On a conjecture by Boyd

On a conjecture b y Bo yd Matild e N. Lal ´ ın ∗ Octob er 30, 2018 Abstract The aim of this note is to pro ve the Mahler measure iden tit y m ( x + x − 1 + y + y − 1 + 5) = 6 m ( x + x − 1 + y + y − 1 + 1) w hic h was conjectured b y Bo yd. The pro of is ac hiev ed b y proving relatio ns hips b et w een regulators of b oth curv es. keywor ds : Mahler measure, elliptic c urve s, elliptic dilogarithm, regulator 2000 Mathematics Subje ct Classific ation : 2000: 11R09, 1 9F27 1 In tro ducti on Bo yd [3] studied the Mahler measure of families of polynomials. In particular, he considered the t w o- v ariable family P k ( x, y ) = x + 1 x + y + 1 y + k . The zeros of P k ( x, y ) correspond, generically to a curv e of g en us 1. Let E k denote the elliptic curv e corresp o nding to the algebraic closure of P k ( x, y ) = 0. Recall that the (logarithmic) Mahler measure of a non-zero Lauren t p oly- nomial, P ( x 1 , . . . , x n ), with complex co efficien ts is defined as m ( P ) = Z 1 0 . . . Z 1 0 log   P  e 2 π i t 1 , . . . , e 2 π i t n    d t 1 . . . t n . ∗ Department of Mathematical and Statistical Sciences, University of Alber ta, 632 Cen- tral Academic Building,Edmo nton, AB T6G 2G1, Canada mlal in@ma th.ua lberta.ca 1 Let us denote m ( k ) := m ( P k ). Bo yd computed m ( k ) for k a p o sitiv e in teger less than or equal to 1 00 (it is easy to see that the Mahler measure do es not dep end on the sign of k for this family). He found that m ( k ) ? = r k L ′ ( E k , 0) , (1) where r k is a ra t io nal num b er and the question mark stands f or an equalit y that has only b een stablished n umerically (t ypically to at least 50 decimal places). The case with k = 1 (resulting in r k = 1) w as considered in detail b y Deninger [5], who f o und an explanation for suc h a form ula by relat ing it to ev aluations of regulator s in the con text of the Blo c h–Beilinson conjectures. Ro driguez-Villegas [8] also considered this family in the context of the Blo c h- Beilinson conjectures, including more general cases where k 2 ∈ Q . He was able to prov e iden tities for the cases where the Blo c h–Beilinson conjectures are known to b e true, suc h as when E k has complex m ultiplication. When the curv es E k 1 and E k 2 are isogenous, their L -f unctions coincide. One can then compare the v alues in equation (1) and conjecture iden tities of the form r k 2 m ( k 1 ) = r k 1 m ( k 2 ). F or example, Theorem 1 m (8) = 4 m (2) , (2) m (5) = 6 m (1) . (3) The first identit y w as pro v ed in [7]. In this note, w e pro ve the second one. 2 F unction al Iden tities F unctional iden tities for m ( k ) hav e b een studied b y Kurok a w a and Oc hiai in [6], and by R o gers and the author in [7]. The simplest ones are g iv en as follo ws: Theorem 2 We have the fol lowin g functional e q uations for m ( k ) : • [6]: F or h ∈ R \{ 0 } : m  4 h 2  + m  4 h 2  = 2 m  2  h + 1 h  . (4) 2 • [7]: I f h 6 = 0 , and | h | < 1 : m  2  h + 1 h  + m  2  i h + 1 i h  = m  4 h 2  . (5) If w e set h = 1 √ 2 in b oth identities , w e obtain m (2) + m (8) = 2 m  3 √ 2  , m  3 √ 2  + m  i √ 2  = m (8) . Similarly , if w e set h = 1 2 , we obtain m (1) + m (1 6) = 2 m (5) , m (5) + m ( − 3i) = m (1 6) . Th us, in order to prov e (2) and (3), w e need to find o ne additional equation for eac h of the ab ov e linear systems. 