aa r X i v : . [ m a t h . N T ] O c t REGULATOR OF MODULAR UNITS AND MAHLER MEASURES
WADIM ZUDILIN
Abstract.
We present a proof of the formula, due to Mellit and Brunault, whichevaluates an integral of the regulator of two modular units to the value of the L -series of a modular form of weight 2 at s = 2. Applications of the formula tocomputing Mahler measures are discussed. Introduction
The work of C. Deninger [7], D. Boyd [3], F. Rodriguez-Villegas [13] and othersprovided us with a natural link between the (logarithmic) Mahler measuresm (cid:0) P ( x , . . . , x m ) (cid:1) := 1(2 πi ) m Z · · · Z | x | = ··· = | x m | =1 log | P ( x , . . . , x m ) | d x x · · · d x m x m of certain (Laurent) polynomials P ( x , . . . , x m ), higher regulators and Be˘ılinson’sconjectures, though it took a while for those original ideas to become proofs ofsome conjectural evaluations of Mahler measures. In this note we mainly discussa recent general formula for the regulator of two modular units due to A. Mellitand F. Brunault, its consequences for 2-variable Mahler measures and some relatedproblems.For a smooth projective curve C given as the zero locus of a polynomial P ( x, y ) ∈ C [ x, y ] and two rational non-constant functions g and h on C , define the 1-form η ( g, h ) := log | g | d arg h − log | h | d arg g ; (1)here d arg g is globally defined as Im(d g/g ). The form (1) is a real 1-form definedand infinitely many times differentiable on C \ S , where S is the set of zeros and polesof g and h . Furthermore, it is not hard to verify that the form (1) is antisymmetric,bi-additive and closed; the latter fact follows fromd η ( g, h ) = Im (cid:18) d gg ∧ d hh (cid:19) = 0 , Date : 14 April 2013.
Revised : 23 September 2013.2010
Mathematics Subject Classification.
Primary 11F67; Secondary 11F11, 11F20, 11G16,11G55, 11R06, 14H52, 19F27.
Key words and phrases.
Regulator, Mahler measure, L -value of elliptic curve.Work is supported by the Australian Research Council. as the curve C has dimension 1. In turn, the closedness of (1) implies that, for aclosed path γ in C \ S , the regulator map r ( { g, h } ) : γ Z γ η ( g, h ) (2)only depends on the homology class [ γ ] of γ in H ( C \ S, Z ).Assuming that the polynomial P ( x, y ) is tempered [2, 13], factorising it as apolynomial in y with coefficients from C [ x ], P ( x, y ) = a ( x ) n Y j =1 ( y − y j ( x )) , and applying Jensen’s formula, we can write [2, 6, 10, 13] the Mahler measure of P in the form m (cid:0) P ( x, y ) (cid:1) = m (cid:0) a ( x ) (cid:1) + 12 π r ( { x, y } )([ γ ]) , (3)where γ := n [ j =1 (cid:8) ( x, y j ( x )) : | x | = 1 , | y j ( x ) | ≥ (cid:9) = { ( x, y ) ∈ C : | x | = 1 , | y | ≥ } (4)is the union of at most n closed paths in C \ S .In case the curve C : P ( x, y ) = 0 admits a parameterisation by means of modular units x ( τ ) and y ( τ ), where the modular parameter τ belongs to the upper halfplane H = { τ ∈ C : Im τ > } , one can change to the variable τ in the integral (2) for r ( { x, y } ); the class [ γ ] in this case [5] becomes a union of paths joining certain cuspsof the modular functions x ( τ ) and y ( τ ). The following general result completes thecomputation of the Mahler measure in the case when x ( τ ) and y ( τ ) are given asquotients/products of modular units g a ( τ ) := q NB ( a/N ) / Y n ≥ n ≡ a mod N (1 − q n ) Y n ≥ n ≡− a mod N (1 − q n ) , q = exp(2 πiτ ) , (5)where B ( x ) = B ( x ) := { x } − { x } + . Theorem 1 (Mellit–Brunault [12]) . For a , b and c integral, with ac and bc notdivisible by N , Z i ∞ c/N η ( g a , g b ) = 14 π L ( f ( τ ) − f ( i ∞ ) , , (6) where the weight modular form f ( τ ) = f a,b ; c ( τ ) is given by f a,b ; c := e a,bc e b, − ac − e a, − bc e b,ac and e a,b ( τ ) := 12 (cid:18) ζ aN − ζ aN + 1 + ζ bN − ζ bN (cid:19) + X m,n ≥ ( ζ am + bnN − ζ − ( am + bn ) N ) q mn , ζ N := exp(2 πi/N ) , (7) are weight level N Eisenstein series.
