BEILINSON-KATO ELEMENTS IN K 2 OF MODULAR CURVES

Transcription

1 BEILINSON-KATO ELEMENTS IN K 2 OF MODULAR CURVES FRANÇOIS BRUNAULT Abstract. This article investigates explicit linear dependence relations in the K 2 -group of modular curves. In particular, it is shown that the Beilinson-Kato elements in K 2 of the modular curve Y N) satisfy the Manin relations when N is not divisible by 3. Similar results are obtained for the modular curves X 1 N) and X 0 N) when N is prime. Finally we exhibit explicit generators of K 2, assuming the Beilinson conjecture. Introduction Let X be a smooth projective curve over Q, and Lh 1 X), s) be the associated L-function. A very special case of Beilinson s conjectures predicts that the special value Lh 1 X), 2) can be expressed in terms of a suitable regulator map on the algebraic K-group K 2 X) see [8] for a nice overview and a precise statement of this conjecture). Beilinson proved a part of his conjecture in the case where X is a modular curve [18]. Beilinson s work was also partially anticipated by Bloch, who studied the particular case of CM elliptic curves [1]. Despite these profound results, the K-group itself remains very mysterious. There s quite an art to constructing special elements in this group and, as soon as the genus of X is not zero, it is not even known whether K 2 X) Z Q is a finite dimensional Q-vector space. I showed in [5] how Beilinson s theorem can be made explicit in the case of the modular curve X 1 N). This raised the question of determining linear dependence relations in the group K 2 X 1 N)) [5, 8]. The main point of this article is to make these relations explicit. Let Y N) be the open modular curve associated to the congruence subgroup ΓN). By taking cup-products of Siegel units, there is a natural map 1) ρ : M 2 Z/NZ) K 2 Y N)) Q. Date: September 4, Mathematics Subject Classification. Primary: 19F99; Secondary: 11F67, 11G18, 11G40. Key words and phrases. K-theory, modular curves, modular symbols, L-functions, Beilinson conjectures. 1

2 2 F. BRUNAULT Under the hypothesis that N is not divisible by 3, I show that ρ satisfies the Manin relations Theorem 1.4). This was also proved by Goncharov [9] using a different method, and his proof works for all N. Thus ρ can be seen as a Manin symbol or modular symbol) with values in K 2 Y N)) Q. This result is similar to constructions of Borisov and Gunnells [2, 3] and Paşol [17] in the case of modular forms. In these works, the product of two Eisenstein series plays the role of the cup-product. I then use this result to study the case of the modular curves X 1 p) and X 0 p), where p is prime Theorems 4.2, 4.4 and 4.8). In particular, the Beilinson conjecture implies that the elements so constructed span the vector space K 2 X 0 p)) Z Q, and I determine all the relations between them. Some questions would deserve further study. I do not know even conjecturally) whether the image of ρ spans K 2 Y N)) Q see Remark 1.7). In view of the arithmetic applications of Kato s Euler system [10], it would be also of interest to describe the action of Hecke correspondences on these elements, in the spirit of Merel s results for modular symbols [15]. 1. The Beilinson-Kato elements in K 2 Let us first state some standard facts on modular curves see [20, 13, 7, 11] for more detailed accounts). Let N 3 be an integer and Y N) be the modular curve classifying elliptic curves E with a level N structure, that is a basis e 1, e 2 ) of E[N] over Z/NZ. The curve Y N) is a smooth projective curve defined over Q, whose affine ring OY N)) contains the cyclotomic field Qζ N ) generated by ζ N := e 2iπ/N. The curve Y N) is not geometrically connected. Indeed, there is an isomorphism Y N)C) = Z/NZ) ΓN)\H), where H is the Poincaré upper half-plane and ΓN) SL 2 Z) is the congruence subgroup of matrices satisfying a b c d mod N). For any z H and λ Q, let us set q = e 2iπz and q λ = e 2iπλz. The curve Y N) has a smooth compactification XN) over Q which is obtained by adding on the cusps. The function field of XN) will be referred to by QXN)). It is naturally embedded into the function field of the compactification of ΓN)\H. There is also a natural inclusion of QXN)) into the field of formal Laurent series Qζ N )q 1/N )), by looking at the q- expansion Siegel units. Let us give the definition of Siegel units see [6, 10, 12] for further reference). The group of modular units of XN) will be denoted

