Brauer-Manin obstruction for Erd\H{o}s-Straus surfaces

We study the failure of the integral Hasse principle and strong approximation for the Erd\H{o}s-Straus conjecture using the Brauer-Manin obstruction.

always has a solution with u 1 , u 2 , u 3 ∈ N. Note that there is always a solution with u 1 , u 2 , u 3 ∈ Z and to prove the conjecture it suffices to consider the case where n is a prime. We refer to Mordell's book [17,Ch. 30] and the more recent paper [7] for further background and history on this problem. In this paper we investigate what modern techniques from arithmetic geometry can say about this conjecture and more generally the structure of the solutions to (1.1). At a first glance, it is not clear how to use tools from modern algebraic geometry to tackle the problem, as N is not a ring. However, this conjecture does indeed have a natural interpretation as a question of strong approximation, stipulating that integer solutions with certain real conditions exist. Our first main result states that there is no Brauer-Manin obstruction in this case (see §1. 2 for a more precise statement and background on the Brauer-Manin obstruction). Despite there being no Brauer-Manin obstruction to the conjecture, it turns out that there is in fact an obstruction to strong approximation at the p-adic places. This obstruction has the following completely explicit description. (In the statement (·, ·) p denotes the Hilbert symbol.) Theorem 1.2. Let n ∈ N be odd and u ∈ N 3 a solution to (1.1). Then p|n (−u 1 /u 3 , −u 2 /u 3 ) p = −1.
In the stated generality, this result does not seem to have been known and gives new conditions which natural number solutions must satisfy. Theorem 1.2 allows one to recover various classically known results in a more systematic and conceptual way, as special cases of a Brauer-Manin obstruction. For example if n is an odd prime, we have the following. Corollary 1.3. Let n = p be an odd prime and u ∈ N 3 a solution to (1.1). Then there exists i = j such that u i /u j ∈ Z * p . For such a solution we have −u i /u j p = −1.
Corollary 1.3 unifies various quadratic reciprocity conditions found by Yamamoto [21] for p ≡ 1 mod 4. We are also able to recover the following result, which was known in some form to Yamamoto [21] (see also [7,Prop. 1.6]). Corollary 1.4. If n is an odd square, then there are no natural number solutions u with n | u 1 , gcd(n, u 2 u 3 ) = 1, or gcd(n, u 1 ) = 1, n | u 2 , n | u 3 .
Corollary 1.4 is really a condition on natural number solutions which is not present for integer solutions (e.g. for n = 9 consider the solutions (− 18,4,4) and (−9, 2, 18)). Similarly, the congruence condition in Corollary 1.3 is also not present for integer solutions in general. For example, consider p = 5 and the solution (−5, 2, 2), where the corresponding Legendre symbol is 1. In fact, for integer solutions which are not natural number solutions, the exact opposite of Theorem 1.2 holds. Theorem 1.5. Let n be an odd integer and u ∈ Z 3 a solution to (1.1) which is not a natural number solution. Then