3 The relati onship w ith the regulator In this section, w e sometimes write x k and y k for x and y , so we can distin- guish them when w e lo ok at differen t curves . After the w orks o f D eninger [5] and Ro driguez-Villegas [8], w e write m ( k ) = 1 2 π r k ( { x k , y k } ) , w ere r k is a perio d of the regulat or in the sym b o l { x k , y k } ∈ K 2 ( E k ). F or our purp oses, w e can reduce to K 2 ( C ( E k )), so that x k , y k are elemen ts of C ( E k ). See [5] and [8 ] for general details, and [7 ] for the sp ecific t r eatmen t of this particular example. In our context, it is enough to tak e in to accoun t that r k ( { x k , y k } ) = α D k (( x k ) ⋄ ( y k )) , where α is a constan t indep endent of k and D k is the elliptic dilogarithm in E k constructed b y Blo c h ( see [2]). 3 W e will briefly explain t he meaning of ( x ) ⋄ ( y ). Let E b e an elliptic curv e with x, y ∈ C ( E ). Consider the divisors ( x ) = X a S ( S ) , ( y ) = X b T ( T ) . No w define ( x ) ⋄ ( y ) = X a S b T ( S − T ) . This is an elemen t in Z [ E ( C )] − = Z [ E ( C )] / ∼ , where the equiv alence relation stands for ( − T ) ∼ − ( T ). Th us, the Mahler measure dep ends just on D k and ( x k ) ⋄ ( y k ). F or exam- ple, if the elliptic curv es are isomorphic, D k do es not c hange and the Mahler measure only depends on ( x k ) ⋄ ( y k ). This idea w as disco ve red by Ro driguez- Villegas [9], and also used by Bertin [1]. W e applied this idea again in [7], to isogenous elliptic curv es, in order to prov e iden tities lik e (5). A W eie rstrass mo del for E k is giv en b y Y 2 = X  X 2 +  k 2 4 − 2  X + 1  , where x = k X − 2 Y 2 X ( X − 1) , y = k X + 2 Y 2 X ( X − 1) . It is not hard to see that E k ( Q ( k )) tor ∼ = Z / 4 Z . T o fix notation, we will denote a g enerato r b y P =  1 , k 2  . Then w e hav e 2 P = ( 0 , 0) . Ev en tually , w e will p erfo rm computations in the curv e with parameter k = h + 1 h . In this curv e, w e will denote Q =  − 1 h 2 , 0  , whic h is a p oint of order 2. Notice that P + Q =  − 1 , h − 1 h  and 2 P + Q = ( − h 2 , 0). In [7] we prov e ( x ) ⋄ ( y ) = 8( P ) . 4 Consider the isomorphism φ : E 2 ( h + 1 h ) → E 2 ( i h + 1 i h ) , ( X, Y ) → ( − X , i Y ) , whic h relates tw o o f the curv es in equation (5). W e use this isomorphism to pull the rational functions x, y ∈ C  E 2 ( i h + 1 i h )  bac k to C  E 2 ( h + 1 h )  : r 2 ( i h + 1 i h ) ( { x, y } ) = r 2 ( h + 1 h ) ( { x ◦ φ, y ◦ φ } ) . On the other hand, it is easy to see tha t ( x ◦ φ ) ⋄ ( y ◦ φ ) = 8 ( P + Q ) . 4 Relationshi ps b e t w een divisors F rom t he previous section, the problem reduces to finding relatio ns b et w een ( P ) and ( P + Q ) in Z h E 2 ( h + 1 h ) ( C ) i − . In order t o do that, w e will lo ok for elemen t s that are trivial in K 2  C  E 2 ( h + 1 h )  . In other words , we will find com binations of Steinberg sym b ols { g , 1 − g } with g ∈ C  E 2 ( h + 1 h )  , suc h that the corresp onding comb inatio n ( g ) ⋄ (1 − g ) yields a linear combination of ( P ) and ( P + Q ). Since { g , 1 − g } is trivial in K -theory , we conclude t ha t ( g ) ⋄ (1 − g ) ∼ 0, yielding a linear combination in v olving ( P ) and ( P + Q ). Consider the function f = Y 2 h +  1 2 − 1 2 h 2  X . W e hav e 1 − f = 1 − Y 2 h −  1 2 − 1 2 h 2  X . Then ( f ) = (2 P ) + 2( P + Q ) − 3 O , (1 − f ) = ( P ) + ( A ) + ( B ) − 3 O , where A =  − 3 + √ 9 − 16 h 2 2 , 7 h 2 − 3 2 h −  h − 1 h  √ 9 − 16 h 2 2  , 5 B =  − 3 − √ 9 − 16 h 2 2 , 7 h 2 − 3 2 h +  h − 1 h  √ 9 − 16 h 2 2  . In particular, for h = 1 √ 2 , w e g et A = 3 P + Q, B = Q, implying ( f ) ⋄ (1 − f ) = 6( P ) − 10( P + Q ) ∼ 0 yielding the exp ected relation. On the other hand, for h = 1 2 , our function f b ecomes f = Y − 3 2 X . In this case, A and B ar e give n b y: A = − 3 − √ 5 2 , − 5 − 3 √ 5 4 ! , B = − 3 + √ 5 2 , − 5 + 3 √ 5 4 ! . In particular, w e ha ve the relations 2 A = 2 B = P , B − A = 2 P , A + B = − P. W e obtain ( f ) ⋄ ( 1 − f ) = ( P ) + (2 P − A ) + (2 P − B ) − 3(2 P ) + 2( Q ) + 2( P + Q − A ) +2( P + Q − B ) − 6( P + Q ) − 3( − P ) − 3( − A ) − 3( − B ) + 9 O = 2( Q + A ) + 2( Q + B ) − 6( P + Q ) + 4( P ) + 2( A ) + 2( B ) . W e need further relations among the divis ors ( A ), ( B ). Th us w e consider the follow ing f unction g = √ 5 − 1 10 Y + 3 + √ 5 20 ( X + 4) , 1 − g = 1 − √ 5 − 1 10 Y − 3 + √ 5 20 ( X + 4) . 