3 BEILINSON-KATO ELEMENTS IN K 2 3 by O Y N)). In order to avoid torsion problems, Siegel units will always be considered in the Q-vector space O Y N)) Z Q. Let B 2 X) = X 2 X + 1 be the second Bernoulli polynomial. 6 Definition 1.1. For any α, β) Z/NZ) 2 {0, 0)} let us define 2) g α,β z) = q 1 2 B 2eα/N) n 0 ) 1 q n q eα/n ζ β N n 1 1 q n q eα/n ζ β N where α Z is the unique representative of α satisfying 0 α < N. By convention g 0,0 = 1. Thus g α,β is a holomorphic function on H. It is known that some power of g α,β in fact g 12N α,β ) is modular with respect to ΓN), and lies in O Y N)) [13, Chap 19 2]. Therefore g α,β is well-defined as an element of O Y N)) Q. Let G be the group GL 2 Z/NZ). It acts from the left on Y N), by the rule 3) a b E, e c d 1, e 2 ) = E, ae 1 + be 2, ce 1 + de 2 ) ) ) a b G. c d This induces on O Y N)) Q a right action of G. It turns out that G acts on the set of Siegel units. More precisely, we have [10, Lemma 1.7] 4) g α,β γ = g α,β) γ γ G). Since 1 G acts trivially on Y N), we get the relation g α, β = g α,β. Kubert and Lang proved that the Siegel units of level N generate O Y N)) Q [12] The construction of Beilinson and Kato. Let us consider the Quillen K-group K 2 Y N)), which enjoys a right action of G by functoriality. Beilinson constructed special elements in it using cup-products of modular units. This motivates the following definition. Definition 1.2. Let ρ be the map 5) ρ : M 2 Z/NZ) K 2 Y N)) Z Q s t {g u v s,t, g u,v }. Remark 1.3. Colmez [6, 1.4.2] constructed an algebraic distribution on M 2 Q Ẑ) with values in K 2, which generalizes Definition 1.2. I shall not use this more conceptual point of view in what follows.

4 4 F. BRUNAULT Let ε resp. σ, τ) be the image of in G resp. 0 1, ) 1 1 Theorem 1.4. The elements ρm) satisfy the following relations 6) ρεm) = ρm) and ρm) + ρσm) = 0 M M 2 Z/NZ)). Let us suppose further that 3 does not divide N. Then we have 7) ρm) + ρτm) + ρτ 2 M) = 0 M M 2 Z/NZ)). Remark 1.5. The Manin relations 6) and 7) have also been established by Goncharov [9, Corollary 2.17], without any assumption on the level N, using a different method. Remark 1.6. The Manin relations 6) and 7) are consistent with the formula of Kato [10, Thm 2.6] giving the regulator of z N = ρi). The element z N plays a prominent role in the construction of Kato s Euler system [10, 5]. Remark 1.7. It would be interesting to know whether the elements ρm) span the Q-vector space K 2 Y N)) Q. A related question is to determine whether K 2 Y N)) is generated by the symbols {u, v} with u, v O Y N)). Since K 2 Y N)) Q is in general not known to be finitedimensional, it is more reasonable to ask whether the Manin relations make up a complete set of relations between the elements ρm). A natural way to tackle this problem would be to compute the Beilinson regulator of ρm). However, the formula of Kato [10, Thm 2.6] seems to indicate that in general ρg) cannot span K 2 Y N)) Q. Proposition 1.8. For any M M 2 Z/NZ) the relations 6) hold. s t Proof. Let M =. We have u v ρεm) = {g s, t, g u,v } = {g s,t, g u,v } = ρm) and ρσm) = {g u, v, g s,t } = {g s,t, g u,v } = ρm), because of the relation g s, t = g s,t and the antisymmetry of the Milnor symbol. The relation 7) can be seen as an analogue of the Manin 3-term relation for modular symbols. The proof of this relation lies deeper, and will be given in the next two sections.

