From the elliptic regulator to exotic relations

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1 An. Şt. Univ. Ovidius Constanţa Vol. 16(), 008, From the elliptic regulator to exotic relations Nouressadat TOUAFEK Abstract In this paper we prove an identity between the elliptic regulators of some -isogenous elliptic curves. This allow us to prove a new exotic relation for the elliptic curve 0A of Cremona s tables. Also we prove the (conjectured) exotic relation for the curve 0B given by Bloch and Graysonin[3]. 1 Introduction For some elliptic curves, the elliptic dilogarithm satisfies linear relations, called exotic by Bloch and Grayson [3]. In [3] a list of elliptic curves that satisfies exotic relations is given. Recently some of these relations are proved by Bertin [1], Touafek [7]. We note that whenever we can find a tempered model of the elliptic curve, the existence of exotic relations is related to elements in the second group of the K-theory K (E) hence to elliptic regulators, so we can prove the exotic relations. Bloch and Grayson conjectured the following fact. Conjecture 1 Suppose that E (Q) tors is cyclic and d = #E (Q) tors >. Write Σ for the number of fibres of type I ν with ν 3 in the Néron model, and suppose [ ] [ d 1 Σ > 1. Then there should be at least d 1 ] Σ 1 exotic relations [ d 1 ] r=1 a r D E (rp) =0, Key Words: Elliptic curves, Elliptic dilogarithm, Elliptic regulator, Exotic relations. Mathematics Subject Classification: 11G05, 14G05, 14H5. Received: April, 008 Accepted: September,

2 118 Nouressadat Touafek where P is a d-torsion point and a r Z. In particular, these conditions are satisfied for the curve 0A of Cremona s tables [4]: E (Q) tors cyclic, d =6andΣ=0. In section 3 we use an identity between regulators and some equalities between the elliptic dilogarithm to prove a new exotic relation for the elliptic curve 0A and also we prove the (conjectured) exotic relation for the curve 0B of Cremona s tables given by Bloch and Grayson in [3]. Preliminaries Let E be an elliptic curve defined over Q. Throughout this paper, the notation E =[a 1,a,a 3,a 4,a 6 ] means that the elliptic curve E is in the Weierstrass form y + a 1 xy + a 3 y = x 3 + a x + a 4 x + a 6..1 The elliptic regulator Let F be a field. By Matsumoto s theorem, the second group of K-theory K (F ) can be described in terms of symbols {f,g}, forf, g F and relations between them. The relations are {f 1 f,g} = {f 1,g} + {f,g} {f,g 1 g } = {f,g 1 } + {f,g } {1 f,f} =0. For example, if v is a discrete valuation on F with maximal ideal M and residual field k, the Tate s tame symbol (x, y) ν ( 1) ν(x)ν(y) x ν(y) y ν(x) mod M defines a homomorphism λ v : K (F ) k. Let Q(E) be the rational function field of the elliptic curve E. To any P E(Q) is associated a valuation on Q(E) that gives the homomorphism λ P : K (Q(E)) Q(P )

3 From the elliptic regulator to exotic relations 119 and the exact sequence 0 K (E) Q K (Q(E)) Q Q(P ) Q... P E(Q) By definition K (E) is defined modulo torsion by K (E) ker λ = P ker λ P K (Q(E)) where λ : K (Q(E)) Q (Q(P ) Q). P E(Q) Definition 1 A polynomial in two variables is tempered if the polynomial of the faces of its Newton polygon has only roots of unity. When drawing the convex hull of points (i, j) Z corresponding to the monomials a i,j x i y j, a i,j 0, you also draw points located on the faces. The polynomial of the face is a polynomial in one variable t which is a combination of the monomials 1, t, t,... The coefficients of the combination are given when going along the face, that is a i,j if the lattice point of the face belongs to the convex hull and 0 otherwise. In particular, the polynomials and P 1 (X 1,Y 1 )=Y 1 +X 1 Y 1 X X 1 P (X,Y )=Y +X Y +Y (X 1) 3 are tempered, so we get {X 1,Y 1 } K (E 1 )and{x,y } K (E ), see Rodriguez-Villegas [5]. Here E 1 is the elliptic curve defined by P 1 (X 1,Y 1 )=0 and E the elliptic curve defined by P (X,Y )=0. Let f and g be in Q(E). Let us define η(f,g) =log f d(arg g) log g d(arg f). Definition The elliptic regulator r of E is given by r : K (E) R {f,g} 1 π γ η(f,g) for a suitable loop γ generating the subgroup H 1 (E,Z) of H 1 (E,Z), where the complex conjugation acts by 1.