Geometric interpretation.
We now explain in more detail how to interpret our results geometrically using the Brauer-Manin obstruction. Consider the corresponding algebraic surface derived from (1.1) This is an affine cubic surface, and geometrically a so-called log K3 surface. Many interesting classical Diophantine equations turn out to concern log K3 surfaces, and their integer points are an active area of research [4,5,10,11,12,15]. Note that U n is singular, with the unique singular point lying at the origin. We let U n denote the natural model for U n given by the same equation in A 3 Z . Note that U 1 ∼ = U n over Q for all n ∈ N, by simply rescaling the u i . The Erdős-Straus conjecture therefore concerns existence of certain integer points on different models over Z for the same surface over Q; in particular this nicely highlights the fact that different models of the same surface can give rise to very different problems in general. Let π 0 (U n (R)) be the set of connected components of U n (R) and A Q,f the ring of finite adeles. We say that U n satisfies strong approximation if U n (Q) has dense is discrete as U n is affine, hence clearly not dense. We let We will show that U n (R) + is a connected component of U n (R), and its complement is also a connected component. We define U n (N) hence stipulates that a special case of strong approximation holds. One can even formulate the conjecture as a problem of strong approximation for U 1 ; here it is equivalent to (1.3) for U 1 and W n for all n ≥ 2, where We now explain how one can use the Brauer group to study this problem (see [18, §8.2] for further background on the Brauer-Manin obstruction). Recall that there is a continuous pairing given by pairing with an element of Br U n and taking the sum of local invariants. For an open subset W ⊂ U n (A Q ) • , we define W Br to be the right kernel of this pairing restricted to W . We have U n (Q) ∩ W ⊂ W Br ; in particular, if W Br = ∅ then U n (Q) ∩ W = ∅ and we say that there is a Brauer-Manin obstruction to strong approximation (cf. (1.3)). We first calculate the Brauer group. Theorem 1.6. We have The algebra in Theorem 1.6 is transcendental, so we will obtain new cases of a transcendental Brauer-Manin obstruction on log K3 surfaces. One novel feature is that there are few examples in the literature where Brauer groups of singular varieties have been computed, as Brauer group computations usually use Grothendieck's purity theorem which requires regularity (or at least a singular locus of large codimension). We prove Theorem 1.6 by first calculating the Brauer group of a desingularisation, then showing that every such Brauer group element comes from the singular surface. This latter property is a special case of a more general result about Brauer groups of normal surfaces with rational singularties (Theorem 2.9), which may be of independent interest.
One could hope to use this Brauer group element to disprove the Erdős-Straus conjecture by showing that (U n (R) + × p U n (Z p )) Br = ∅; our next result says that this does not happen. (1.5) The first equation (1.4) is a more precise version of Theorem 1.1. The second equation (1.5) says that nonetheless there is always a Brauer-Manin obstruction to strong approximation for natural number solutions (as manifested by Theorems 1.2 and 1.5).
Despite there being a Brauer-Manin obstruction to strong approximation, it turns out that not every failure of strong approximation is explained by the Brauer-Manin obstruction.
We prove this by showing that U n (Z) is not Zariski dense using real considerations. The conclusion then follows from the fact that Br U n / Br Q is finite.
Outline of the paper. In §2 we study the geometry of Erdős-Straus surfaces over a field k of characteristic zero. We calculate the desingularistaion, the Picard group, and the Brauer group (Theorem 1.6). In §3 we apply our knowledge of the Brauer group to prove the remaining results from the introduction. The appendix explains in more detail how Corollary 1.3 relates to results of Yamamoto [21].
Notation. For a field k, we denote by µ(k) the group of roots of unity in k. For a scheme X, we denote by Br X = H 2 (X, G m ) its (cohomological) Brauer group.
Acknowledgements. We thank Yang Cao, Jean-Louis Colliot-Thélène, and Christian Elsholtz for useful comments and references. This work was undertaken at the Institut Henri Poincaré during the trimester "Reinventing rational points". The authors thank the organisers and staff for ideal working conditions. The second-named author is supported by EPSRC grant EP/R021422/1.

Geometry of Erdős-Straus surfaces
In this section we study the geometry of the surfaces U n from (1.2). We work over a field k of characteristic 0 with algebraic closurek. The primary aim of this section is to prove Theorem 1.6. We also prove a result of independent interest on Brauer groups of rational surface singularities (Theorem 2.9).

The Cayley cubic and its lines. We let
be the closure of U n in P 3 k ; this is isomorphic to the Cayley cubic surface over k. The surface S n has 4 singularities, each of type A 1 , given by setting all but one coordinate equal to 0; we let P = (1 : 0 : 0 : 0) be the singularity in U n . The Cayley cubic has 9 lines overk. This induces 6 lines on U n , of which we are interested in the following 3 lines

2.2.
Desingularisation. Let U n be the desingularisation of U n given by blowing up P once, with exceptional curve E ⊂ U n . By abuse of notation, we denote by L i,j the strict transform of the relevant lines in U n . We have the equation With respect to this equation, the curves of interest to us are Parametrisation. Any cubic surface with a singular rational point is rational, with a birational parametrisation given by projecting away for the singular point. Applying this to the singularity P , we obtain the birational map to P 2 . On the desingularisation, this becomes the birational morphism U n → P 2 , (u 1 , u 2 , u 3 ; y 1 : y 2 : y 3 ) → (y 1 : y 2 : y 3 ).
Note that the boundary is the disjoint union of the lines L i,j The following important observation will be used numerous times.
Proof. That V n ∼ = G 2 m follows from the fact that the map (2.2) becomes an isomorphism onto its image when restricted to V n . The second part follows from the fact that the invertible regular functions on G 2 m are generated by characters and non-zero constants.
Proof. By Lemma 2.1 any invertible regular function must be a non-trivial product of powers of y 1 /y 3 and y 2 /y 3 , modulo constants. However, such a function cannot be invertible on U n since its divisor is always non-trivial by (2.1).

Picard group.
Lemma 2.3. We have Pic U n = Pic U n,k ∼ = Z generated by L 1,2 .
Proof. By Lemma 2.1 and (2.1) we have the exact sequence where the second map associates to a rational function its divisor and the third map associates to a divisor its class. But Pic V n,k = 0 by Lemma 2.1. The result now follows from (2.1).