5 BEILINSON-KATO ELEMENTS IN K Weierstrass units For any z H, we let z, u) be the Weierstrass -function associated to the lattice Λ z = Zz + Z C. It is defined for u C Λ z. Definition 2.1. For any a = a 1, a 2 ) Z/NZ) 2 {0, 0)}, let us define 8) a z) = z, ã1z + ã ) 2 z H), N where ã 1 and ã 2 are any representatives of a 1 and a 2 in Z. We use these functions to construct the Weierstrass units. This classical construction is undertaken in [12, Chap 2 6]. We give some details for the sake of completeness. Theorem [12]). Let a, b, c, d be four nonzero elements of Z/NZ) 2 satisfying a ±b and c ±d. The function 9) defines an element of O Y N)). a b c d Proof. The function a is holomorphic on H and defines a modular form of weight 2 for the group ΓN). For any z H, we have a z) = b z) if and only if a = ±b. Thus a b )/ c d ) is well-defined and does not vanish on H. The fact that it belongs to QXN)) is a consequence of results of Shimura [19, 4], [20, Chap 6]). It essentially amounts to express a b )/ c d ) in terms of the x-coordinates of N-torsion points of the universal elliptic curve over Y N). The fact that 9) is a modular unit is proved in [12, Chap 2 Thm 6.1]. Now we express the Weierstrass units in terms of Siegel units. Once again this is done in [12, Chap 2 6]. Proposition 2.2. Let a, b, c, d be four nonzero elements of Z/NZ) 2 satisfying a ±b and c ±d. Then the following identity holds in O Y N)) Q 10) a b = g a+bg a b gc 2 gd 2. c d gag 2 b 2 g c+d g c d Proof. We start with the following classical formula from the theory of elliptic functions [13, Chap 18 Thm 2] σz, u + v)σz, u v) 11) z, u) z, v) = σ 2 z, u)σ 2 z, v) z H),

6 6 F. BRUNAULT where σ refers to the Weierstrass sigma function. For any a 1, a 2 ) Z 2, let us define in the same way as 8) σ a1,a 2 z) = σz, a 1z + a 2 ) z H). N We write abusively a = a 1, a 2 ) and b = b 1, b 2 ) for representatives of a and b in Z 2. The formula 11) can then be rewritten as a b = σ a+bσ a b. σaσ 2 b 2 Using the expression of σ as an infinite q-product [13, Chap 18 Thm 4], we get the following formula compare with [12, p. 29 and 51]) a b = 2iπ) 2 q b 1/N ζ b 2 N where γ is defined by 1 q n ) 4 n 1 γq, a + b)γq, a b) γ 2 q, a)γ 2 q, b) γq, a 1, a 2 ) = n 01 q n q a1/n ζ a 2 N ) 1 q n q a1/n ζ a 2 N ). n 1 Using the obvious notation for c and d, this gives a b = q b 1 d 1 )/N ζ b 2 d 2 N c d γq, a + b)γq, a b) γ 2 q, a)γ 2 q, b) Using the expression 2) for Siegel units, we get the equation γ 2 q, c)γ 2 q, d) γq, c + d)γq, c d). a b = ζ b 2 d 2 g a+b g a b gc 2 gd 2 N. c d gag 2 b 2 g c+d g c d It is a priori a relation between q-products, but raising it to an appropriate power yields an equality in Qζ N )q 1/N )) and thus in O Y N)). Therefore the formula 10) is valid in O Y N)) Q. 3. The three-term relation Weierstrass units 9) satisfy additive relations. These have already been used by Kubert and Lang to get diophantine results on modular curves [12, Chap 8]. In fact the whole proof of 7) is based on the following simple identity 12) a b a c + b c a c = 1. The relation 12) also has applications to the S-unit equation and is connected to the arithmetic of Fermat curves see the nice introduction of [12, Chap 8] for precise statements and references).

