Explicit Arithmetic on Algebraic Surfaces

Transcription

1 Explicit Arithmetic on Algebraic Surfaces Anthony Várilly-Alvarado Rice University University of Alberta Colloquium January 30th, 2012

2 General goal Let X be an algebraic variety defined over Q. Assume that X is nice: smooth, projective, and geometrically integral. Example (Swinnerton-Dyer) X = {x 4 + 2y 4 z 4 4w 4 = 0} P 3 Q General goal: describe the set X (Q) of Q-valued points on X. In the example above, we have [1 : 0 : 1 : 0] X (Q). Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

3 General goal Let X be an algebraic variety defined over Q. Assume that X is nice: smooth, projective, and geometrically integral. Example (Swinnerton-Dyer) X = {x 4 + 2y 4 z 4 4w 4 = 0} P 3 Q General goal: describe the set X (Q) of Q-valued points on X. In the example above, we have [1 : 0 : 1 : 0] X (Q). Question Is X (Q) infinite? Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

4 General goal Let X be an algebraic variety defined over Q. Assume that X is nice: smooth, projective, and geometrically integral. Example (Swinnerton-Dyer) X = {x 4 + 2y 4 z 4 4w 4 = 0} P 3 Q General goal: describe the set X (Q) of Q-valued points on X. In the example above, we have [1 : 0 : 1 : 0] X (Q). Question Is X (Q) infinite? No one knows... Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

5 Qualitative Questions Is X (Q) nonempty? 1 YES. Is X (Q) finite or infinite? dense for the Zariski topology? dense for the adelic topology? 2 NO. Why not? Local obstructions? Cohomological obstructions? Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

6 Birational invariance The answers to most of these questions depend only on the birational class of X. Definition Two nice varieties X and Y over Q are birational if they contain open sets U and V that are isomorphic as varieties over Q. Example (Lang-Nishimura) Let X and Y be two birational nice varieties over Q. Then X (Q) Y (Q). This suggests we let classification results from birational geometry guide our choice of varieties on which to explore the above questions. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

7 Birational classification of Algebraic surfaces Let X be a smooth projective minimal algebraic surface over Q. Write κ(x ) for the Kodaira dimension of X. { κ(x ) = : in this case X is ruled or (incl. del Pezzo) rational an abelian surface, or κ(x ) = 0: in this case X is a K3 surface, or an Enriques surface, or a bielliptic surface. κ(x ) = 1: in this case X is a properly elliptic surface. κ(x ) = 2: in this case X is a surface of general type. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

8 Necessary local conditions for X (Q) Let Ω := {p N : p prime} { }. Let Q p be the field of p-adic numbers (completion of Q with respect to the p-adic absolute value). Write Q := R. Observation The embeddings Q Q p give inclusions X (Q) X (Q p ). Hence X (Q) = X (Q p ) for all p Ω. Write X (A) := p Ω X (Q p ) Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

9 Hasse principle and weak approximation Definition We say X satisfies the Hasse principle if X (A) = X (Q). Topologize X (A) by taking the product topology of the p-adic topologies of the X (Q p ). Since X is projective, we call this the adelic topology. Definition We say X satisfies weak approximation if X (A) and the image of X (Q) X (A) is dense for the adelic topology. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

10 Manin s obstruction sets In 1970, Manin observed that any subset S of the Brauer group Br(X ) gives rise to an intermediate obstruction set between X (Q) and X (A): X (Q) X (A) S X (A). The set X (A) S already contains the closure of X (Q) for the adelic topology: X (Q) X (A) S X (A). These sets often explain the failure of the Hasse principle and weak approximation on many kinds of varieties. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

11 Brauer-Manin obstructions Definition We say that X is a counter-example to the Hasse principle explained by a Brauer-Manin obstruction if for some S Br(X ). X (A) and X (A) S = Definition We say that X is a counter-example to weak approximation explained by a Brauer-Manin obstruction if for some S Br(X ). X (A) \ X (A) S Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

12 The Brauer group of a field Fix a field k. The Brauer group of k is Br(k) = {central simple algebras over k} / This is a group under tensor product. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