EGULATOR OF MODULAR UNITS AND MAHLER MEASURES 3
The L -value on the right-hand side of (6) is well defined because of subtractingthe constant term f ( i ∞ ) = 12 (cid:18) ζ bN − ζ bN ζ bcN − ζ bcN − ζ aN − ζ aN ζ acN − ζ acN (cid:19) = − (cid:18) cot πbN cot πbcN − cot πaN cot πacN (cid:19) in the q -expansion f ( τ ) = f ( i ∞ ) + P n ≥ c n q n . Furthermore, if a linear combination f ( τ ) = X ( a,b,c ) ∈M λ a,b,c f a,b ; c ( τ ) , λ a,b,c ∈ C , happens to be a cusp form (and this corresponds to application of Theorem 1 toMahler measures), then formula (6) produces the evaluation X ( a,b,c ) ∈M λ a,b,c Z i ∞ c/N η ( g a , g b ) = 14 π L ( f ( τ ) , . Note as well that the theorem allows one to integrate between any cusps c/N and d/N with the help of R d/Nc/N = R i ∞ c/N − R i ∞ d/N .Here is a sketch of the proof of Theorem 1; details are given in Section 2. Weparameterise the contour of integration by τ = c/N + it , 0 < t < ∞ , and notethat the M¨obius transformation τ ′ := ( cτ − ( c + 1) /N ) / ( N τ − c ) preserves thecontour: τ ′ = c/N + i/ ( N t ). Then the logarithms of g a ( τ ) and g b ( τ ), hence theirreal and imaginary parts — everything we need for computing the form (1), canbe written as explicit Eisenstein series of weight 0 in powers of exp( − πt ) andexp( − π/ ( N t )). Finally, executing an analytical change of variable from [14] (asdetailed in [18, Section 3]) the integrand becomes a linear combination of pairwiseproducts of weight 1 Eisenstein series in powers of exp( − πt ) integrated against theform t d t along the line 0 < t < ∞ .Applications of Theorem 1 to Boyd’s and Rodriguez-Villegas’ conjectural eval-uations of 2-variable Mahler measures are discussed in Section 3, while Section 4highlights some open problems related to 3-variable Mahler measures.2. Proof of the Mellit–Brunault formula
The two auxiliary lemmas indicate particular modular transformations of themodular functions (5) and the Eisenstein series (7). Lemma 1 also describes theasymptotic behaviour of the modular functions (5) in a neighbourhood of a cuspwith Re τ = 0; it is used in the form (10) to determine the integration contours (4)for our applications in Section 3. WADIM ZUDILIN
Lemma 1.
For a, c integers, log g a ( c/N + it ) = πicB ( a/N ) − πt N B ( a/N ) − X m,n ≥ n ≡ a ζ acmN m exp( − πmnt ) − X m,n ≥ n ≡− a ζ − acmN m exp( − πmnt )= − πi πia ( c + 1)( N − ac ) + πicB ( ac/N ) − πB ( ac/N ) N t − X m,n ≥ n ≡ ac ζ − amN m exp (cid:18) − πmnN t (cid:19) − X m,n ≥ n ≡− ac ζ amN m exp (cid:18) − πmnN t (cid:19) , where t > .Proof. First note that definition (5) implieslog g a ( τ ) = πiτ N B ( a/N ) + X n ≥ n ≡ a log(1 − q n ) + X n ≥ n ≡− a log(1 − q n )= πiτ N B ( a/N ) − X m,n ≥ n ≡ a q mn m − X m,n ≥ n ≡− a q mn m . Therefore, the substitution τ = c/N + it , equivalently q = ζ cN exp( − πt ), results inthe first expansion of the lemma.Secondly, the modular units (5) are particular cases of the ‘generalized Dedekindeta functions’ [17, eq. (3)]. Applying [17, Theorem 1] with the choice h = 0 and γ = (cid:0) c − c − − c (cid:1) we deduce that g a ( τ ) = e g a,c (cid:18) cτ − ( c + 1) /NN τ − c (cid:19) , where e g a,c ( τ ) := exp( − πi/ πia ( c + 1)( N − ac )) q NB ( ac/N ) / × Y n ≥ n ≡ ac mod N (1 − ζ − a ( c +1) N q n ) Y n ≥ n ≡− ac mod N (1 − ζ a ( c +1) N q n ) . On the other hand, τ ′ := cτ − ( c + 1) /NN τ − c (cid:12)(cid:12)(cid:12)(cid:12) τ = c/N + it = cN + iN t , so thatlog e g a,c ( τ ′ ) = − πi πia ( c + 1)( N − ac ) + πicB ( ac/N ) − πB ( ac/N ) N t − X m,n ≥ n ≡ ac ζ − a ( c +1) m + cmnN m exp (cid:18) − πmnN t (cid:19) − X m,n ≥ n ≡− ac ζ a ( c +1) m + cmnN m exp (cid:18) − πmnN t (cid:19) , EGULATOR OF MODULAR UNITS AND MAHLER MEASURES 5 and it remains to use the congruences n ≡ ac and n ≡ − ac to simplify the exponentsof the roots of unity. (cid:3) Lemma 2.