6 W e hav e ( g ) = ( Q ) + ( A ) + ( − Q − A ) − 3 O , (1 − g ) = ( − P ) + 2( B ) − 3 O . The diamond op eration yields a new relation: ( g ) ⋄ (1 − g ) = ( Q + P ) + 2( Q − B ) − 3( Q ) + ( A + P ) + 2( A − B ) − 3 ( A ) +( − Q − A + P ) + 2( − Q − A − B ) − 3( − Q − A ) − 3( P ) − 6( − B ) + 9 O = 3( Q + P ) − 2( Q + B ) − 3( A ) + 4( Q + A ) − 3( P ) + 5( B ) . In order to get more relations, we apply the G alois conjug a te, ( g σ ) ⋄ (1 − g σ ) = 3( Q + P ) − 2( Q + A ) − 3( B ) + 4( Q + B ) − 3( P ) + 5 ( A ) . The last tw o equations yield ( g ) ⋄ (1 − g )+( g σ ) ⋄ (1 − g σ ) = 6( Q + P )+2( Q + A )+2( Q + B )+2( A )+2( B ) − 6( P ) . Finally , we obtain ( f ) ⋄ (1 − f ) − ( g ) ⋄ (1 − g ) − ( g σ ) ⋄ (1 − g σ ) = − 12( Q + P ) + 10( P ) ∼ 0 . 5 Conclus ion of the pro of Giv en a relatio nship of the fo rm a ( P ) ∼ b ( P + Q ) , w e get ar 2 ( h + 1 h ) n x 2 ( h + 1 h ) , y 2 ( h + 1 h ) o = br 2 ( i h + 1 i h ) n x 2 ( i h + 1 i h ) , y 2 ( i h + 1 i h ) o , and am  2  h + 1 h  = bm  2  i h + 1 i h  . Th us, for h = 1 √ 2 , we recov er m (8) = 8 5 m  3 √ 2  = 8 3 m  i √ 2  = 4 m (2) . 7 F or h = 1 2 , w e conclude m (16) = 11 6 m (5) = 11 5 m ( − 3i) = 11 m (1) . m (5) = 6 m (1) .  Questions that remain o p en are how to predict iden tities suc h as ( 2 ) and (3) and, more precisely , to list all such identities . Ac kno wled gmen ts The author w ould lik e to thank Herb ert Ga ngl for his encouragemen t with this problem. The author is also grateful to David Boy d, Mathew Rogers, and F ernando Ro driguez-Villegas f o r helpful discussions . Thanks are also due to the referee whose constructiv e commen ts hav e impro ve d the exp osition of the pap er. This researc h w as supp o r t ed b y Univ ersit y of Alb erta F ac. Sci. Startup Gran t N0310 00610 and NSERC Discov ery Grant 355412 -2008 References [1] M. J. Bertin, Mesure de Mahler d’une famille de p olynˆ omes. J. R eine A ngew . Math. 569 (200 4), 175–188. [2] S. J. Bloch, Higher regulators, algebraic K -theory , and zeta functions of elliptic curv es. CRM Monograph Series, 11. Americ an Mathema tic al So cie ty, Pr o v i d enc e , RI , 2000. x+97 pp. [3] D . W. Boy d, Mahler’s measure and sp ecial v alues o f L-functions, Exp er- iment. Math. 7 (19 98), 37-82. [4] J. W. S. Cassels, Lectures on elliptic curv es. London Mathematical So- ciet y Student T exts, 2 4. Cambridge University Pr ess, Cambridg e, 1991. vi+137 pp. 8 [5] C. Deninger, Deligne p erio ds of mixe d motiv es, K -theory and the entrop y of certain Z n -actions, J. A mer. Math. So c . 10 (1997) , no. 2, 25 9–281. [6] N. Kurok a wa and H. Ochiai, Mahler measures via crystalization, Com- men tarii Mathematici Univ ersitatis Sa ncti P auli, 54 (2 005), 12 1-137. [7] M. N. Lal ´ ın, M. D . Rog ers, F unctional equations f or Mahler measures of gen us-one curv es, Algebr a Numb er The ory 1 (2007), no. 1, 87–117. [8] F. Ro drig uez-Villegas, Mo dular Mahler measures I, T opics in n um b er theory (Univ ersit y Park, P A, 19 97), 17– 4 8, Math. Appl., 4 6 7, Kluw er Acad. Publ., Dordrec ht, 1 9 99. [9] F. Ro drig uez-Villegas, Iden tities b et we en Mahler measures, Numb er the- ory for the mil le nnium, I I I (Urb ana, IL, 2000) , 223 – 229, A K Pe ters, Natic k, MA, 2002. 9