Brauer group.
2.5.1. Brauer group of U n . We denote by Br 1 X = ker(Br X → Br Xk) the algebraic Brauer group of a variety X/k.
Proof. Lemma 2.2 and the Hochschild-Serre spectral sequence give an injection Br 1 U n / Br k ֒→ H 1 (k, Pic U n,k ). But Pic U n,k = Z with trivial Galois action by Lemma 2.3, hence this Galois cohomology group is trivial.
Proposition 2.5. The natural map Br U n,k → Br V n,k , induced by the inclusion V n ⊂ U n , is an isomorphism. In particular Br U n,k ∼ = Q/Z(−1) as a Galois module, and its elements are represented by the cyclic algebras Proof. The explicit description of Br V n,k follows from Lemma 2.1 and the fact that Br G 2 m,k ∼ = Q/Z(−1), given by the stated cyclic algebras. So let b = (u 1 /u 3 , u 2 /u 3 ) ζ . It suffices to show that b is unramified along the boundary (2.4). The L i,j are regular and disjoint, hence Grothendieck's purity theorem [8, Cor. 6.2] yields the exact sequence where the last map is the residues along the L i,j,k . However L i,j,k ∼ = A 1 k is simply connected, so the corresponding residues are trivial. The result follows.
We next show that every Galois-invariant element of Br U n,k in fact descends to the ground field k. To do this, we make use of the relation derived from (1.1). (This relation will also appear in other parts of the paper).
Proposition 2.6. The natural map Br U n → (Br U n,k ) Gal(k/k) is surjective. A complete set of representatives for the elements of Br U n / Br k is given by the cyclic algebras α ζ = (−u 1 /u 3 , −u 2 /u 3 ) ζ , ζ ∈ µ(k).
These algebras have the following equivalent representations in the Brauer group Proof. By Proposition 2.5, we have (Br U n,k ) Gal(k/k) ∼ = (Q/Z(−1)) Gal(k/k) , and this is (non-canonically) isomorphic to µ(k) [5,Lem. 2.4]. By Proposition 2.5, the cyclic algebras α ζ therefore give a complete set of representatives for the Galois-invariant elements. It thus suffices to show that these descend to k. The different representations are easily checked to hold in the Brauer group of the function field of U n , using (2.1) and the relation (a, b) = (−b/a, 1/a), which holds in the K 2 of any field of characteristic 0. By (2.1), we need to show that α ζ is unramified along the L i,j . By symmetry, it suffices to show that α ζ is unramified along L 2,3 . However, by (2.1) and standard formulae for residues [9, Prop. 7.5.1, Ex. 7.1.5], the residue of α ζ along L 2,3 is where m is the order of ζ. But using the relation (2.5), we have so that the residue is in fact equal to 1 along L 2,3 as u 3 = 0 here. This shows that α ζ ∈ Br U n , as required.
Note that Proposition 2.6 shows that Br U n / Br k is finite; something which is not a priori obvious.
Remark 2.8. Note that the "obvious" Galois-invariant element (u 1 /u 3 , u 2 /u 3 ) does not descend to Q. Despite being unramified overQ, it ramifies over the lines L i,j with constant (non-trivial) residue. We have multiplied this element by some ramified algebraic Brauer group elements to kill these constant residues.
2.5.2. Brauer group of U n . We calculated the Brauer group of the desingularisation U n using Grothendieck's purity theorem. This method uses that U n is smooth and does not apply directly to U n . To calculate Br U n we use the following general result, which does not seem to have been noticed before. Theorem 2.9. Let U be a normal surface over k with rational singularities and f : U → U a desingularisation. Then the induced map Br U → Br U is surjective.
Proof. We will compute the higher direct images R q f * G m with respect to the étale topology and use the Leray spectral sequence; the necessary material can be found in [14, §III, §IV].
Let P 1 , . . . , P r be the closed points at which U is singular, with residue fields k j = κ(P j ), and let E j be the exceptional divisor above P j . LetP j be a geometric point above P j , and letĒ j the fibre aboveP j . By [1, Prop. 1],Ē j is a tree of P 1 s. By [14,Prop. 11.1], PicĒ j is isomorphic to Z d j , where d j is the number of irreducible components ofĒ j , with the absolute Galois group of k j permuting the factors as it permutes the irreducible components.
Let O sh U,P j be a strict Henselisation of the local ring of U at P j . The standard calculation of the stalks of higher direct images shows that (R 1 f * G m )P j is isomorphic to Pic( U × U Spec O sh U,P j ). The natural map Pic( U × U Spec O sh U,P j ) → PicĒ j is injective by [14, Thm. 12.1] and surjective by [14,Lem. 14.3], so is an isomorphism. We deduce that (R 1 f * G m )P j and PicĒ j are isomorphic as Galois modules over k j . Let i j : P j → U be the inclusion. Given that R 1 f * G m is supported at the points P j , we have computed (2.6) It follows that since PicĒ j is an induced module. We now show that the stalks (R 2 f * G m )P j are torsion-free. The Kummer sequence on U gives, for any m ≥ 1, an exact sequence Proper base change [16,Cor. VI.2.7] shows where the last isomorphism follows from the Kummer sequence ofĒ j , as BrĒ j = 0 by [8, Cor. 1.2]. Therefore (R 1 f * G m )P j surjects onto (R 2 f * µ m )P j by (2.6), showing that (R 2 f * G m )P has no non-trivial m-torsion. Using (2.6) and (2.7), the Leray spectral sequence now gives an exact sequence Since U is regular, Br U is a subgroup of Br k( U) and is therefore torsion. Thus the rightmost arrow is zero. This proves that Br U → Br U is surjective.
In the case of Erdős-Straus surfaces, we obtain the following stronger result.
Corollary 2.10. The natural map Br U n → Br U n is an isomorphism.
Proof. By Theorem 2.9, it suffices to show that the stated map is injective. The exact sequence (2.8) here reads But Pic U n → Pic E is surjective as the strict transform of L 1,2 has intersection number 1 with the exceptional divisor E. This completes the proof.
Corollaries 2.7 and 2.10 in particular prove Theorem 1.6.
Remark 2.11. The map in Theorem 2.9 need not be an isomorphism in general. If X is the Cayley cubic surface over C, then Br X ∼ = Z/2Z [2, Tab. 2], but the Brauer group of the desingularisation is clearly trivial.
Remark 2.12. We have calculated Br U n for completeness; however, we could just have chosen to work on the desingularisation instead. Namely, consider the Brauer group element α ∈ Br U n . Restricting α to the exceptional divisor E ∼ = P 1 Q , we find that α is constant along E as Br P 1 Q = Br Q (in fact our choice of α is even trivial along E). Therefore, we could have chosen to instead define as pairing with α is independent of the choice of lift of adelic point from U n to U n . This is essentially the approach advocated in [6, §8] for dealing with the Brauer-Manin obstruction on singular varieties. (Note that in our case the smooth points are dense in U n (Q v ) for all v, so U n (Q v ) = U n (Q v ) cent in the notation of loc. cit.)

