On K2 T (E) for tempered elliptic curves

Transcription

1 On K T 2 (E) for tempered elliptic curves Department of Mathematics, Nanjing University, China guoxj@nju.edu.cn hrqin@nju.edu.cn Janurary NTU

2 1. Preliminary and history

3 K-theory of categories Let C be an exact category and QC the new category which has the same objects with C but a morphism X Y is given by an isomorphism classes of diagrams X Z Y. Let BQC be the classifying space of C. Then the K-groups of C is defined as K i (C) = π i+1 (BQC).

4 K-theory of schemes Let X be a Noetherian scheme. Let P(X ) be the exact category of locally free O X -modules of finite ranks. Let M(X ) be the exact category of coherent O X -modules. Then the K-theory and K -theory of the X is defined as K i (X ) =K i (P(X )), K i (X ) =K i (M(X )).

5 The K 2 of fields Let R be a ring. The definition of K 2 R was given by Milnor in For a field F, Matsumoto proved in 1969 that K 2 F is the quotient of F F by the subgroup generated by the elements a (1 a), where a 0, 1. We use the Steinberg symbol {a, b} to denote the image of a b in K 2 F.

6 K 2 of elliptic curves Let Let E be an elliptic curve defined over Q, E its Neron model. K T 2 (E) = ker(k 2 (F ) D F(D) ), where D runs through all irreducible curves on E and F(D) denotes the residue field at D. The component map corresponding to D is T D : {a, b} ( 1) v D(a)v D (b) av D(b) b v D(a) (modulo D). The group K2 T (E)/torsion is called the integral part of K 2 (E). K2 T (E) is the image of K2 T (E) K 2 (E). Hence we have Xuejun that Guo and Hourong Qin

7 Beilinson Conjecture In 1980 s, A. A. Beilinson conjectured that for a general non-singular projective curve of genus g defined over the number field k, the dimension of K 2 (E) Q should be g[k : Q].

8 Bloch and Grayson s modification In 1986, Bloch and Grayson s work suggested Beilinson s conjecture should be modified to K2 T (E) Q, i.e., for a general non-singular projective curve of genus g defined over the number field k, the dimension of K2 T (E) Q should be g[k : Q].

9 Quillen s localization long exact sequence By Quillen s localization long exact sequence, K 2 (k x ) K 2 (E) K 2 F k x, x E x E we have K 2 (E) Q (ker(k 2 (F ) k x )) Q. x E

10 Beilinson s work In 1986, Beilinson proved that for modular elliptic curves defined over Q, the rank of K 2 (E) 1. Later it is also proved that the rank of K2 T (E) 1. His proof did not give the explicit expression of the non-torsion element of K 2 (E). Since all elliptic curves defined over Q are modular, we always have that the rank of K2 T (E) 1.

11 Bloch s construction of elements in K 2 (E) We assume that O, P, Q E(Q) div(f ) = d(p) d(o), div(g) = e(q) e(o), f (Q) = 1, g(p) = 1. Then {f, g} ker(k 2 (F ) k x ).(By the product formula) x E

12 Ross work In 1992, Ross proved that the K 2 of all but finitely many elliptic curves defined over Q and possessing a rational point of order at least 3 has strictly positive rank. At that time, it was not known if all elliptic curves defined over Q are modular.

13 Ji and Qin s work In 2010, Ji and Qin gave the explicit expression of the non-torsion elements of K 2 (E) when E has a rational point of order at least 4. When E has a rational point of exact order 3, they gave the explicit expression of the non-torsion elements of (K 2 (E)), except at most one R-isomorphism class of E.

14 In 2006, T. Dokchitser, R. De Jeu and D. Zagier studied the same problem for certain hyperelliptic curves, y 2 = cx 2g+1 + (b g x g b 1 x + b 0 ) 2 (genus g with 2g + 1 torsion) and y 2 = c 2 x 2g+2 + (cx g+1 + b g x g b 1 x + b 0 ) 2. (genus g with 2g + 2 torsion) All these works depend essentially on Bloch s construction. The elliptic curves or hyperelliptic curves need to have enough rational torsions.

