On the Mahler measure of hyperelliptic families

We prove Boyd’s “unexpected coincidence” of the Mahler measures for two families of two-variate polynomials defining curves of genus 2. We further equate the same measures to the Mahler measures of polynomials y3-y+x3-x+kxy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^3-y+x^3-x+kxy$$\end{document} whose zero loci define elliptic curves for k≠0,±3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ne 0,\pm 3$$\end{document}.


Introduction
In his pioneering systematic study [2] of the Mahler measures of two-variate polynomials Boyd has distinguished several special families, for which the measures are related to the L-values of the curves defined by the zero loci of the polynomials. The two particular families P k (x, y) = (x 2 + x + 1)y 2 + kx(x + 1)y + x(x 2 + x + 1) and Q k (x, y) = (x 2 + x + 1)y 2 + (x 4 + kx 3 + (2k − 4)x 2 + kx + 1)y + x 2 (x 2 + x + 1) are nicknamed in [2] as Family 3.2 and Family 3.5B, respectively. Generically, both P k (x, y) = 0 and Q k (x, y) = 0 define curves of genus 2 whose jacobians are isogenous to the product of two elliptic curves. Computing the Mahler measures of P k (x, y) and Q k (x, y) numerically and identifying them as rational multiples of the L-values L (E k , 0), where is isomorphic to one of the elliptic curves in the product for each of the two families, Boyd observes the "unexpected coincidence" m(P k ) = m(Q k+2 ) for integer k in the range 4 ≤ k ≤ 33 (but not for k ≤ 3). The primary goal of this note is to confirm Boyd's observation.
Note that for k = 0, ±3 the curve E k is elliptic and it is isomorphic to the elliptic curve R k (x, y) = 0, where the polynomial is tempered-all the faces of its Newton polygon are represented by cyclotomic polynomials. The elliptic origin of the family R k (x, y) and Beilinson's conjectures predict [2,5] that, apart from a finite set of k, the measure m(R k ) is Q-proportional to the L-value L (E k , 0) for k ∈ Z (in fact, even for k such that k 2 ∈ Z as in any such case the curve R k (x, y) = 0 possesses the model defined over Z). Our next result unites the predictions with the findings of Boyd in [2].
Noticing that P −k (x, y) = P k (x, −y) and R −k (x, y) = R k (−x, −y) we conclude that m(P |k| ) = m(P k ) and m(R |k| ) = m(R k ), hence it is sufficient to establish the identity in Theorem 2 and analyse the two polynomial families for positive real k only.
Our analysis of the three polynomial families is performed in Sects. 1-3, each section devoted to one family. We compute the derivatives of the corresponding Mahler measures with respect to the parameter k and make use of the easily seen asymptotics as |k| → ∞, to conclude about the equality of the Mahler measures themselves. This is a strategy we have successfully employed before in [1]. Our findings provide one with the reasons of why the ranges for k in Theorems 1 and 2 cannot be refined, and in Sect. 4 we discuss some further aspects of this "expected noncoincidence." One of our reasons for linking the Mahler measures of hyperelliptic families P k (x, y) and Q k (x, y) to that of elliptic family R k (x, y), not previously displayed, is a hope to actually prove m(R k ) = c k L (E k , 0) with c k ∈ Q × for some values of k. Armed with the recent formula for the regulator of modular units [7] and its far-going generalisation for the regulator of Siegel units [4] established by Brunault, such identities are expected to be automated in the near future. The main obstacle to produce a single example for m(R k ) is of purely computational nature: the smallest conductor of the elliptic curve E k one gets for k > 3, k 2 ∈ Z, is 224 = 2 5 × 7 when k = 4. We further comment on this circumstance and on a related conjecture of Boyd for m(Q −1 ) in the final section.

The first family
We use the equality m(P |k| ) = m(P k ) to reduce our analysis in this section to that for k ≥ 0. Write and > 0, when we order the zeroes y 1 (x), y 2 (x) appropriately. In the latter case and |y 2 (x)| = min{|y 1 (x)|, |y 2 (x)|} < 1.
Using Jensen's formula and the symmetry y 1 (x) = y 1 (x −1 ), we obtain Im x>0 Im x>0 Im x>0 Re log The derivative of the result with respect to k is which is a complete elliptic integral. Performing additionally the change c = (4 − v)/16 we obtain in particular, we have the following.

The second family
The analysis here is very similar to the one we had in the paper [1]. First introduce where In the latter case With the help of Jensen's formula we obtain Im x>0 Note that for k > 0 we have if k ≥ 4.

Remark 1 The appearance of incomplete elliptic integrals
for k < 4 hints on why the Mahler measures q(k + 2) are possibly not related to the corresponding L-values (see the question marks and the "half-Mahler" measures m in [2, Table 9]). Our next statement refers to the situation when incomplete elliptic integrals do not occur.
for k > 3. On comparing the integrals in (3) and (4) this implies the required coincidence.
Proof of Theorem 1 Proposition 2 implies that p(k) = q(k + 2) + C for k ≥ 4, with some constant C independent of k. On using the asymptotics (2) we conclude that C = 0, and the theorem follows.