For a, b integers not divisible by N , N τ e a,b (cid:18) − N τ (cid:19) = e e a,b ( τ ) := X m,n ≥ m ≡ a, n ≡ b q mn − X m,n ≥ m ≡− a, n ≡− b q mn . Proof.
In [16, Section 7] the following general Eisenstein series of weight 1 and level N are introduced: G a,c ( τ ) = G N, c,a ) ( τ ) := − πiN κ a,c + X m,n ≥ n ≡ c mod N ζ amN q mn/N − X m,n ≥ n ≡− c mod N ζ − amN q mn/N ! , where κ a,c :=
12 1 + ζ aN − ζ aN if c ≡ N , − (cid:26) cN (cid:27) if c N .
Then for γ = (cid:0) A BC D (cid:1) ∈ SL ( Z ) we have G a,c ( γτ ) = ( Cτ + D ) G aD + cB,aC + cA ( τ ) . (8)The partial Fourier transform from [8, Chapter III] applied to G a,c results in b G a,b ( τ ) := N − X c =0 ζ bcN G a,c ( τ ) = − πiN (cid:18) ζ aN − ζ aN + 1 + ζ bN − ζ bN (cid:19) − πiN X m,n ≥ ( ζ am + bnN − ζ − ( am + bn ) N ) q mn/N . On the other hand, taking γ = (cid:0) −
11 0 (cid:1) in (8) we find that τ − b G a,b ( − /τ ) = N − X c =0 ζ bcN G − c,a ( τ )= − πiN N − X c =0 ζ bcN − (cid:26) aN (cid:27) + X m,n ≥ n ≡ a ζ − cmN q mn/N − X m,n ≥ n ≡− a ζ cmN q mn/N ! = − πi X m,n ≥ n ≡ a, m ≡ b q mn/N − X m,n ≥ n ≡− a, m ≡− b q mn/N ! . Using now b G a,b ( N τ ) = − πi e a,b ( τ ) /N we obtain the desired transformation. (cid:3) The next two statements are to take care of integrating the constant terms ofauxiliary Eisenstein series.
WADIM ZUDILIN
Lemma 3.
For a, b integers not divisible by N , Z ∞ (cid:18) e a,b ( it ) + e a, − b ( it ) − ζ aN − ζ aN (cid:19) t d t = i Cl (cid:18) πaN (cid:19) B (cid:18) bN (cid:19) , where Cl ( x ) := X m ≥ sin mxm denotes Clausen’s ( dilogarithmic ) function.Proof. The integral under consideration is equal to Z ∞ X m,n ≥ ( ζ am + bnN − ζ − ( am + bn ) N + ζ am − bnN − ζ − ( am − bn ) N ) exp( − πmnt ) t d t = Z ∞ X m,n ≥ ( ζ amN − ζ − amN )( ζ bnN + ζ − bnN ) exp( − πmnt ) t d t. On using the Mellin transform Z ∞ exp( − πkt ) t s − d t = Γ( s )(2 π ) s k s for Re s > , (9)the integral of the double sum evaluates to14 π X m ≥ ζ amN − ζ − amN m X n ≥ ζ bnN + ζ − bnN n = iπ Cl (cid:18) πaN (cid:19) X n ≥ cos(2 πnb/N ) n . It remains to use X n ≥ cos nxn = π B (cid:18) x π (cid:19) , and the required evaluation follows. (cid:3) Lemma 4.