4 10 Nouressadat Touafek. The elliptic dilogarithm We have two representations for E(C) E(C) C /(Z + τz) C / q Z ( (u), (u)) u( mod Λ) z = e πiu where is the Weierstrass function, Λ = {1,τ} the lattice associated to the elliptic curve and q = e πiτ. Definition 3 The elliptic dilogarithm D E [] is defined by D E (P )= n=+ n= D(q n z), where P E(C) is the image of z C, z = e πiu, u = ξτ + η and D is the Bloch-Wigner dilogarithm D(x) :=ILi (x)+log x arg(1 x). Remark 1 1. The Bloch-Wigner dilogarithm is a function univalued, real analytic in P 1 (C)\{0, 1, }, continuous in P 1 (C) [9].. There is a second representation of the elliptic dilogarithm given by Bloch [], [10] in terms of Eisenstein-Kronecker series D E (P )= (Iτ) π R( exp( πi(nξ mη)) m,n Z (m,n) (0,0) ). (1) (mτ + n) (m τ + n) 3. The elliptic dilogarithm can be extended to divisors on E(C) D E ((f)) = i n i D E (P i ), where (f) = i n i [P i ]..3 The diamond operation Let Z [E(C)] be the subgroup of Z [E(C)] modulo the equivalence relation cl ([ P ]) = cl ([P ]).

5 From the elliptic regulator to exotic relations 11 Definition 4 The diamond operation is defined by : Z [E(C)] Z [E(C)] Z [E(C)] ((f), (g)) i,j n im j cl([p i P j ]) where (f) = i n i [P i ] and (g) = j m j [P j ]. The following theorem [], establishes the relation between the elliptic regulator and the elliptic dilogarithm. Theorem 1 The elliptic dilogarithm D E can be extended to a morphism Z [E(C)] R. If f, g are functions on E and {f,g} K (E), then πr({f,g}) =D E ((f) (g)); in particular D E ((f) (1 f)) = 0. 3 An identity between regulators Let E 1 be the elliptic curve, isomorphic to the curve 0A, with equation Y 1 +X 1Y 1 = X 3 1 X 1 and let E be the elliptic curve, isomorphic to the curve 0B, with equation Y +X Y +Y =(X 1) 3. Proposition 1 We have πr({x 1,Y 1 }) = 4D E1 (P 1 ) 4D E1 (P 1 ) πr({x,y }) = 6D E (P )+6D E (P ) where P 1 =( 1, ) is the 6-torsion point of the curve E 1 =[, 0, 0, 1, 0] and P =(5, 4) is the 6-torsion point of the curve E =[, 3,, 3, 1]. Proof. We need to compute the following divisors (X 1 )= [3P 1 ] [O 1 ] (Y 1 )= [3P 1 ]+[4P 1 ]+[5P 1 ] 3[O 1 ] (X )= [3P ] [O ] (Y )= 3[P ] 3[O ]

6 1 Nouressadat Touafek hence (X 1 ) (Y 1 )= 4cl([P 1 ]) 4cl([P 1 ]) and (X ) (Y )=6cl([P ]) + 6cl([P ]) so, by theorem 1 we get πr({x 1,Y 1 })= 4D E1 (P 1 ) 4D E1 (P 1 ) and πr({x,y })=6D E (P )+6D E (P ). Remark Using the previous proposition and formula (1), we have find by the computer, r ({X,Y })? = r ({X 1,Y 1 }), where the notation A? = B, means A is conjectured to be equal to B, that is A and B are numerically equal to at least 5 decimal places. Theorem We have the following identity r ({X,Y })=r ({X 1,Y 1 }). Proof. Let Ξ 1 =0A be the elliptic curve with equation S 1 = T T 1 T 1 and Ξ be the elliptic curve, isomorphic to 0B, with equation It is easy to check that and S = T 3 T +5T. give isomorphisms Ξ 1 E 1, Ξ E. Also we have the -isogeny [6] given by T 1 = X 1, S 1 = Y 1 + X 1 () T = X, S = Y + X +1 (3) Φ: Ξ 1 Ξ (T 1,S 1 ) ( S 1 T1 ), S1(T 1 +1). T1 (4)

7 From the elliptic regulator to exotic relations 13 Using (), (3) and (4) we can see Φ as Φ: E 1 E (X 1,Y 1 ) (X Φ,Y Φ )=( (Y1+X1), (Y1+X1+1)(Y1+X X1 1 +X1) ). X1 The elliptic curve E 1 can be considered as a double cover of P 1 by π X1 : E 1 P 1 ramified at the zeros of X1 3 + X 1 X 1, i.e. 0, 1+ 5, 1 5. The closed curve σ 1 = π 1 X 1 ([0, 1+ 5 ]) generates H 1 (E 1, Z). The elliptic curve E can be considered as a double cover of P 1 by π X : E P 1 ramified at the zeros of X 3 X +5X, i.e. 0, 1+i, 1 i. The closed curve σ = π 1 X ([1 i, 1+i]) generates H 1 (E, Z). Using the -isogeny we get where P 1, 5P 1 Φ P P, Q Φ 3P P 1, 4P 1 Φ 4P 3P 1, O 1 Φ O P =( ϕ, ϕ), Q =( 1 ϕ, 1 ϕ ), Q = P +3P 1, ϕ = Also, when (X,Y ) describes σ,(x Φ,Y Φ ) describes twice the closed curve σ = {(X 1,Y 1 ) E 1 (C)/ X 1 =1}, which generates H 1 (E 1, Z), because it s in the same homology class as σ 1. Hence, r ({X,Y })=± 1 r ({ X Φ,Y Φ}). (5) We have (1 + X 1 + Y 1 ) = [P 1 ]+[5P 1 ] 3[O 1 ] (X 1 + Y 1 ) = [3P 1 ]+[P]+[Q] 3[O 1 ] (Y 1 + X 1 + X1 ) = [3P 1]+[5P 1 ]+[P 1 ] 4[O 1 ].