Brauer-Manin obstruction
We now study the Brauer-Manin obstruction in our case over Q. Let n ∈ N.

Local invariants.
We begin by calculating the local invariants of the element α = (−u 1 /u 3 , −u 2 /u 3 ). We take the convention that the local invariants lie in µ 2 , rather than Z/2Z. Thus for a place v of Q the local invariant map is given by the Hilbert symbol Then the U n (R) + and U n (R) − are both connected and Proof. We first show that U n (R) has two connected components. Consider This map is not surjective; indeed, we rearrange the equation (2.5) to obtain So the image misses every point on the hyperbola u 1 + u 2 − 4u 1 u 2 /n = 0, except the origin which is the image of the singular point. The hyperbola splits the plane into 3 components, but one segment passes through the origin and hence the image of (3.2) has 2 components. The fibres of (3.2) are connected, being a single point or R over the origin. Hence U n (R) has 2 connected components. These are easily checked to be the two components stated in the lemma. The local invariants are then calculated by standard formulae for Hilbert symbols.

Preliminaries.
Lemma 3.2. Let p be an odd prime with v p (n) ≤ 1 and u ∈ U n (Z p ) with u 1 u 2 u 3 = 0. Then there exists i = j such that u i /u j ∈ Z * p .
Proof. Write u i = a i p b i and n = n ′ p b where p ∤ n ′ a i . The equation (1.2) becomes 4a 1 a 2 a 3 p b 1 +b 2 +b 3 = n ′ (a 1 a 2 p b 1 +b 2 +b + a 1 a 3 p b 1 +b 3 +b + a 2 a 3 p b 2 +b 3 +b ).
Without loss of generality 0 ≤ b 1 ≤ b 2 ≤ b 3 . If b 2 = 0, then the result is clear. So assume for a contradiction that 1 ≤ b 2 < b 3 . But as b ≤ 1, we then have Thus p | a 1 a 2 , which contradicts the fact that the a i are units, as required.
Remark 3.3. Note that Lemma 3.2 fails in general if n has a prime divisor with valuation at least 2. For example, for n = 9 we have the solution (4, 6, 36). This will mean that for general n one cannot expect a simpler statement than the one already given in Theorem 1.2.
Lemma 3.4. Let p and u ∈ U n (Z p ) be such that u 2 /u 3 ∈ Z * p . Then .
Proof. The element α is easily seen to extend to the base change (U n ) Zp . The result then follows from general facts, namely that α(u) ∈ Br Z p = 0. For completeness, we give a direct proof which is essentially a p-adic version of the residue calculations from the proof of Proposition 2.6.
By continuity, we may assume that u 1 u 2 u 3 = 0. Up to permuting coordinates, , then the invariant is 1 by Lemma 3.4. So assume v p (u 1 ) = v p (u 3 ), so that p | u 1 u 2 u 3 . But from the equation (1.2), it is clear that p cannot divide only one of the u i since p ∤ 2n. As As p | u 3 , v p (u 1 ) = v p (u 3 ), and the left hand side is a p-adic unit, we must have v p (u 3 /u 1 ) > 0. Thus −u 2 /u 3 ≡ 1 mod p, and so the local invariant is again trivial by Lemma 3.4.