7 BEILINSON-KATO ELEMENTS IN K 2 7 Since the canonical bilinear map O Y N)) O Y N)) K 2 Y N)) enjoys Steinberg relations [16, 9.8], the identity 12) implies the following relation in K 2 Y N)) 13) { a b, } b c = 0. a c a c Using the expression of Weierstrass units in terms of Siegel units gives linear dependence relations between the elements ρm) in K 2 Y N)) Q. The main task will be to show that the 3-term relation is a consequence of these relations. Let a, b, c be three nonzero elements of Z/NZ) 2 such that a ±b, b ±c and c ±a. Using 10) and 13) we have the following identity in K 2 Y N)) Q { ga+b g a b g ag 2 c 2, g } b+cg b c g ag 2 c 2 = 0. gag 2 b 2 g a+c g a c gb 2g2 c g a+c g a c Expanding this and using the relation g a = g a, we get the more symmetric identity 14) {g a+b g a b g 2 c, g b+c g b c g 2 a} + {g b+c g b c g 2 a, g c+a g c a g 2 b } + {g c+a g c a g 2 b, g a+b g a b g 2 c } = 0. We remark that when a = 0 the relation 14) still makes sense and holds. Similarly it holds in the cases b = 0, c = 0, a = ±b, b = ±c or c = ±a. Thus 14) is true for any values of a, b, c Z/NZ) 2. We now wish to write 14) as a linear combination of 3-term relations. Let us define ψm) ) = ρm) + ρτm) + ρτ 2 M) for any M M 2 Z/NZ). s t Let M =. An elementary computation yields u v 15) ψm) = {g s,t, g u,v } + {g u,v, g s u,t v } + {g s u,t v, g s,t }. a For any two elements a and b of Z/NZ) 2, let us write for the 2 by 2 b) matrix with row vectors a and b. Then 15) can be rewritten as a a b a b 16) ψ = ρ + ρ + ρ. b) b) a b a We also have a a b a + b 17) ψ = ρ + ρ + ρ. b) b) a + b a

8 8 F. BRUNAULT Lemma 3.1. For any a, b, c Z/NZ) 2, the left hand side of the relation 14) can be written as 18) a 2ψ b) a + 2ψ b) + ψ b + a b + c b + 2ψ c) + ψ b + a b c b + 2ψ c ) + ψ ) b a b + c c + 2ψ a) ) + ψ c + 2ψ a b a. b c ) Proof. By expanding 14) completely, we obtain 19) {g a+b, g b+c } + {g b+c, g c a } + {g c a, g a+b } + {g a+b, g b c } + {g b c, g c+a } + {g c+a, g a+b } + 2{g a+b, g a } + 4{g a, g b } + 2{g b, g a+b } + {g a b, g b+c } + {g b+c, g c+a } + {g c+a, g a b } + {g a b, g b c } + {g b c, g c a } + {g c a, g a b } + 2{g a b, g a } + 2{g b, g a b } + 2{g c, g b+c } + 2{g b+c, g b } + 4{g b, g c } + 2{g c, g b c } + 2{g b c, g b } + 4{g c, g a } + 2{g a, g c+a } + 2{g c+a, g c } + 2{g a, g c a } + 2{g c a, g c } = 0. In most lines of 19) we recognize an expression of type 16) or 17), but there are incomplete terms. We can arrange the picture by splitting the terms with a coefficient 4 and moving them to the right places. This gives exactly 18). We now make use of the relation 14) with a particular choice of a, b and c. Let us assume that c = a + b. This gives for any choice of a and b) a a b b 2ψ + 2ψ + 2ψ + 2ψ b) b) a + b a b a + b a + b a + b 20) + 2ψ + 2ψ + ψ a a a + 2b a + b a + b a + b + ψ + ψ + ψ = 0. a a + 2b a a Using the notation M = and letting T resp. T b) ) be the image of resp. ) in G, we can rewrite 20) as

9 BEILINSON-KATO ELEMENTS IN K 2 9 2ψM) + 2ψ εm) + 2ψ τεm) + 2ψ τt 2 M) + 2ψτ 2 εm) ψτ 2 T 2 M) + ψ τ 2 T 2 M) + ψ M) + ψτ M) = 0. Since ψm) = ψ M) = ψτm) for any M, this simplifies to ) 3ψM) + 6ψεM) + 3ψT 2 M) + 3ψT 2 M) + ψ M) = Let us consider the formal linear combination of matrices in Z[M 2 Z NZ )] 1 1 DM) = 3[M] + 6[εM] + 3[T 2 M] + 3[T 2 M] + [ M] By assumption, we have det = Z NZ ). Lemma 3.2. The elements DM) span Q[M 2 Z )] when M runs through NZ M 2 Z ). NZ Proof. We ) remark that DM) is congruent mod 3 to the single matrix 1 1 M. Therefore the determinant of the vectors DM) in the canonical basis of Z[M 2 Z )] is not zero mod 3, and thus a nonzero integer. 1 2 NZ Using 21) and Lemma 3.2 gives ψm) = 0 for any M M 2 Z/NZ), which concludes the proof of Theorem Varying the modular curve In this section I study special elements in the groups K 2 X 1 N)) Q and K 2 X 0 N)) Q, in the case of prime level. In particular, I make explicit the link between the Beilinson-Kato elements and the elements which come up during my PhD thesis [4]. Let us first recall the definition of particular modular units on X 1 N) [5, 95)]. Let Y 1 N) be the modular curve over Q classifying elliptic curves E with a point P of order N, and let X 1 N) be the smooth compactification of Y 1 N). The set of cusps of X 1 N)C) is identified with Γ 1 N)\P 1 Q), and with this convention the cusp [0] is defined over Q. Let W N : X 1 N) X 1 N) be the Atkin-Lehner involution, which is defined over Qζ N ). For any λ Z/NZ), the Diamond operator λ associated to λ is defined by E, P ) E, λp ). On the complex points of X 1 N) we have ) λ [z] = [m λ z] λ 1 0 where m λ SL 2 Z) is any matrix congruent to mod N. 0 λ