For a, b integers not divisible by N , Z ∞ iN t d X m ≥ ζ amN − ζ − amN m X n ≥ n ≡ b − X n ≥ n ≡− b ! exp (cid:18) − πmnN t (cid:19) = − i Cl (cid:18) πaN (cid:19) ζ bN − ζ bN . Proof.
Performing the change of variable u = 1 / ( N t ) in the integral, it becomesequal to 2 πNi Z ∞ X m ≥ ( ζ amN − ζ − amN ) X n ≥ n ≡ b − X n ≥ n ≡− b ! n exp( − πmnu ) u d u, EGULATOR OF MODULAR UNITS AND MAHLER MEASURES 7 and applying (9) with s → + it evaluates to Nπ X m ≥ sin(2 πam/N ) m lim s → + X n ≥ n ≡ b − X n ≥ n ≡− b ! n s = 1 π Cl (cid:18) πaN (cid:19) · (cid:0) ψ (1 − { b/N } ) − ψ ( { b/N } ) (cid:1) = 1 π Cl (cid:18) πaN (cid:19) π cot πbN , where ψ ( x ) is the logarithmic derivative of the gamma function. It remains to usecot( πb/N ) = − i (1 + ζ bN ) / (1 − ζ bN ). (cid:3) Proof of Theorem . To integrate the 1-form η ( g a , g b ) along the interval τ ∈ ( c/N, i ∞ )we make the substitution τ = c/N + it , 0 < t < ∞ . It follows from Lemma 1 thatlog | g a ( τ ) | = − πB ( ac/N ) N t − X m ≥ ζ amN + ζ − amN m X n ≥ n ≡ ac + X n ≥ n ≡− ac ! exp (cid:18) − πmnN t (cid:19) (10)andd arg g a ( τ ) = − i d X m ≥ ζ acmN − ζ − acmN m X n ≥ n ≡ a − X n ≥ n ≡− a ! exp( − πmnt )= 12 i d X m ≥ ζ amN − ζ − amN m X n ≥ n ≡ ac − X n ≥ n ≡− ac ! exp (cid:18) − πmnN t (cid:19) . This computation implies η ( g a , g b ) = − πB ( ac/N )2 iN t d X m ≥ ζ bmN − ζ − bmN m X n ≥ n ≡ bc − X n ≥ n ≡− bc ! exp (cid:18) − πmnN t (cid:19) + 14 i X m ≥ ζ am N + ζ − am N m X n ≥ n ≡ ac + X n ≥ n ≡− ac ! exp (cid:18) − πm n N t (cid:19) × d X m ≥ ζ bcm N − ζ − bcm N m X n ≥ n ≡ b − X n ≥ n ≡− b ! exp( − πm n t ) WADIM ZUDILIN + πB ( bc/N )2 iN t d X m ≥ ζ amN − ζ − amN m X n ≥ n ≡ ac − X n ≥ n ≡− ac ! exp (cid:18) − πmnN t (cid:19) − i X m ≥ ζ bm N + ζ − bm N m X n ≥ n ≡ bc + X n ≥ n ≡− bc ! exp (cid:18) − πm n N t (cid:19) × d X m ≥ ζ acm N − ζ − acm N m X n ≥ n ≡ a − X n ≥ n ≡− a ! exp( − πm n t ) . The terms involving double sums only can be integrated with the help of Lemma 4,and we obtain Z i ∞ c/N η ( g a , g b ) = πi ζ bcN − ζ bcN Cl (cid:18) πbN (cid:19) B (cid:18) acN (cid:19) − πi ζ acN − ζ acN Cl (cid:18) πaN (cid:19) B (cid:18) bcN (cid:19) − π i X m ,m ≥ ( ζ am N + ζ − am N )( ζ bcm N − ζ − bcm N ) X n ≥ n ≡ ac + X n ≥ n ≡− ac ! X n ≥ n ≡ b − X n ≥ n ≡− b ! − X m ,m ≥ ( ζ bm N + ζ − bm N )( ζ acm N − ζ − acm N ) X n ≥ n ≡ bc + X n ≥ n ≡− bc ! X n ≥ n ≡ a − X n ≥ n ≡− a !! × n m Z ∞ exp (cid:18) − π (cid:18) m n N t + m n t (cid:19)(cid:19) d t. Now we execute the change of variable u = n t/m , interchange integration andquadruple summation and use Lemma 2: Z i ∞ c/N η ( g a , g b ) = πi ζ bcN − ζ bcN Cl (cid:18) πbN (cid:19) B (cid:18) acN (cid:19) − πi ζ acN − ζ acN Cl (cid:18) πaN (cid:19) B (cid:18) bcN (cid:19) − π i Z ∞ X m ,m ≥ ( ζ am N + ζ − am N )( ζ bcm N − ζ − bcm N ) exp( − πm m u ) × X n ≥ n ≡ ac + X n ≥ n ≡− ac ! X n ≥ n ≡ b − X n ≥ n ≡− b ! exp (cid:18) − πn n N u (cid:19) − X m ,m ≥ ( ζ bm N + ζ − bm N )( ζ acm N − ζ − acm N ) exp( − πm m u ) × X n ≥ n ≡ bc + X n ≥ n ≡− bc ! X n ≥ n ≡ a − X n ≥ n ≡− a ! exp (cid:18) − πn n N u (cid:19) d u EGULATOR OF MODULAR UNITS AND MAHLER MEASURES 9 = πi ζ bcN − ζ bcN Cl (cid:18) πbN (cid:19) B (cid:18) acN (cid:19) − πi ζ acN − ζ acN Cl (cid:18) πaN (cid:19) B (cid:18) bcN (cid:19) − π i Z ∞ (cid:18) e a,bc ( iu ) − e a, − bc ( iu ) − ζ bcN − ζ bcN (cid:19)(cid:0)e e b,ac ( i/ ( N u )) + e e b, − ac ( i/ ( N u )) (cid:1) − (cid:18) e b,ac ( iu ) − e b, − ac ( iu ) − ζ acN − ζ acN (cid:19)(cid:0)e e a,bc ( i/ ( N u )) + e e a, − bc ( i/ ( N u )) (cid:1) d u = πi ζ bcN − ζ bcN Cl (cid:18) πbN (cid:19) B (cid:18) acN (cid:19) − πi ζ acN − ζ acN Cl (cid:18) πaN (cid:19) B (cid:18) bcN (cid:19) + π Z ∞ (cid:18) e a,bc ( iu ) − e a, − bc ( iu ) − ζ bcN − ζ bcN (cid:19)(cid:0) e b,ac ( iu ) + e b, − ac ( iu ) (cid:1) u − (cid:18) e b,ac ( iu ) − e b, − ac ( iu ) − ζ acN − ζ acN (cid:19)(cid:0) e a,bc ( iu ) + e a, − bc ( iu ) (cid:1) u d u = πi ζ bcN − ζ bcN Cl (cid:18) πbN (cid:19) B (cid:18) acN (cid:19) − πi ζ acN − ζ acN Cl (cid:18) πaN (cid:19) B (cid:18) bcN (cid:19) + π Z ∞ (cid:0) e a,bc ( iu ) e b, − ac ( iu ) − e a, − bc ( iu ) e b,ac ( iu ) (cid:1) u − (cid:18) ζ bcN − ζ bcN (cid:0) e b,ac ( iu ) + e b, − ac ( iu ) (cid:1) − ζ acN − ζ acN (cid:0) e a,bc ( iu ) + e a, − bc ( iu ) (cid:1)(cid:19) u d u (we apply Lemma 3)= π Z ∞ (cid:18) f a,b ; c ( iu ) + 12 1 + ζ aN − ζ aN ζ acN − ζ acN −
12 1 + ζ bN − ζ bN ζ bcN − ζ bcN (cid:19) u d u, and the result follows by appealing to (9). (cid:3) Applications
The modularity theorem guarantees that an elliptic curve C : P ( x, y ) = 0 canbe parameterised by modular functions x ( τ ) and y ( τ ), whose level N is necessarilythe conductor of C , such that the pull-back of the canonical differential on C isproportional to 2 πif ( τ ) d τ = f ( τ ) d q/q , where f is (up to an isogeny) a normalisednewform of weight 2 and level N , which automatically happens to be a cusp form anda Hecke eigenform. Computing the conductor of C and producing the cusp form f of this level give an efficient strategy to determine successively the coefficientsin the q -expansions of x ( τ ) = ε q − M + · · · and y ( τ ) = ε q − M + · · · subject to P ( x ( τ ) , y ( τ )) = 0, where ε and ε are suitable nonzero constants. The particularform of q -expansions only fixes a normalisation of x ( τ ) and y ( τ ) up to the action ofthe corresponding congruence subgroup Γ ( N ). Finally, it remains to verify whether x ( τ ) and y ( τ ) just found are modular units — modular functions whose all zeroesand poles are at cusps (so that they admit eta-like product expansions); if this is thecase, we can use Theorem 1 to compute the Mahler measure m( P ( x, y )). Note thatthe property of being a modular unit imposes a strong condition on the q -expansion of the logarithmic derivative — it can be easily detected in practice by examining acouple of (hundred) terms in the q -expansion of the latter.In this section we touch the ‘classical’ family of Mahler measuresm( xy + ( x + kx + 1) y + x ) = m (cid:16) k + x + 1 x + y + 1 y (cid:17) , k ∈ Z \ { , } , which goes back to the works [3, 7, 13]. Namely, we will see that Theorem 1 appliesin the cases when the corresponding zero locus E : k + x + 1 x + y + 1 y = 0 (11)can be parameterised by modular units. For this family of tempered Laurent poly-nomials, equation (3) assumes the formm (cid:16) k + x + 1 x + y + 1 y (cid:17) = m( y + ( k + x + x − ) y + 1) = 12 π r ( { x, y } )([ γ ]) , (12)where γ is a single closed path on E \ { (0 , } corresponding to the zero y ( x ) of y + ( k + x + x − ) y + 1 which satisfies | y ( x ) | ≥ k = 2 i ; this is Example 2 below. Themodular functions x and y satisfying (11) are searched in the form x ( τ ) = ( εq ) − + · · · and y ( τ ) = − ( εq ) − + · · · , where ε ∈ Z [ k ] is chosen so that k/ε is a positive integer.The condition on the pull-back of the canonical differential on E takes the form q (d x/ d q ) εx ( y − /y ) = f, where f ( τ ) is the corresponding Hecke eigenform of weight 2.The computational part of the examples below was accomplished in sage and gp-pari , which allowed us to compute as many terms in the q -expansions of amodular parameterisation of a given elliptic curve as requested. Assisted with thissoftware, we were normally able to relate occurring modular forms and functions(for example, their product expansions) by computing and examining sufficientlymany terms in their q -expansions.Below we will have occasional appearance of Dedekind’s eta-function η ( τ ) := q / Q ∞ n =1 (1 − q n ). We hope that this extra eta notation does not cause any confusionwith (1), as it depends here on a single variable, which is always a rational multipleof τ from the upper halfplane. Example 1.
