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# The 1729 *K*3 surface

- Ken Ono
^{1}and - Sarah Trebat-Leder
^{1}Email author

**Received:**5 October 2015**Accepted:**7 September 2016**Published:**17 October 2016

The Erratum to this article has been published in Research in Number Theory 2017 3:12

## Abstract

*K*3 surface with Picard number 18, one which can be used to obtain infinitely many cubic twists over \(\mathbb {Q}\) with rank \(\ge 2\).

## Mathematics Subject Classification

- 14G05
- 14H52

## 1 Background

### 1.1 1729 and Ramanujan

Srinivasa Ramanujan is said to have possessed an uncanny memory for idiosyncratic properties of numbers. J. E. Littlewood famously remarked that “every number is a personal friend of Ramanujan.” This opinion is supported by the famous story of 1729, the so-called “Hardy–Ramanujan” number after the famous anecdote of G. H. Hardy concerning a visit to the hospital to see his collaborator Srinivasa Ramanujan: (page 12 of [6]):

*I remember once going to see him when he was ill at Putney. I had ridden in taxi-cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”*

- (1)In 1913, he published the identityin [8], which after dividing both sides by 27 gives \(12^3 = (-1)^3 + 10^3 + 9^3\).$$\begin{aligned} (6A^2 - 4AB + 4B^2)^3= & {} (3A^2 + 5AB - 5B^2)^3 + (4A^2 - 4AB + 6B^2)^3 \\&+ (5A^2 - 5AB - 3B^2)^3 \end{aligned}$$
- (2)In entry 20 (iii) of Chapter 18 of his second notebook (see [2]), he gave the identityand explicitly wrote down (1.1) as part of a list of examples.$$\begin{aligned}&(M^7 - 3M^4(1 + P) + M(3(1 + P)^2 - 1))^3 \\&\quad +\, (2M^6 - 3M^3(1 + 2P) + (1 + 3P + 3P^2))^3\\&\quad +\, (M^6 - (1 + 3P + 3P^2))^3 = (M^7 - 3M^4 P + M(3P^2 - 1))^3 \end{aligned}$$
- (3)
In 1915, he asked for solutions to \(X^3 + Y^3 + Z^3 = U^6\) and \(X^3 + Y^3 + Z^3 = 1\) in [9] and [10]. He included (1.1) as examples he wanted the parametrized solutions to account for.

- (4)
On pg. 341 of his lost notebook (see [1]), he offered a remarkable method for finding two infinite families of solutions \(x_i^3 + y_i^3 = z_i^3 \pm 1\), which can be found in the Appendix and involves expanding rational functions at zero and infinity. He included (1.1) as part of a list of examples.

### Remark

His families in (4) give infinitely many near misses to Fermat’s Last Theorem for exponent 3.

### Remark

In our modern perspective, the equation \(X^3 + Y^3 = Z^3 + W^3\) gives a cubic surface, which is in fact a rational elliptic surface.

### 1.2 Elliptic curves

Another classical question is which rational numbers *d* can be written as the sum of two rational cubes. For example, if \(d = 1\), then this is equivalent to the exponent 3 case of Fermat’s last theorem and was proven by Euler. By Ramanujan’s time, there were many values of *d* for which this equation was known to have no solutions. For example, in the 1800s, Sylvester conjectured that if \(p \equiv 5 \pmod {18}\) and \(q \equiv 11 \pmod {18}\) are primes, then the equation has no solutions when \(d = p, 2p, 4p^2, 4q, q^2, 2q^2\). Pepin proved these claims and more (see [4] for more on the history of this problem).

We now know that the equation \(E{:}\;X^3 + Y^3 = 1\) is an elliptic curve with *j*-invariant 0, and for cube-free *d*, \(E_d{:}\;X^3 + Y^3 = d\) is the cubic twist of *E* by *d*. Since \(E_d\) is torsion free for all integral \(d > 2\), this classical question is really one about ranks of elliptic curves in families of cubic twists.

### Theorem 1.1

### Remark

While proving Theorem 1.3, we will also show that the rank of \(E_{k(T)}\) over \(\overline{\mathbb {Q}}(T)\) is 4. In fact, rank 4 happens already over \(\mathbb {Q}(\sqrt{-3})(T)\).

### Corollary 1.2

All but finitely many \(t \in \mathbb {Q}\) have that \(E_{k(t)}/\mathbb {Q}\) has rank \(\ge 2\).

In a certain sense, elliptic curves with rank at least two are very rare. Most specialists believe, based on both numerical evidence and heuristic considerations, that \(50\,\%\) of all elliptic curves have rank 0 and \(50\,\%\) have rank 1, leaving all curves of higher rank to fall in the remaining \(0\,\%\). More precisely, the Birch and Swinnerton-Dyer conjecture implies that the rank of an elliptic curve is equal to the ordering of vanishing of the associated *L*-series at \(s = 1\). This implies that the rank is even or odd according to the sign of the functional equation of this series, which is referred to as the parity conjecture. It is believed that this sign is \(+\) or − with equal frequency, and it is expected that almost all curves have the smallest rank compatible with the sign of their functional equation. However, numerical data of Zagier and Kramarz [14] suggests that this might not be the case for the family \(X^3 + Y^3 = d\). Their numerical data suggests that a positive proportion have rank at least 2, and a positive proportion have rank at least 3. For \(n = 2, 3\), let \(S_n(X)\) be the number of cube free *d* with \(|d| \le X\) and rank of \(E_d\) at least *n*. Assuming the parity conjecture, Mai [7] proved \(S_2(X) \gg X^{2/3 - \epsilon }\). Stewart and Top [12] then unconditionally showed that \(S_2(X) \gg X^{1/3}\) and \(S_3(X) \gg X^{1/6}\). Assuming the parity conjecture, the exponent on their rank 3 result can be improved to 1 / 3.

