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Mahler measure of some singular K3-surfaces

We study the Mahler measure of the three-variable Laurent polynomial x + 1/x + y + 1/y + z + 1/z - k where k is a parameter. The zeros of this polynomial define (after desingularization) a family of K3-surfaces. In favorable cases, the K3-surface has Picard number 20, and the Mahler measure is related to its L-function. This was first studied by Marie-Jose Bertin. In this work, we prove several new formulas, extending the earlier work of Bertin.

This formula shows, in particular, that the Mahler measure of a polynomial with integral coefficients is the logarithm of an algebraic number.
The situation for several variable polynomials is very different. There are several formulas for specific polynomials yielding special values of L-functions. The first examples were computed by Smyth in the 1970s [Sm71,Bo81] and give special values of the Riemann zeta function and Dirichlet L-series: L(χ −3 , 2) = L ′ (χ −3 , −1), m(x + y + z + 1) = 7 2π 2 ζ(3). Then, in the mid 1990s, Boyd [Bo98] (after a suggestion of Deninger) looked at more complicated polynomials such as the family where k is an integral parameter. For most k, the zero set P k (x, y) = 0 is a genus-one curve which we will denote by E (k) . Boyd found several numerical formulas of the same shape: where s k is a rational number and the question mark means that both sides of the equation are equal to at least 25 decimal places. In fact, it suffices to consider k natural since one can easily see that m(P k ) = m(P −k ). In particular, for k = 1, (1.2) m x + 1 x + y + 1 y − 1 = 15 4π 2 L(E 15 , 2) = L ′ (E 15 , 0), which was recently proven by Rogers and Zudilin [RZ11].
The connection with the L-function of the elliptic curve defined by the zeros of the polynomial was explained by Deninger [De97] in a very general context and Rodriguez-Villegas [RV97] for some specific formulas in terms of Beilinson's conjectures. Beilinson's conjectures predict that special values of L-functions (coming from an arithmetic-geometric structure) are given by certain values of the regulator associated to the structure up to a rational number. In favorable cases, Mahler measure can be related to the regulator. In particular, this allowed Rodriguez-Villegas to prove the formulas for the case where E has complex multiplication, since in this case Beilinson's conjectures are known to be true.
More generally, let P (x, y) be a polynomial in two variables with integer coefficients and suppose that P does not vanish on the 2-torus T 2 . If P defines an elliptic curve E and the polynomials of the faces P F of P (defined in terms of the Newton polygon of P ) are cyclotomic (in other words, they have measure zero), then the following relation between m(P ) and the L-series of the elliptic curve E is conjectured to hold: m(P ) ? = qN 4π 2 L(E, 2) = qL ′ (E, 0), where N is the conductor of E and q is a rational number.
A natural extension to this connection involves polynomials whose zeros define Calabi-Yau varieties. Onedimensional Calabi-Yau varieties are elliptic curves, while 2-dimensional Calabi-Yau varieties are elliptic K3-surfaces. For example, it is natural to consider the family of polynomials resulting from adding an extra variable to the equation in (1.1). Bertin has been pursuing this program [Be06,Be08a,Be10] with the families Relating these examples back to the elliptic curve case, one may ask for a natural condition on the faces of the Newton polytope for the polynomials P k in order to expect relationships between m(P k ) and the L-series of the associated surface. The situation is more complicated than in the elliptic curve case, since the faces in the above examples have nonzero Mahler measure. This question remains open.
The first step in Bertin's work is to generalize Rodriguez-Villegas's expression of the Mahler measure in terms of Eisenstein-Kronecker series for these two families of polynomials defining K3-surfaces. For example, in [Be06] Bertin proves Here k = w + 1 w and where η denotes the Dedekind eta function.
For exceptional values of k, the corresponding K3-surface Y k is singular (or extremal) and τ is imaginary quadratic. The Eisenstein-Kronecker series can be split into two sums, one with the Re 1 (jmτ +n) 3 (jmτ +n) terms and the other with the 1 (jmτ +n) 2 (jmτ +n) 2 terms. The first one is related to the L-series of the surface, while the second one is either zero or may be expressed in terms of a Dirichlet series related to the Mahler measure of the 2-dimensional faces of the Newton polytope of the polynomial P k .
Bertin obtained where Y k denotes the K3-surface associated to the zero set P k (x, y, z) = 0, T denotes its transcendental lattice, and L(g N , 3) the L-series at s = 3 of a modular form of weight 3 and level N .
In this note, we continue the work of Bertin and prove The case with k = 18 is particularly difficult because the corresponding K3-surface has an infinite section that is defined over a quadratic field rather than being defined over Q. The method we use to find this infinite section should be useful in other cases.