The most classical example corresponds to the choice k = 1, when theelliptic curve in (11) has conductor N = 15 and can be parameterised by modularunits x ( τ ) = 1 q ∞ Y n =0 (1 − q n +7 )(1 − q n +8 )(1 − q n +2 )(1 − q n +13 ) = g ( τ ) g ( τ ) ,y ( τ ) = − q ∞ Y n =0 (1 − q n +4 )(1 − q n +11 )(1 − q n +1 )(1 − q n +14 ) = − g ( τ ) g ( τ ) , EGULATOR OF MODULAR UNITS AND MAHLER MEASURES 11 so that q (d x/ d q ) x ( y − /y ) = f ( τ ) := η ( τ ) η (3 τ ) η (5 τ ) η (15 τ )and the path of integration γ in (12) corresponds to the range of τ between the twocusps − / / (15). Therefore, Theorem 1 results inm (cid:16) x + 1 x + y + 1 y (cid:17) = 12 π (cid:16)Z i ∞− / − Z i ∞ / (cid:17) η ( g /g , g /g )= 18 π L (2 f , − − f , − − f , − + 2 f , − , π L ( f , , which is precisely Boyd’s conjecture from [3] first proven in [15].Note that this evaluation implies some other Mahler measures, namely [9, 10]m (cid:16) x + 1 x + y + 1 y (cid:17) = 6m (cid:16) x + 1 x + y + 1 y (cid:17) m (cid:16)
16 + x + 1 x + y + 1 y (cid:17) = 11m (cid:16) x + 1 x + y + 1 y (cid:17) , m (cid:16) i + x + 1 x + y + 1 y (cid:17) = 5m (cid:16) x + 1 x + y + 1 y (cid:17) , though the corresponding elliptic curves k + x + 1 /x + y + 1 /y = 0 for k = 5, 16 and3 i are not parameterised by modular units. Example 2 ([11]) . The modular parameterisation of (11) for k = 2 i (the conductorof elliptic curve is then N = 40) and the corresponding Mahler measure evaluationm (cid:16) i + x + 1 x + y + 1 y (cid:17) = 10 π L ( f , , where f ( τ ) := η ( τ ) η (8 τ ) η (10 τ ) η (20 τ ) η (5 τ ) η (40 τ ) + η (2 τ ) η (4 τ ) η (5 τ ) η (40 τ ) η ( τ ) η (8 τ ) , were given in Mellit’s talk [11]. He identifies x ( τ ) and y ( τ ) with infinite productswhich are fully expressible by means of Ramanujan’s lambda function λ ( τ ) = q / ∞ Y n =1 (1 − q n )( n ) = q / ∞ Y n =1 (1 − q n − )(1 − q n − )(1 − q n − )(1 − q n − ) ;namely, x ( τ ) = − i λ (4 τ ) λ ( τ ) λ (8 τ ) = − i g g g g g g g g g g g g g g ,y ( τ ) = i λ ( τ ) λ (2 τ ) λ (8 τ ) = i g g g g g g g g g g in the notation (5) with N = 40. The corresponding range of τ for the path γ in(12) is from 1 /
10 to − / Example 3.
The elliptic curve (11) for k = 2 has conductor N = 24 and admitsparameterisation by modular units x ( τ ) = g g g g g g , y ( τ ) = − g g g g . Theorem 1 applies and produces the evaluationm (cid:16) x + 1 x + y + 1 y (cid:17) = 12 π (cid:16)Z i ∞− / − Z i ∞ / (cid:17) η (cid:18) g g g g g g , g g g g (cid:19) = 6 π L ( f , , where f ( τ ) := η (2 τ ) η (4 τ ) η (6 τ ) η (12 τ ), conjectured in [3] and established in [14]. Example 4.
For N = 17, the pair of modular units x ( τ ) = − i g g g g , y ( τ ) = i g g g g parameterise the elliptic curve i + x + 1 /x + y + 1 /y = 0. Applying Theorem 1 for τ ranging from 3 /
17 to − /
17, we obtainm (cid:16) i + x + 1 x + y + 1 y (cid:17) = 172 π L ( f , , where f ( τ ) := q (d x/ d q ) ix ( y − /y ) = q − q − q − q + 4 q + 3 q − q + 2 q − q − q − q + q + O ( q ) . This Mahler measure evaluation was conjectured in [13, Table 4].
Example 5.