### Remark

We are able to tie some of Stewart and Top’s results using our polynomial *k*(*t*). In particular, using Theorem 1 of [12], we are able to get that \(S_2(X) \gg X^{1/3}\). Additionally, assuming the parity conjecture and using its explicit form for cubic twists as given in Sect. 12 of [12], we can also show that \(S_3(X) \gg X^{1/3}\).

### 1.3
*K*3-surfaces

We can also view \(X^3 + Y^3 = k(T)\) as an elliptic surface, which turns out to be a *K*3 surface. *K*3 surfaces, which were defined by André Weil in 1958, have become fundamental objects in string theory, moonshine, arithmetic geometry, and number theory.

### Theorem 1.3

*K*3-surface with Picard number 18 over \(\overline{\mathbb {Q}}\).

### Remark

In Stewart and Top’s case, the elliptic *K*3-surface has Picard number 20.

### Remark

In [5], Dolgachev, van Geemen, and Kondo associate *K*3 surfaces to nodal cubic surfaces (i.e. cubic surfaces with at worst nodes as singularities). These *K*3 surfaces admit a natural elliptic fibration and all have an automorphism of order 3. Our *K*3 surface is part of this family.

We will give the definition of a *K*3 surface and its Picard number in Sect. 3.1.

In Sect. 2, we will prove Theorem 1.1 and Corollary 1.2. In Sect. 3, we will use Theorem 1.1 and results on elliptic *K*3 surfaces to prove Theorem 1.3.

## 2 Proof of Theorem 1.1 and its corollary

### 2.1 Background

### 2.2 Rank at least two

### 2.3 Rank at most two

*p*. Over a function field \(\mathbb {F}_p(T)\), it is known that the rank is at most the order of vanishing of the

*L*-function, and so we can get an upper bound on the rank of the reduced curve by computing its

*L*-function in Magma. For the prime of good reduction \(p = 17\), the

*L*-function factors as

Since reducing mod *p* cannot decrease the rank, this shows that the rank over \(\mathbb {Q}(T)\) is at most 2.

### 2.4 Studying the cubic twists

If \(E/\mathbb {Q}(T)\) is an elliptic curve which is not isomorphic over \(\mathbb {Q}(T)\) to an elliptic curve defined over \(\mathbb {Q}\), we can specialize *T* to a rational number *t*. A result of Silverman [11] gives that for all but finitely many rational numbers *t*, this specialization map \(\phi _{t}{:}\;E(\mathbb {Q}(T)) \rightarrow E_{t}(\mathbb {Q})\) is an injective homomorphism. Therefore, all but finitely many values of *t* yield elliptic curves \(E_{k(t)}(\mathbb {Q})\) with rank at least 2.

## 3 A *K*3 surface

### 3.1 Background on K3 surfaces

A *K*3 surface is a smooth minimal complete surface that is regular and has trivial canonical bundle. Some examples of *K*3 surfaces include intersections of three quadrics in \(\mathbb {P}^5\), intersections of a quadric and a cubic in \(\mathbb {P}^4\), and a non-singular degree 4 surface in \(\mathbb {P}^3\). André Weil named them in honor of Kummer, Kähler, Kodaira, and the mountain K2.

*X*is the group of divisors modulo algebraic equivalence, i.e.

*X*. For

*K*3 surfaces over characteristic zero fields, this Picard Number is always \(\le 20\).

### 3.2 Proof of Theorem 1.3

The criterion given in [3] on pages 276–277 show that it’s a *K*3 surface.

*r*the \(\overline{\mathbb {Q}}(T)\)-rank of the elliptic curve defined by our equation. We know that the \(\mathbb {Q}(t)\)-rank is 2. Also, note that \(E_{(k(T)}\) has CM over \(\mathbb {Q}(\sqrt{-3})\), and so the action of the endomorphism ring on our two independent points gives a \(\mathbb {Z}\)-module of rank 4.

By our computation of the *L*-function for \(E_{k(T)}\) reduced mod 17 in (2.3), we see that the rank over \(\overline{\mathbb {F}}_{p}\) is at most 4, and hence the rank over \(\overline{\mathbb {Q}}(T)\) is too.

Therefore, the rank over \(\overline{\mathbb {Q}}(T)\) is exactly 4, and hence the Picard number is 18.

## Notes

## Declarations

### Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

### Acknowledgements

The first author thanks the National Science Foundation (Grant Nos. DMS0964844 and DMS1250467) and the Asa Griggs Candler Fund for their support. The second author is supported by an NSF Graduate Research Fellowship. The authors would also like to thank Bartosz Naskrȩcki for correcting an error in an earlier version of the paper, and Doug Ulmer for his many helpful comments.

## Authors’ Affiliations

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