Background on K3-surfaces
A K3-surface is a complete smooth surface Y that is simply connected and admits a unique (up to scalars) holomorphic 2-form ω. We list here some useful facts about K3-surfaces along with notation that will be used throughout. See [Yu04] for general results about Calabi-Yau manifolds including K3-surfaces.
• H 2 (Y, Z) is a free group of rank 22. If ρ(Y ) = 20, we say that the K3-surface is singular. • The transcendental lattice is defined by 2.1. L-functions. Let Y be a surface. The zeta function is defined by where N n (Y ) denotes the number of points on Y in F p n . If Y is a K3-surface defined over Q, then Y gives a K3-surface over F p for almost all p and where deg P 2 (u) = 22. In fact, where the polynomial R p (u) comes from the algebraic cycles and Q p (u) comes from the transcendental cycles. Hence, for a singular K3-surface, deg Q p = 2 and deg R p = 20. Finally, we will work with the part of the L-function of Y coming from the transcendental lattice, which is given by where ( * ) represents finite factors coming from the primes of bad reduction.
2.2. Elliptic surfaces. An elliptic surface Y over P 1 is a smooth projective surface Y with an elliptic fibration, i.e., a surjective morphism Φ : Y → P 1 such that almost all of the fibers are smooth curves of genus 1 and no fiber contains an exceptional curve of the first kind (with self-intersection −1). Here we list some facts about elliptic surfaces. See [SS10] for a comprehensive reference containing these results.
The group of global sections of the elliptic surface is called the Mordell-Weil group and can be naturally identified with the group of points of the generic fiber. Its rank r can be found from the formula due to Shioda [Sh90]. Here m ν denotes number of irreducible components of the corresponding singular fiber and h is the number of singular fibers. Global sections can be also thought as part of the Néron-Severi group NS(Y ) given by the divisors modulo algebraic equivalence. It is finitely generated and torsion-free. Intersection of divisors yields a bilinear pairing which gives NS(Y ) the structure of an integral lattice.
The trivial lattice T(Y ) is the subgroup of NS(Y ) generated by the zero section and the fiber components. Its determinant is given by where m The Mordell-Weil group can also be given a lattice structure MWL(Y ). Then where E is the generic fiber. The bilinear pairing induced by intersection can be used to construct a height that satisfies where χ(Y ) is the arithmetic genus (χ(Y ) = 2 for K3-surfaces), P · O ≥ 0, and the (always nonnegative) correction terms contr ν (P ) measure how P intersects the components of the singular fiber over ν. This height is the canonical height that one obtains by thinking about the elliptic surface as an elliptic curve over a function field [Sh90].

2.3.
A particular family of K3-surfaces. In this note, we consider the family of polynomials The desingularization of P k = 0 results in a K3-hypersurface Y k . We homogenize the numerator of P k : x 2 yz + xy 2 z + xyz 2 + t 2 (xy + xz + yz) − kxyzt, and then get an elliptic fibration by setting t = s(x + y + z).

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To study the components of the singular fibers, one expresses the K3-surface Y k as a double covering of a well-known rational elliptic surface given by Beauville [Bea82] (2.6) (x + y)(x + z)(y + z) + uxyz = 0.
By analyzing the structure of the singular fibers, we can compute the rank of the group of sections r. In the case of Beauville's surface, the singular fibers are given by To conclude this section, we summarize some results from Peters and Stienstra [PS89] on this family of K3-surfaces. For generic k, the Picard number is ρ(Y k ) = 19. We focus on the singular K3-surfaces -that is, on k values for which ρ(Y k ) = 20. The transcendental lattice T of the general family Y k has a Gram matrix of the form Having Picard number ρ = 20 is equivalent to having a relation between the generic basis {γ 1 , γ 2 , γ 3 } of transcendental periods; that is, becomes algebraic for some choice of p, q, r. Now, let k = w + 1 w . Then w can be represented as a modular function: where H denotes the upper half-plane. Furthermore, a period is algebraic precisely when it is orthogonal to γ 1 + τ γ 2 − 6τ 2 γ 3 . Combining these facts yields a quadratic equation for τ : (2.9) −6pτ 2 + 12qτ + r = 0.
Thus to find k-values such that Y k is a singular K3-surface, we look for k values yielding an imaginary quadratic τ . Here are a few such values: k 0 2 3 6 10 18 Given τ , one may find the parameters p, q, and r, and then find the discriminant of T up to squares by taking the determinant of the resulting Gram matrix. See Section 4 for details in the cases where k = 3, k = 6, and k = 18.