Another conjecture in [13, Table 4],m (cid:16) √ x + 1 x + y + 1 y (cid:17) = 72 π L ( f , , corresponds to k = √ Z of conductor N = 56.The conjecture follows from parameterisation of the curve by the couple x ( τ ) = 1 √ η ( τ ) η (4 τ ) η (7 τ ) η (28 τ ) η (2 τ ) η (8 τ ) η (14 τ ) η (56 τ ) ,y ( τ ) = − √ η (2 τ ) η (4 τ ) η (14 τ ) η (28 τ ) η ( τ ) η (7 τ ) η (8 τ ) η (56 τ ) , so that f ( τ ) := q (d x/ d q ) √ x ( y − /y ) = q + 2 q − q − q − q + 2 q − q + 8 q − q + 6 q + 8 q + O ( q ) , and integration in Theorem 1 for τ ∈ ( − / , − / ∪ (5 / , / EGULATOR OF MODULAR UNITS AND MAHLER MEASURES 13
It is not clear whether there are finitely or infinitely many cases of the parameter k in (11) subject to parameterisation by modular units. A possible approach incases when such parameterisation is not available is writing down algebraic rela-tions between any two standard modular units (5) of a given level N and sieving therelations which may be used in producing the Mahler measures of 2-variable poly-nomials which are potentially linked to the wanted Mahler measures by K -theoreticmachinery [6, 9, 10].Finding what curves C : P ( x, y ) = 0 can be parameterised by modular units isan interesting question itself. F. Brunault notices some heuristics to the fact thatthere are only finitely many function fields F of a given genus g over Q which embedinto the function field of a modular curve such that F can be generated by modularunits; for g ≥ It would be desirable to have an analogue of Theorem 1 for 3-variable Mahlermeasures of (Laurent) polynomials P ( x, y, z ) such that the intersection of the zeroloci P ( x, y, z ) = 0 and P (1 /x, /y, /z ) = 0 defines an elliptic curve E , and m( P )is presumably related to the L -series of E evaluated at s = 3. No example of thistype is established, and one of the simplest evaluations is Boyd’s conjecture [4]m (cid:0) (1 + x )(1 + y ) − z (cid:1) ? = 2 L ′ ( E , −
1) = 2254 π L ( E , . On the surface (1 + x )(1 + y ) − z = 0 we have x ∧ y ∧ z = x ∧ y ∧ (1 + x )(1 + y ) = x ∧ y ∧ (1 + x ) + x ∧ y ∧ (1 + y )= − x ∧ (1 + x ) ∧ y + y ∧ (1 + y ) ∧ x = − ( − x ) ∧ (1 + x ) ∧ y + ( − y ) ∧ (1 + y ) ∧ x. Applying the machinery described in [6, Section 5.2] to the 3-variable polynomial P ( x, y, z ) = (1 + x )(1 + y ) − z we obtain m ( P ) = 14 π Z γ (cid:0) ω ( − x, y ) − ω ( − y, x ) (cid:1) , where ω ( g, h ) := D ( g ) d arg h + 13 (cid:0) log | g | d log | − g | − log | − g | d log | g | (cid:1) log | h | (13)and γ := { ( x, y, z ) : | x | = | y | = | z | = 1 } ∩ { ( x, y, z ) : (1 + x )(1 + y ) − z = 0 }∩ { ( x, y, z ) : (1 + x )(1 + y ) z − xy = 0 } . Note that { (1 + x )(1 + y ) − z = 0 } ∩ { (1 + x )(1 + y ) z − xy = 0 } is the double coverof an elliptic curve of conductor 15. Indeed, eliminating z we can write (one half of)its equation as (1 + x )(1 + y ) + x y = 0 in variables x = √ x , y = √ y , or x + 1 /x + y + 1 /y + 1 = 0in variables x = x y , y = x /y . Using the parameterisation of the latter equationby the modular units from Example 1 we find out that m ( P ) = 12 π Z / − / (cid:0) ω ( X, Y ) − ω ( Y, X ) (cid:1) where X ( τ ) := g ( τ ) g ( τ ) g ( τ ) g ( τ ) = q − + O ( q − ) and Y ( τ ) := g ( τ ) g ( τ ) g ( τ ) g ( τ ) = 1 + O ( q ) . Also note that1 − X ( τ ) = − g ( τ ) g ( τ ) g ( τ ) g ( τ ) = − q − + O ( q − ) and 1 − Y ( τ ) = g ( τ ) g ( τ ) g ( τ ) g ( τ ) = q + O ( q )are modular units.The problem with integrating the form (13) is that it is, roughly speaking, inte-grating the product of three modular components: two of them are logarithms ofmodular functions (hence of weight 0) and one is the logarithmic derivative of amodular function (hence of weight 2). On the other hand, the expected data forapplying the method from [14] used in our proof of Theorem 1 in Section 2 wouldbe integrating a product of two Eisenstein series of weights − Acknowledgements.
This note would be hardly possible without constant pa-tience of A. Mellit in describing details of his work with F. Brunault. I am pleasedto thank Mellit for all those lectures he delivered to me in person and by e-mail,as well as for providing me with the sketch [12] of proof of what is stated here asTheorem 1. I am deeply grateful to F. Brunault, M. Rogers and J. Wan for theirhelpful assistance on certain parts of this work. Finally, I thank the anonymousreferee for her valuable comments and healthy criticism that helped to improve theexposition.
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