−)] We know that the Galois cocycle for the actual module [HCHE,l 1 (Ql (−1), Ql (0)χ D ) is given by σ −c1 (L) − −c1 (L) , i.e. we can in Ext. MGAL. play the same game with the inverse. But here we have to be careful as we have to normalise the generator after dualising, see Remark 1.12, i.e. we have to multiply.
(30) Realisations of Kummer–Chern–Eisenstein-motives. with − 4ζ F 1(−1) . Therefore we get (ε2 ) )α −1 (σ ) σ → χ D (σ ) τ ε (σ−1 . 0. α. − 4ζ. 1 F (−1). 47. =. ε and our Galois representation is . (σ ). Remark 2.6. Since the restriction of L 1 ⊗ L 2 is trivial on. S∞ × Q (Lemma 1.2) and its Chern class generates the (+1)-eigenspace Q(0) (Remark 1.10), we get a three dimensional representation σ →. 1 0 0 0 χ D (σ ) τ ε (σ )α −1 (σ ) −1 0 0 α (σ ). , that is induced by. 2 0 → Ql (0) ⊕ Ql (0)χ D → HCHE,l (S, Q(1)) → Ql (−1) → 0.. Let us note that this is the realisation of the Kummer motive K . ε attached to 2 our. ε, i.e. K . εl = [HCHE,l (S, Q(1))(−)] =. ε. 3. The Hodge-de Rham Realisation 2 (S, Q(1))(−)] of the In this chapter, we compute the extension class [HCHE,∞ 1 Hodge-de Rham realisation, which is an element in ExtMHd RQ (Q(−1), Q(0)χ D ). In Sect. 1.4 we give the recipe that describes such an element. We must understand the sections s B and sd R in our setting. Recall that Corollary 1.8 is the starting point of the construction of the motive, i.e. we have the sequence 0 → H!2 (S, Q(1)) → H 2 (S, Q(1)) → Q(−1) → 0. And the top Q(−1) is given by the toroidal comS, R j∗ Q/j! Q) ⊗ Q(1). Now we consider the complex points pactification. S as H 2 (. S(C) and let ∂. S∞ (C) be the boundary of a suitable neighbourhood of. S∞ (C). By [10], 5, we have the exact sequence. S∞ (C), Q) → Hc2 (S(C), Q) → H 2 (S(C), Q) → H 2 (∂. S∞ (C), Q) → 0. 0 → H 1 (∂. For the vanishing of H 1 (S(C), Q), resp. Hc3 (S(C), Q) see Lemma 1.6. So the S∞ (C), Q) is isomorphic to H 2 (. S(C), R j∗ Q/j! Q). By [1], I.5, or cokernel H 2 (∂. [10], 3 and 5, we know that ∂. S∞ (C) is isomorphic to the boundary ∂ S(C) B S of the Borel–Serre compactification S(C) → S(C) B S . This put us in the position to describe the sections s B and sd R by Eisenstein cohomology, and that is done in the next section.. 3.1. The extension class as eisenstein class All this bases on Harder’s notes [15], see also [9] and [12]. Consider the short exact sequence 0 → H!2 (S(C), C) → H 2 (S(C) B S , C) → H 2 (∂ S(C) B S , C) → 0, which now comes from the Borel–Serre compactification S(C) → S(C) B S . The homotopy equivalence between S(C) and S(C) B S induces the isomorphism between H 2 (S(C), C) and H 2 (S(C) B S , C). For simplicity, we restrict the general setting concerning Eisenstein cohomology to our situation, i.e. to the case of our group G = Res F/Q (GL2 /F) and the cohomology in degree two H 2 (S(C), C).