Main results and the general strategy for the proof
Theorem 3.1. We have the following formulas: where Y k is the K3-hypersurface defined by the zeros of P k (x, y, z), T(Y k ) is its transcendental lattice, and g N is a CM modular form of level N .
The strategy for proving these formulas is as follows: • Understand the transcendental lattice and the group of sections.

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• Relate the Mahler measure m(P k ) to the L-function of a modular form.
• Relate the L-function of the surface Y k to the L-function of that same modular form.

The Transcendental Lattice and the Rank
We will prove the following: 4.1. The transcendental lattice and the rank for Y 6 . When k = 6, we see from the table on page 5 that τ = 1 √ −6 . Thus, it satisfies the equation −6τ 2 − 1 = 0, so in equation (2.9) we take p = 1, q = 0, and r = −1. By equation (2.8), the vector γ 1 − γ 3 becomes algebraic over To find the transcendental lattice, we use the Gram matrix (2.7) to find vectors orthogonal to v. A simple computation yields: {γ 2 , γ 1 + γ 3 }; hence these span a sublattice of T. We again use (2.7), this time to find the Gram matrix for the space spanned by these two vectors: Thus the discriminant of T, up to a square, is equal to 24. It remains to decide if it is 6 or 24. Equation (2.5) expresses Y 6 as a double-covering of the Beauville surface (2.6), with u = (s 2 − 6s + 1)/s 2 .
Since we know the singular fibers of the Beauville surface, we easily find the singular fibers of Y 6 : I 2 over u = 1, s = ∞ I 2 over u = 1, and s = 1 3 I 2 double over u = −8.
By the work of Miranda and Persson [MP89], since the rank is 1 and χ = 2, the torsion must have order 6, and therefore it must be generated by ρ 6 . With the aid of Pari/gp or Sage we also find the following point in each fiber: Since this point is not generically among the torsion points of each fiber, it must give an infinite section, which is in particular defined over Q. In fact, this point is a generator of the infinite section, but we do not need this fact for our computation.
The determinant of this matrix is 120, so the discriminant of the transcendental lattice is either 30 or 120.
By the remark following (4.1), | det T(Y 18 )| = 30 or 120. Hence either | det T(Y 18 )| = 30 and h(P ) = 5/2 or | det T(Y 18 )| = 120 and h(P ) = 10. Finding the infinite section for Y 18 is more difficult than for Y 3 because the infinite section is not defined over Q. Details of the method used to find the infinite section, prove that we have a generator, and compute its height are in Section 7. The outcome of the computations is a generator p σ defined over Q( √ −3) satisfying h(p σ ) = 10; hence | det T(Y 18 )| = 120.

Relating the Mahler Measure to a newform
The main ingredient we use to relate Mahler mesure to newforms is the following result from [Be06]. (1 − e 2πinτ ).