8 14 Nouressadat Touafek Performing the necessary computation, we obtain (X Φ ) (Y Φ ) = 1cl([P 1 ]) + 1cl([P 1 ]) + 1cl([P P 1 ]) + 1cl([P Q]) +1cl([Q P 1 ]) + 1cl([Q P ]) and (1 + X 1 + Y 1 ) (X 1 + Y 1 ) = 5cl([P 1 ]) cl([p 1 ]) 3cl([P P 1 ]) 3cl([P Q]) 3cl([Q P 1 ]) 3cl([Q P ]). Using the fact that D E ((1 + X 1 + Y 1 ) (X 1 + Y 1 )) = 0 we get D E1 (X Φ Y Φ )=8D E1 (P 1 )+8D E1 (P 1 ) so by Theorem 1 and Proposition 1 By (5), (6) and remark we get r ({ X Φ,Y Φ}) = r ({X 1,Y 1 }). (6) r ({X,Y })=r ({X 1,Y 1 }). Let E 1, E be as above. We have the following theorem [8]. Theorem 3 We have the following equalities 1) D E1 (P 1 ) = D E (P )+3D E (P ) ) D E1 (P 1 ) = D E (P )+D E (P ). Proof. The proof follow the same way of the proof of Theorem 3. in [7] Now, we are able to give a new exotic relation for the curve 0A. Corollary 1 We have the linear relation 16D E1 (P 1 ) 11D E1 (P 1 )=0. Proof. It results from Proposition 1 and Theorem that so by theorem 3, we get 4D E1 (P 1 ) 4D E1 (P 1 )=6D E (P )+6D E (P ); 16D E1 (P 1 ) 11D E1 (P 1 )=0.

9 From the elliptic regulator to exotic relations 15 Remark 3 1. In turn, by Theorem 3, the relation 16D E1 (P 1 ) 11D E1 (P 1 )=0 becomes 5D E (P ) 13D E (P )=0. This achieves the proof of the (conjectured) exotic relation for the curve 0B given by Bloch and Grayson in [3].. In [3] only elliptic curves with negative discriminant are considered, so our new exotic relation does not appear in the list of Bloch and Grayson because the curve 0A have a positive discriminant. References [1] M.J. Bertin, Mesure de Mahler et régulateur elliptique: Preuve de deux relations exotiques, CRMProc. LecturesNotes, 36 (004), 1-1. [] S. Bloch, Higher regulators, algebraic K-theory and zeta functions of ellptic curves, ( Irvine Lecture, 1977) CRM-Monograph series, vol.11, Amer.Math.Soc., Providence, RI, 000. [3] S. Bloch and D. Grayson, K and L-functions of elliptic curves computer calculations, I, Contemp. Math. 55 (1986), [4] J.E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, nd edition, [5] F. Rodriguez-Villegas, Modular Mahler measures I, Topics in Number Theory ( S. D. Ahlgren, G. E. Andrews, and K. Ono, eds), Kluwer, Dordrecht, 1999, pp [6] J.H. Silverman, The arithmetic of elliptic curves, Springer Verlag, Berlin and New York, [7] N. Touafek, Some equalities between elliptic dilogarithm of -isogenous elliptic curves, Int. J. Algebra, (008), [8] N. Touafek, Thèse de Doctorat, Université de Constantine, Algeria, 008. [9] D. Zagier, Dedekind Zeta functions, and the Algebraic K-theory of Fields, Arithmetic algebraic geometry (Texel, 1989), , Progr. Math., 89, Birkhuser Boston, Boston, MA, [10] D. Zagier and H. Gangl, Classical and elliptic polylogarithms and special values of L-series, the Arithmetic and Geometry of Algebraic cycles, Nato, Adv. Sci. Ser.C. Math.Phys.Sci., Vol. 548, Kluwer, Dordrech, 000, pp LaboratoiredePhysiqueThéorique Equipe de Théorie des Nombres Université de Jijel, Algeria nstouafek@yahoo.fr

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