15 A new construction In 1994,D. Cooper, M. Culler, H. Gillet, D. D. Long and P. B. Shalen gave a new construction. Let E be an elliptic curve defined by f (x, y) = c (i,j) x i y j = 0. Let be the Newton polygon of the polynomial f (x, y) = 0. Let τ be a side of. We will always parameterize a side clockwise around and in such a way that τ(0), τ(1),... are the consecutive lattice points in τ. To every side τ, we associate a one-variable polynomial P τ = c τ(k) t k. k=0

16 We call polynomial P tempered if for every side τ, the zeroes of P τ consist of roots of unity only. Theorem (CCGLS) Let E be an elliptic curve defined by a tempered polynomial. Then there is an N such that the steinberg symbol {x, y} N K 2 (E). Note that this construction does not depend on the torsion points. We will see that there are tempered elliptic curves without rational torsion elements. It is essentially different from Bloch s method.

17 Mahler Measures The logarithmic Mahler measure of a Laurent polynomial is defined as m(p) = 1 0 P C[x 1 ±1 ±1,..., x n ] 1 and its Mahler measure as M(P) = e m(p). 0 log P(e 2πiθ 1,..., e 2πiθn ) dθ 1 dθ n

18 Rodriguez Villegas Theorem Theorem Let E be an elliptic curve defined by a tempered polynomial P. We assume that the E T 2 =, where T 2 is the torus {x x = 1} {y y = 1}. Then m(p) = 1 2π γ η{x, y}, where γ is a closed path not going through poles or zeroes of x or y which generates the subgroup H 1 (E, Z) 1 of H 1 (E, Z) where complex conjugation acts by 1, and η({x, y}) = log x dargy log y dargx.

19 Corollary Let E be an elliptic curve defined by a tempered polynomial P. If m(p) 0, then there is an N such that the steinberg symbol {x, y} N K 2 (E) is not torsion. Theorem (Boyd, Lawton, Smyth) Let f (x, y) Z[x, y]. Then m(f ) = 0 if and only if there is a cyclotomic polynomial g such that f (x, y) = ±x a y b g(x c y d ) for some naturalnubers a, b, c, d.

20 Main results Let E be an elliptic curve defined by a tempered polynomial P. Then there is always an N such that the steinberg symbol {x, y} N K 2 (E) is not torsion. Furthermore we have checked that for the following three families x 3 + y kxy = 0, k Z; x 2 y + y 2 x + x + y kxy = 0, k Z; (x y)(x 1)(y 1) kxy = 0, k Z. there is always an N such that the steinberg symbol {x, y} N K2 T (E).

21 An application Our explicit expression of the elements of K 2 (E) gives another explanation of one of Zagier and Gangl s results in Classical and elliptic polylogarithms and special values of L-series Recall the definition of elliptic dilogarithm. Li 2 (z) = z 0 log(1 t) dt t L 2 (z) : = ImLi 2 (z) + arg(1 z) log z. L 2,q (z) : = n Z L 2 (q n z), then extend L 2,q (z) by linearity to the set of all divisors on E(C).

22 In 1991, Zagier and Gangl suggested a list ( supported by computer calculation) for the elliptic curve E : y 2 y = x 3 x, 8πL 2,q (P 3 ) 37L(E, 2) = , 8πL 2,q (P 4 ) 37L(E, 2) = where P k = (kp) k(p) k3 k 6 ((2P) 2(P)), P = [0, 0]. We can show that all these identities are true (in fact there are equivalent which is already prove by Zagier).

23 Sketch of the proof 1. By our explicit expression, it is easy to see that each P k comes from the {x, y} N for suitable natural number N. We conjecture that {x, y} is the generator of K 2 (E)/torsion. 2. E is the modular curve X 1 (11), both x and y can be written in the form of modular units u, v. One can see Yifan Yang s paper for details. 3. For modular curves X 1 (p), one can use Brunault s explicit formula (cf. Brunault, Valeur en 2 de fonctions L de formes modulaires de poids 2 : theoreme de Beilinson explicite, 2007, BSMF) to express L(X 1 (11), 2) in terms of the Beilinson regulator of steinberg symbol {u, v}, where u, v are modular units.

24 4. Use Zagier s expression of the Beilinson regulator of steinberg symbol in terms of suitable elliptic dilogarithm of some divisor P. 5. Find the relations between the divisor P and the divisor (u) (v) = i,j a ib j (P i Q j ) in step 4, where (u) = i a i(p i ), (v) = j b j(q i ).

25 Thanks!