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Then m(P k ) = Im τ 8π 3 m,n −4 2 Re 1 (mτ + n) 3 (mτ + n) + 1 (mτ + n) 2 (mτ + n) 2 +16 2 Re 1 (2mτ + n) 3 (2mτ + n) The evaluation of the Eisenstein-Kronecker series often leads to Hecke L-functions. Let K be an imaginary quadratic number field and m be an ideal of O K . A Hecke character of K modulo m with ∞-type ℓ is a homomorphism φ on the group of fractional ideals of K which are prime to m such that for all α ∈ K * with α ≡ 1 mod m, φ((α)) = α ℓ . The ideal m is called the conductor of φ if it is minimal in the following sense: if φ is defined modulo m ′ , then m|m ′ . Let The Mellin transform gives a Hecke eigenform: A theorem of Hecke and Shimura implies that f φ has weight ℓ + 1 and level ∆ K N (m). If ℓ is even, where −∆ K is the discriminant of the field, and χ K is its quadratic character.
A newform f = a n q n ∈ S k (Γ 1 (N )) is said to have complex multiplication (CM) by a Dirichlet charac- By a result of Ribet, a newform has CM by a quadratic field K iff it comes from a Hecke character of K. In particular, K is imaginary and unique. Schütt [Sc08] proves that there are only finitely many CM newforms with rational coefficients for certain fixed weights (including 3) up to twisting, and he gives a comprehensive table for these.
5.1. The relation with a newform for P 6 . From Theorem 5.1, This summation can be viewed (see [Be06]) as a Hecke L-series on the field Q( √ −6). This field has discriminant −24 and class number 2, with the nontrivial class represented by (2, √ −6). That is, we have m(P 6 ) = 24 By the results of Hecke and Shimura, we look for a correspondence to a (quadratic) twist of a newform of weight 3 and level 24. According to Schütt's table [Sc08], there is only one newform (up to twisting) of weight 3 and level 24. The twist must be of the form d p for d dividing 24, and we can compute the twist exactly by comparing the first few coefficients, as shown in the following table. We find that the twist is given by −3 p . Therefore, 5.2. The relation with a newform for P 3 . This case was also considered in [Be06] as a Hecke L-series on the field Q( √ −15). This field has discriminant −15 and class number 2, with the nontrivial class represented by 2, 1+ There is only one newform of level 15 and weight 3 in Schütt's table. We compare the first few coefficients.
There is only one newform of weight 3 and level 120 in Schütt's table. The final results yields The main tool for this section is the following result from [Sc08].
Theorem 6.1 (Schütt). The following classification of singular K3-surfaces over Q are equivalent.
• By the discriminant d of the transcendental lattice of the surface up to square.
• By the discriminant −d of the Néron-Severi lattice of the surface up to square.
• By the associated newform up to twisting.
• By the level of the associated newform up to square.
• By the CM field Q( √ −d) of the associated newform.
This theorem depends on Livné's modularity theorem for singular K3-surfaces that predicts that L(T(Y ), s) is modular and that the corresponding modular form has weight 3.
The first step in finding the corresponding modular form is to compute the first few coefficients A p from L(T(Y ), s); then the coefficients are compared to the tables that can be found in [Sc08] in order to identify the corresponding CM newform. Tackling the first step requires the following result from [Be10].
Now suppose that r(Y ) = 1 and that there is an infinite section defined over Q( √ d). Then Notice that the result stated in [Be10] requires a generator of MWL(Y ) to be defined over Q( √ d). But it is not hard to see that it suffices to find any element of infinite order to be defined over Q( √ d).
6.1. Relating L(T(Y 6 ), s) to a newform. We know from Section 4.1 that r(Y 6 ) = 0 and that | det T(Y 6 )| = 24, so we use equation (6.1). With the help of Pari/gp or Sage we compute several coefficients A p and compare them to the coefficients of the newform of level 24 from Schütt's table in [Sc08]. 6.2. Relating L(T(Y 3 ), s) to a newform. In this case, r(Y 3 ) = 1 and the infinite section is defined over Q. We apply equation (6.2) to compute the A p values and compare with the table from [Sc08] in order to obtain As a final note, we remark that one could have started the computations from this subsection without knowing that the infinite section is defined over Q( √ −3). Computing several values of A p with equation (6.1) and comparing with the table from [Sc08] will reveal the necessary correction factor. This allows one to predict that the infinite section is defined over Q( √ −3), and armed with this knowledge the infinite section is more easily computed (see Section 7.1).

Infinite section for Y 18
We now describe the computations used to find an infinite section p σ for the elliptic surface given in equation (4.2), show that our p σ is a generator for the infinite part of the group of sections, and prove that h(p σ ) = 10. 7.1. Finding the infinite section. As noted above, we can predict that the infinite section is defined over Q( √ −3). Therefore, we twist equation (4.2) by −3 in order to get an elliptic surface with the infinite section defined over Q. We denote this twist Y −3 (we drop the Y 18 notation in this case because there is no ambiguity). Applying the general formula for a quadratic twist [Co99, Chapter 4], we have Y −3 : y 2 + (σ 2 − 18σ + 1)xy = x 3 + (−σ 4 + 36σ 3 − 329σ 2 + 90σ + 2)x 2 + 9σ(−σ + 18)x.
7.2. Proof that p σ is a generator. Let K = Q( √ −3)(σ). To prove that p σ is indeed a generator of the infinite section, we need to see that we cannot write p σ + kρ 6 = [2]P for any P ∈ E(K) and k = 0, . . . , 5. In fact, it suffices to prove that p σ + kρ 6 = [2]P has no solution P ∈ E(K) for k = 0, 3. We will use the following theorem.

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To compute q ± as in Theorem 7.1, we first find It is then a simple matter to compute and neither of these are squares in K.
Finally, note that the fibers over α 2 and β 2 are of type I 1 , so we know that the infinite section cuts Θ α2,0 and Θ β2,0 because that is the only choice.