10 10 F. BRUNAULT Definition 4.1. For any λ Z/NZ), let u λ O Y 1 N)) Q be the unique modular unit satisfying 22) divu λ ) = λ [0] [0] and u λ W N is normalized. Note that we use the cusp [0] instead of [ ]. It essentially amounts to the same thing, because the two definitions are related by W N. In [5, Prop 6.1] I show that the element {u λ, u µ } belongs to K 2 X 1 N)) Q for any choice of λ, µ Z/NZ). From now on, let us suppose that N = p is an odd prime. In [5, 8] I remark that the Beilinson conjecture should imply some linear dependence relations between the elements {u λ, u µ }. It turns out that these relations can be worked out explicitly and even rigorously proved, as follows. Let B 2 : R/Z R be the 1-periodic function obtained from B 2 by defining B 2 t) = B 2 t) for any 0 t 1. For any u, v Z/pZ), let us define 23) γu, v) = λ,µ Z/pZ) B 2 λu p )B 2 µv p ){u λ, u µ } K 2 X 1 p)) Q. By convention, we put γu, v) = 0 when u = 0 or v = 0. Theorem 4.2. The elements γu, v) u, v Z/pZ) satisfy the following relations 24) 25) 26) γu, v) = γ±u, v) = γu, ±v), γu, v) + γv, u) = 0, γu, v) + γv, u v) + γ u v, u) = 0. Proof. Since B 2 is an even function and u λ = u λ, we have the relations γ±u, v) = γu, ±v) = γu, v). The antisymmetry of the Milnor symbol yields γv, u) = γu, v), which proves 25). In order to prove the three-term relation 26), we jump to Xp). We have a finite morphism π : Y p) Y 1 p) which is defined over Q, given by E, e 1, e 2 ) E, e 2 ). Let Mp) be the field of meromorphic functions on the compactification of Γp)\H. It is a Galois extension of Cj) with Galois group SL 2 Z/pZ)/±1. We say that a function f Cq 1/n )) for some n 1) is normalized when the leading coefficient of its q-expansion is one. This definition extends naturally to Cq 1/n )) Q. Two functions f, g Mp) coincide if and only if their divisors are equal and f/g is normalized. Since we have an

11 BEILINSON-KATO ELEMENTS IN K 2 11 inclusion O Y p)) Mp), we will apply this principle to check equality between modular units in O Y p)) Q. The set of cusps of Γp)\H is identified with Γp)\P 1 Q), and the restriction of π to the cusps is the natural projection Γp)\P 1 Q) Γ 1 p)\p 1 Q). The inverse image of a cusp [x] by π is given by p 1 π [x] = [x + k] x P 1 Q)). k=0 The set of cusps Γp)\P 1 Q) can be identified with the set of nonzero column vectors of Z/pZ) [ 2 quotiented by ±1, the bijection being induced by a [a/c] P 1 Q) for any two relatively prime integers a and c. We now c] consider π u λ O Y p)) Q Mp) Q. Its divisor is given by p 1 27) div π u λ = π div u λ = [ λ 0 + k] [k] = k=0 k Z/pZ [ k λ] [ k 1]. On the other hand, the order of the Siegel unit g α,β at the cusp [ ] can be deduced from the q-product 2). Since q 1/p is a uniformizing parameter at [ ], we have ord [ ] g α,β = p 2 B 2 α ) α, β) 0, 0). p Using the transformation formula 4), we deduce the order of g α,β at any cusp : ord [ a c ] g α,β = p 2 B αa + βc 2 α, β) 0, 0). p A straightforward computation gives 28) div g 0,β = p 4 λ Z/pZ) k Z/pZ B 2 βλ [ ] k p ) + p λ 24 From 27) and 28), it follows that the divisor k Z/pZ) [ k 0] β 0). div g 0,β p B 2 βλ 4 p ) div π u λ λ Z/pZ) does not depend on β Z/pZ). Moreover, we have g 0,β 1 pz ) = g β,0pz) in C Q z H). and g β,0 pz) is a normalized function. Since each u λ W p is normalized, we can write