(31) 48. A. Caspar. with the constant coefficient system C. Let us start with the de Rham theorem ([12], Satz 3.7.8.2) H ∗ (∂ S(C) B S , C) H ∗ (g, K ; C ∞ (B(Q)\G(A))), with the Lie algebra g = Lie(G ∞ ) of G ∞ , the group K = K ∞ and the standard Borel subgroup B ⊂ G, where the notation are as in the introduction. We define (as in ∞ ∞ by Eis(ψ) := [12], 5.2, or [9], 2) Eis : C (B(Q)\G(A)) → C 2(G(Q)\G(A)) {g → a∈B(Q)\G(Q) ψ(ag)}. This induces a map H (g, K ; C ∞ (B(Q)\G(A))) → H 2 (g, K ; C ∞ (G(Q)\G(A))) and a section H 2 (∂ S(C) B S , C) → H 2 (S(C), C) of the restriction map to the boundary, see loc. cit. To describe this in more detail, we go back to the Lie algebra cohomology. We have the isomorphism (again by the de Rham theorem) Hom K (2 (g/k), C ∞ (B(Q)\G(A))) =
(32) 2 (B(Q)\G(A)/K ∞ K f ) =
(33) 2 (∂ S(C) B S ), where k := Lie(K ∞ ) is the Lie algebra of K ∞ . The product G(R) GL2 (R) × GL2 (R) induces a Hodge decomposition 2 (g/k) = 2 (g1 /k1 ) ⊕ (1 (g1 /k1 ) ⊗ 1 (g2 /k2 )) ⊕ 2 (g2 /k2 ) for the exterior algebra 2 (g/k). Now we can choose a dual basis ω+, j , ω−, j of 1 (g j /k j ), which corresponds to dz j , dz j or d x j ± idy j , j = 1, 2, see for example [10], 3, or [14], 4.3.3. To simplify the notation, we denote δz j := ω+, j and δz j := ω−, j , this means for the cohomology classes [dz j ] = [δz j ]. Since {dz 1 ∧ dz 2 , dz 1 ∧ dz 2 , dz 1 ∧ dz 2 , dz 1 ∧ dz 2 } generate the same C-vector space as {d x1 ∧ d x2 , d x1 ∧ dy2 , d x2 ∧ dy1 , dy1 ∧ dy2 }, we get in same manner elements {δx1 ∧δx2 , δx1 ∧δy2 , δx2 ∧δy1 , δy1 ∧δy2 } in Hom K (2 (g/k), C ∞ (B(Q)\G(A))), which form a dual basis. Now we have to consider the finite part of the cohomology of the boundary H 2 (∂ S(C) B S , C). Since we know that this is one dimensional, the finite adelic part is just generated by a normed standard spherical function ψ f , i.e. ψ f (1) = 1. More precisely, we have the Lemma 3.1. The cohomology class [(δx1 ∧ δx2 ) ⊗ ψ f ] ∈ H 2 (∂ S(C) B S , C) of (δx1 ∧ δx2 ) ⊗ ψ f ∈ Hom K (2 (g/k), C ∞ (B(Q)\G(A))) generates the cohomology H 2 (∂ S(C) B S , C) of the Borel-Serre boundary ∂ S(C) B S . Proof. See e.g. [9], Proposition 1.1.. . Let us denote this generator ω0 by ω0 := δx1 ∧ δx2 . Then we have Lemma 3.2. Let ω = δx1 ∧ δx2 + α1 · δx1 ∧ δy2 + α2 · δx2 ∧ δy1 + β · δy1 ∧ δy2 be a form, which defines a closed form ω ⊗ ψ f ∈
(34) 2 (∂ S(C) B S ). Then its cohomology class [ω ⊗ ψ f ] in H 2 (∂ S(C) B S , C) equals [ω0 ⊗ ψ f ], i.e. it is independent of the coefficients α1 , α2 and β. If we apply the Eisenstein operator to ω ⊗ ψ f , we get a closed form Eis(ω ⊗ ψ f ) ∈ Hom K (2 (g/k), C ∞ (G(Q)\G(A))), and the class [Eis(ω ⊗ ψ f )] ∈ H 2 (S(C), C) restricted to the boundary ∂ S(C) B S is again [Eis(ω ⊗ ψ f )]|∂ S(C) B S = [ω ⊗ ψ f ] ∈ H 2 (∂ S(C) B S , C), i.e. independent of the coefficients α1 , α2 and β. And Eis is indeed a section for the restriction map. Proof. See e.g. [9], Theorem 2.1.. . Let us compute the Eisenstein class. Theorem 3.3. Let ω = δx1 ∧ δx2 + α1 · δx1 ∧ δy2 + α2 · δx2 ∧ δy1 + β · δy1 ∧ δy2 be a closed two form ω ∈
(35) 2 (∂ S(C) B S ). Then [Eis(ω ⊗ ψ f )] ∈ H 2 (S(C), C) is.
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