The third family
Since m(R |k| ) = m(R k ), we assume that k ≥ 0 throughout the section.
For the elliptic family we write This time the zeroes y 1 (x) and y 2 (x) of the quadratic polynomial R k (x, y) satisfy Lemma 1 If k ≥ 3 then k (x) ≥ 0, so that both y 1 (x) and y 2 (x) are real. If 0 ≤ k < 3 then y 1 (x) and y 2 (x) are complex conjugate to each other for Proof Note that 16 cos 2 θ (3 − 4 cos 2 θ) ≤ max 0≤c≤1 16c(3 − 4c) = 9, hence The second part of the statement is a mere computation.
Proof Denote c = cos 2 θ for x = exp(iθ), so that our task is to show that for 0 ≤ c ≤ 1. If k 2 − 48c + 64c 2 ≥ 0, meaning that either k ≥ 3 and c ∈ [0, 1] or 2 , the inequality (6) is equivalent to The latter inequality holds automatically when the right-hand side is nonpositive, that is, , and the required inequality follows.
for 0 ≤ c ≤ 1, we first notice that the inequality is trivially true for c ≥ k 2 /16 since the right-hand side is then nonpositive. If c < k 2 /16, the inequality (7) after squaring becomes equivalent to 8 √ c(1 − c) ≤ k. The latter inequality holds true because the maximum of √ c(1 − c) is attained at c = 1/3 and is equal to 2/(3 √ 3). .

(8)
Proof Using the two lemmas above we conclude that for values of k ≥ 16/(3 √ 3) Jensen's formula gives us It remains to perform the change c = t 2 .
If 0 < k < 16/(3 √ 3) then the cubic polynomial f (t) = 8t 3 − 8t + k has two real zeroes on the interval 0 < t < 1, since Proof Note that for the values of x corresponding to t 1 (k) and t 2 (k) we always have k (x) ≥ 0, so that both y 1 (x) and y 2 (x) are real. The solutions of |y 1 (x)| = 1 and |y 2 (x)| = 1 correspond to solving where t = | cos θ | = |x + x −1 |/2. By elementary manipulations the latter equation reduces to 8t 3 − 8t + k = 0, and the remaining task is to distinguish whether we get |y 1 (x)| = 1 or |y 2 (x)| = 1. We do not reproduce this technical but elementary analysis here.
Proof To each x on the unit circle we assign the real parameter θ such that x = e iθ and real parameter t = |x + x −1 |/2 = | cos θ | ∈ [0, 1]. The analysis of Lemmas 1-4 shows that the ranges of t that correspond to |y 1 (x)| ≥ 1 and |y 2 (x)| ≥ 1 are as follows: if 0 < k < 2 √ 2 then Therefore, ; here we have observed that the additionally occurring integrals in the process of differentiating vanish because Re log y j (x) = log |y j (x)| = 0 by Lemma 4 in the corresponding cases.
Note that for both 0 < k < 2 √ 2 and 2 √ 2 ≤ k < 16/(3 √ 3) the result is the same: To complete the proof we apply the substitution t 2 = c.

Proposition 5 For k positive real, k
(10) Proof Applying the substitution to the integral on the left-hand side we obtain .

Now the substitution
into the latter integral results in the the right-hand side in (10).

Remark 3
For k > 0, k = 3, the identity in Proposition 5 relates the periods of the elliptic curves E k in (1) (which is isomorphic to u 2 = (v + 12)(v 2 + k 2 v − 4k 2 )) and The curves E k and E k are not isomorphic but the latter one happens to be a quadratic twist of the former.

Proof of Theorem 2
The equality of elliptic integrals in (10) means that the derivatives of p(k) and r (k) coincide for k ≥ 16/(3 √ 3). Thus p(k) = r (k) + C for the range of k, and the asymptotics (2) implies that C = 0 and finishes the proof of the theorem.

Accurateness of Theorem 2 and related comments
Though our Remarks 1 and 2 are aimed at explaining the choice of ranges for k in Theorems 1 and 2, in conclusion we would like to specifically address the difference between m(P 3 ) and m(R 3 ). The choice k = 3 corresponds to a simultaneous degeneration in the families of curves P k (x, y) = 0 and R k (x, y) = 0. The curve P 3 (x, y) = (x 2 + x + 1)y 2 + 3x(x + 1)y + x(x 2 + x + 1) = 0 has genus 1; it is isomorphic to the conductor 15 elliptic curve y 2 + x y + y = x 3 + x 2 which has Cremona label 15a8 On the other hand, denotes the Bloch-Wigner dilogarithm. Then by results in [3] the Mahler measure of A(x, y) is equal to The resulting measure m(R 3 ) = 1.01151388 . . . visually appears to be different from (11) confirming that m(P k ) = m(R k ) at least for k = 3. Furthermore, m(R 3 ) does not seem to be a Q-linear combination of L (χ −3 , −1) and L (χ −15 , −1). It would be interesting to establish the expected evaluation m(R 4 ) = − 1 3 L (E 224a , 0), hence also for m(P 4 ) and m(Q 6 ), by using the recent formula of Brunault [4] for the regulator of Siegel units. Note that the elliptic curve R 4 (x, y) = 0 does not possess a modular-unit parametrisation (so that the formula from [7] is not applicable) and it is isomorphic to the curve y 2 = x 3 + x 2 − 8x − 8 which has Cremona label 224a2 [6, Curve224.a1].