12 12 F. BRUNAULT g 0,β = h λ Z/pZ) π u λ p 4 B 2 βλ p )) where h O Y p)) Q is well-defined and independent of β. We then have 29) { g 0,u h, g 0,v h } = p2 16 π γu, v) u, v Z/pZ) ). We are now ready to prove 26). Since the map π : K 2 Y 1 p)) Q K 2 Y p)) Q is injective, it suffices to work in the latter vector space. The cases u = 0, v = 0 and u + v = 0 are easily treated. In the general case, we write 30) { g 0,u h, g 0,v h } = {g 0,u, g 0,v } + {h, g 0,u g 0,v }. Thanks to Theorem 1.4, we already know that u, v) {g 0,u, g 0,v } satisfies the three-term relation. Since {h, g 0,u g 0,v } is a boundary element, we get the desired result. Remark 4.3. In general, the relations 24), 25) and 26) between the elements γu, v) do not make up a complete set of relations. It can be seen by working out the case p = 5 explicitly. In that case X 1 p) is isomorphic to P 1 over Q and K 2 X 1 p)) Q is known to be 0. In the general case however, if we average under the action of Diamond operators see below), we can produce special elements in K 2 X 0 p)) Q together with a full set of relations. A theorem of Schappacher and Scholl [18, iii)] implies that γu, v) belongs to the integral subsapce K 2 X 1 p)) Z Q, and we can ask about the span of the elements γu, v). Let 31) r p : K 2 X 1 p)) Z Q Hom Q Ω 1 X 1 p)), R) be the Beilinson regulator map, as defined in [5, 1]. Theorem 4.4. The Beilinson conjecture for Lh 1 X 1 p)), 2) implies that K 2 X 1 p)) Z Q is generated by the elements γu, v), with u, v Z/pZ). Proof. Beilinson s conjecture predicts that r p is injective and that its image is a Q-structure of the target vector space. We already know that Beilinson s conjecture implies that K 2 X 1 p)) Z Q is generated by the elements {u λ, u µ } [5, 8]. It is sufficient to show that each {u λ, u µ } is a Q-linear combination of the elements γu, v). Let us consider

13 BEILINSON-KATO ELEMENTS IN K 2 13 θ = B 2 λ p )[λ] Q[Z/pZ) / ± 1]. λ Z/pZ) For every even Dirichlet character χ : Z/pZ) C, we have 32) χθ) = λ Z/pZ) B 2 λ p )χλ) = { 1 p 6p χ = 1) τχ) Lχ, 2) π 2 χ 1) where τχ) = p 1 a=1 χa)e2iaπ/p is the Gauß sum of χ. But for any character χ, we have Lχ, 2) 0, so that θ is invertible in the group algebra Q[Z/pZ) / ± 1]. We finally investigate the group K 2 X 0 p)) Q. The natural morphism X 1 p) X 0 p) identifies K 2 X 0 p)) Q with the fixed part of K 2 X 1 p)) Q under the Diamond operators. Definition 4.5. For any x Z/pZ), let 33) γ 0 x) = Besides, we define γ 0 0) = γ 0 ) = 0. u Z/pZ) γu, ux). Lemma 4.6. For any x Z/pZ), we have γ 0 x) K 2 X 0 p)) Q. Proof. It suffices to prove that π γ 0 x) is invariant under any matrix t = a b GL 0 d 2 Z/pZ). Because of 4), we have g 0,β t = g 0,dβ. Using 29), we remark that which is clearly invariant under t. p 2 16 π γ 0 x) = { g 0,u h, g 0,ux h } u Z/pZ) = u Z/pZ) {g 0,u, g 0,ux } Remark 4.7. The element γ 0 x) K 2 X 0 p)) Q is defined only implicitly. By this I mean that the actual definition uses Milnor symbols with functions on X 1 p), and not on X 0 p), which only contains two cusps. It is posible to rewrite γ 0 x) as follows 34) γ 0 x) = u Z/pZ) = ν Z/pZ) B 2 λu p )B 2 µux p ){u λ, u µ } λ,µ Z/pZ) B 2 u p )B 2 uνx ) p ) ) {u λ, u λν }. u Z/pZ) λ Z/pZ)