Bad odd primes.
Lemma 3.6. Let p | n be an odd prime. Then the map is surjective.
Proof. We first consider the case where p n. Write n = pn ′ where p ∤ n ′ and substitute u 1 = pa 1 . The equation (1.2) becomes Modulo p this is As p is odd, there exists a solution with 4a 1 ≡ n ′ mod p and u 2 , u 3 arbitrary modulo p. Geometrically, the equation (3.3) defines the union of three planes which is non-singular away from the common points of intersection. Providing that u 2 u 3 ≡ 0 mod p, we may therefore use Hensel's lemma to lift to a p-adic solution. Thus, we have shown that we may choose p-adic solutions such that p u 1 , p ∤ u 2 u 3 and both possibilities may be realised. The result in this case therefore follows from Lemma 3.4. We now consider the general case. Let n = p b n ′ where p ∤ n and b > 1. We take a p-adic solution u ∈ U pn ′ (Z p ) as constructed in the previous case, and consider the solution p b−1 u ∈ U n (Z p ). The quotients u 1 /u 3 , u 2 /u 3 are unchanged, hence the result follows from the previous case and (3.1).

The prime 2.
Lemma 3.7. Suppose that n is even. Then the map is surjective.
Proof. It suffices to prove the result for n = 2, since then we can just obtain the result for all even n by rescaling, as in the proof of Lemma 3.6. Here our equation is There is the natural number solution (1, 2, 2) which is easily seen to have local invariant −1. Next, one verifies that the solution Surprisingly, for odd n the local invariant is always trivial at 2.
Proof. Without loss of generality, we have |u 1 | ≤ |u 2 | ≤ |u 3 |. Then by (3.6) we have |u 1 | ≤ 3n/4, so there are only finitely many choices for u 1 . If 4/n = 1/u 1 , then we obtain the solution u 2 = −u 3 , which is being excluded. Hence we have 4 n − 1 u 1 = 1 u 2 + 1 u 3 and the left hand size is non-zero and takes only finitely many values. But then as in (3.6), one finds that u 2 and u 3 take only finitely many values, as required.
Lemma 3.11. For all but finitely many primes p, the map U n (Z) → U n (F p ) is not surjective.
Proof. Follows from Lemma 3.10 and the Lang-Weil estimates [13] If the map U n (Q) → U n (A Q ) Br • had dense image then, as Br U n / Br Q is finite (Theorem 1.6), it follows from [5, Lem. 6.5] (applied to U n ) that the map U n (Z) → U n (Z p ) has dense image for all finitely many primes p; however this clearly contradicts Lemma 3.11, and shows Theorem 1.8.
Remark 3.12. Let X be a smooth variety over Q which contains a dense torus T with H 0 (XQ, G m ) =Q * and Pic XQ torsion free. If the action of T on itself extends to X, i.e. X is a toric variety, then in [3,20] it is shown that the Brauer-Manin obstruction is the only one to strong approximation away from ∞. However this result need not hold if the action of T does not extend to the whole variety, since U n contains G 2 m but does not satisfy this result by Theorem 1.8.

Appendix A. Comparison with previous results
In [21] (see also [17, p. 290]), Yamamoto shows numerous quadratic reciprocity requirements for solutions to (1.1) when n = p is prime, with various hypotheses. In this appendix we explain how these are all special cases of Corollary 1.3.
There are two types of solutions to (1.1) (see [17,Ch. 30] and [7,Prop. 2.11]). Type 1 is when p exactly divides one of the u i to valuation 1, and Type 2 is when p divides exactly two of the u i to valuation 1.