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 $y^3-y+x^3-x+kxy$ whose zero loci define elliptic curves for $k\ne0,\pm3$.


Introduction
In his pioneering systematic study [2] of the Mahler measures of two-variate polynomials D. 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 E k : y 2 = x 3 + (k 2 − 24)x 2 − 16(k 2 − 9)x (1) 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 R k (x, y) = y 3 − y + x 3 − x + kxy 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 [2,6] 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 Sections 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 Section 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 fargoing generalisation for the regulator of Siegel units [4] established by F. 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.
Using Jensen's formula and the symmetry y 1 (x) = y 1 (x −1 ), we obtain 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 dp(k) 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 > 0, when we order the zeroes y 1 (x), y 2 (x) appropriately. In the latter case With the help of Jensen's formula we obtain Note that for k > 0 we have Performing the change of variable t = (v + 2k(k + 1))/(v − 4k) we then obtain 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.
Proof. We will show that 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. 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 .

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 to 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 and 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.

Remark 2. The integral in
Proposition 5. For k positive real, k = 3, Proof. Applying the substitution to the integral on the left-hand side we obtain . Now the substitution u = 2(v + 12) −k 2 + 24 + k √ k 2 + 16 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 E k : 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 quartic polynomial has exactly two complex conjugate zeroes .