14 14 F. BRUNAULT In 34), each sum over λ already lies in K 2 X 0 p)) Q. recognize the sum over u to be a Dedekind sum. Moreover, we For any x P 1 Z/pZ), let ξx) H 1 )X 0 p)c), cusps, Z) be the modular a b symbol {g x 0, g x } where g x = SL c d 2 Z) is any matrix satisfying ) x = c/d mod p. Let ξ ± x) = 2 1 ξx) + ξ x). For any cusp form f S 2 Γ 1 p)), we define ξ f x) = ω ξx) f and ξ ± f x) = ω ξ ± x) f where ω f = 2iπfz)dz. Theorem ) For any newform f S 2 Γ 0 p)), we have 35) r p γ 0 x)), f = 8p 1) Lf, 2)ξ + f pπ x) x Z/pZ) ). 2) For any x P 1 Z/pZ), the following relations hold 36) γ 0 x) = γ 0 x) γ 0 x) + γ 0 1/x) = 0 γ 0 x) + γ 0 1 x 1 ) + γ 01 1 x ) = 0. 3) The equations 36) make up a complete set of relations for the elements γ 0 x). Proof. The point 1) will be a consequence of the explicit computation of Beilinson s regulator for the modular curve X 1 p) [5, Thm 1.1]. Let X be the set of even non-trivial characters of Z/pZ). For any χ X, we define a modular unit u χ O Y 1 p)) C by 37) u χ = λ Z/pZ) u λ Lχ, 2)χλ) 2π 2 ). Now let us compute the following element in K 2 X 1 p)) C γ x = χ X χx){u χ, u χ } x Z/pZ) ). Using 37) gives 38) γ x = 1 4π 4 λ,µ Z/pZ) χ xµ ){u λ )Lχ, 2)Lχ, 2) λ, u µ }. χ X The inner sum can be computed using the formula 32), which gives

15 BEILINSON-KATO ELEMENTS IN K ) π 4 p 1) 2p α,β Z/pZ) αxµ=±βλ B 2 α p )B 2 β p ) π4 p α,β Z/pZ) B 2 α p )B 2 β p ). The second term of 39) contributes to zero in 38) by antisymmetry of the Milnor symbol. Finally, we get γ x = p 1 4p α,β,λ,µ Z/pZ) αxµ=βλ B 2 α p )B 2 β p ){u λ, u µ } = p 1 4p γ 0x). In order to use [5, Thm 1.1], we have to take care of the Atkin-Lehner involution W p. Let wf) be the W p -eigenvalue of f. We let temporarily ũ χ resp. ũ λ ) be the modular unit defined in [5, 5)] resp. in [5, 95)]). We have u λ W p = ũ λ 1 and for any χ X {u χ, u χ } W p = = Lχ, 2)Lχ, 2) 4π 4 Lχ, 2)Lχ, 2) 4π 4 = {ũ χ, ũ χ } χλ)χµ){u λ, u µ } W p λ,µ Z/pZ) χλ/µ){ũ λ, ũ µ } λ,µ Z/pZ) because of [5, Prop 5.4]. Let f S 2 Γ 0 p)) be a newform and wf) be the W p -eigenvalue of f. Using [5, Thm 1.1], we have r p {u χ, u χ }), f = r p {u χ, u χ } W p ), W p f = wf) r p {ũ χ, ũ χ }), f = A classical computation [14] yields 2p 1)wf) Lf, 2)Lf, χ, 1). pπτχ) Lf, χ, 1) = wf)τχ) χa)ξ + f p a) a Z/pZ) By taking the sum over characters χ, we obtain χ X). 2p 1)2 r p γ x ), f = Lf, 2)ξ + p 2 f π x). This proves 35). The relations 36) are an easy consequence of Theorem 4.2 and the definition 33) of γ 0 x). Note that they are consistent with the regulator formula 35). Finally, for the point 3), let γ 0 be the map

16 16 F. BRUNAULT γ 0 : Q[Z/pZ) ] K 2 X 0 p)) Q [x] γ 0 x). Let R be the kernel of γ 0. We wish to show that R is generated by the relations 36). For this we use the theory of Manin symbols. For any x Z/pZ), the cycle ξx) has trivial boundary. Thus we have a map ξ + : Q[Z/pZ) ] H + 1 X 0 p)c), Q). Manin s theorem implies that the kernel of ξ + is generated by the relations 36), so that ker ξ + R. In order to prove the reverse inclusion, it suffices to consider the dimensions. Let gx 0 p)) be the genus of X 0 p). From 35) we know that the image of γ 0 has dimension at least gx 0 p)). Manin s theorem implies that the dimension of the image of ξ + is precisely gx 0 p)) the element ξ0) = {0, } = ξ ) has non-trivial boundary). We conclude that dim R dim ker ξ +, so that R is generated by 36). References [1] S. J. Bloch. Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, volume 11 of CRM Monograph Series. American Mathematical Society, Providence, RI, [2] L. A. Borisov and P. E. Gunnells. Toric modular forms and nonvanishing of L- functions. J. Reine Angew. Math., 539: , [3] L. A. Borisov and P. E. Gunnells. Toric modular forms of higher weight. J. Reine Angew. Math., 560:43 64, [4] F. Brunault. Étude de la valeur en s = 2 de la fonction L d une courbe elliptique. PhD thesis, Université Paris 7, décembre arxiv:math/ v1. [5] F. Brunault. Valeur en 2 de fonctions L de formes modulaires de poids 2 : théorème de Beilinson explicite. Bull. Soc. Math. France, 1352): , [6] P. Colmez. La conjecture de Birch et Swinnerton-Dyer p-adique. Astérisque, 294: , [7] P. Deligne and M. Rapoport. Les schémas de modules de courbes elliptiques. In Modular functions of one variable, II Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages Lecture Notes in Math., Vol Springer, Berlin, [8] T. Dokchitser, R. de Jeu, and D. Zagier. Numerical verification of Beilinson s conjecture for K 2 of hyperelliptic curves. Comp. Math., 1422): , [9] A. B. Goncharov. Euler complexes and geometry of modular varieties. Geom. Funct. Anal., 176): , Special volume dedicated to J. Bernstein. [10] K. Kato. p-adic Hodge theory and values of zeta functions of modular forms. Astérisque, 295: , Cohomologies p-adiques et applications arithmétiques III). [11] N. M. Katz and B. Mazur. Arithmetic moduli of elliptic curves, volume 108 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, [12] D. S. Kubert and S. Lang. Modular units, volume 244 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science]. Springer- Verlag, New York, [13] S. Lang. Elliptic functions, volume 112 of Graduate Texts in Mathematics. Springer- Verlag, New York, second edition, With an appendix by J. Tate.

17 BEILINSON-KATO ELEMENTS IN K 2 17 [14] Y. I. Manin. Parabolic points and zeta functions of modular curves. Math. USSR Izvestija, 61):19 64, [15] L. Merel. Universal Fourier expansions of modular forms. In On Artin s conjecture for odd 2-dimensional representations, volume 1585 of Lecture Notes in Math., pages Springer, Berlin, [16] J. Milnor. Introduction to algebraic K-theory. Princeton University Press, Princeton, N.J., Annals of Mathematics Studies, No. 72. [17] V. Paşol. A Modular Symbol with Values in Cusp Forms. arxiv:math/ v1, November [18] N. Schappacher and A. J. Scholl. Beilinson s theorem on modular curves. In Beilinson s conjectures on special values of L-functions, volume 4 of Perspect. Math., pages Academic Press, Boston, MA, [19] G. Shimura. Correspondances modulaires et les fonctions ζ de courbes algébriques. J. Math. Soc. Japan, 10:1 28, [20] G. Shimura. Introduction to the arithmetic theory of automorphic functions, volume 11 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton, NJ, Reprint of the 1971 original, Kano Memorial Lectures, 1. address: brunault@umpa.ens-lyon.fr URL: Université de Lyon, E.N.S. Lyon, U.M.P.A., 46 allée d Italie, F , France