13 Examples of Brauer groups Br(C) = 0. More generally, Br(k) = 0. Br(R) = Z/2Z. The nontrivial class is represented by Hamilton s quaternions: R{1, i, j, k} where i 2 = j 2 = 1, k = ij = ji. Br(Q p ) = Q/Z via the invariant map inv p : Br(Q p ) Q/Z from local class field theory. Class field theory: 0 Br(Q) p Ω Br(Q p ) P p invp Q/Z 0. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

14 Quaternion algebras Quaternion algebras: Fix a field k. Let a, b k. Define the k-algebra k{1, i, j, k} where i 2 = a, j 2 = b, k = ij = ji. This algebra is usually denoted (a, b). When k = Q p, there are explicit formulas that allow us to evaluate the map inv p : Br(k) Q/Z on quaternion algebras. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

15 The Brauer group of a variety Informally, an Azumaya algebra A on X is an organized collection of central simple algebras over X. For a point x X, A gives a central simple algebra A (x) over the residue field k(x) of x. The Brauer group of X is Br(X ) = {Azumaya algebras over X } / This is a group under tensor product. The Brauer group is birationally invariant! Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

16 The Brauer group of a variety Theorem (Grothendieck; 1968) If X is a nice variety over a field k, then there is an injection Br(X ) Br(k(X )). Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

17 Brauer-Manin sets Fix a class A Br(X ). For a field K, there is an evaluation map ev A : X (K) Br K, x A x OX,x K. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

18 Brauer-Manin sets Fix a class A Br(X ). For a field K, there is an evaluation map ev A : X (K) Br K, x A x OX,x K. We obtain a commutative diagram X (Q) p Ω X (Q p) Q φ A ev A eva P 0 Br Q p Ω Br Q p invp p Q/Z 0 Commutativity implies that X (Q) φ 1 A (0) (!) Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

19 Brauer-Manin sets Definition Let S Br(X ). The Brauer-Manin obstruction set determined by S is As promised, we have inclusions X (A) S := A S φ 1 A (0). X (Q) X (A) S X (A). Recall: this means we obtain potential obstructions to the Hasse principle and weak approximation. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

20 Three kinds of Brauer elements Constant elements Br 0 X := im(br Q Br X ) No obstructions: X (A) {A} = X (A) for all A Br 0 X Algebraic elements Br 1 X := ker(br X Br X ) There is an isomorphism Br 1 X Br 0 X H 1( Gal(Q/Q), Pic X ) (Hochschild-Serre spectral sequence) If Pic X = Z then Br 1 X gives no obstructions Transcendental elements Br X \ Br 1 X. geometric: they survive base-change to an algebraic closure Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

21 Del Pezzo surfaces Definition A del Pezzo surface is a nice surface such that K X is ample. The degree of X is d := K 2 X. Del Pezzo surfaces are geometrically rational surfaces, and their degree lies in the range 1 d 9. Lower degree = more complicated geometry. Example A smooth cubic surface in P 3 Q is a del Pezzo surface (of degree 3). Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

22 Del Pezzo surfaces Hasse principle and weak approximation for del Pezzo surfaces: d 5 d = 4 d = 3 d = 2 d = 1 HP [BSD75] [CG66] [KT04] WA [CTSSD87] [SD62] [KT08]? (1) Check mark ( ) means: phenomenon holds. (2) A reference points to a counterexample in the literature. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

23 Del Pezzo surfaces Hasse principle and weak approximation for del Pezzo surfaces: d 5 d = 4 d = 3 d = 2 d = 1 HP [BSD75] [CG66] [KT04] WA [CTSSD87] [SD62] [KT08] [VA08] (1) Check mark ( ) means: phenomenon holds. (2) A reference points to a counterexample in the literature. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

24 Two ways to think about Del Pezzo surfaces of degree 1 The anticanonical model: X is isomorphic to a smooth sextic hypersurface in P Q (1, 1, 2, 3) := Proj(Q[x, y, z, w]), e.g., w 2 = z 3 + Ax 6 + By 6, A, B Q. Conversely, any smooth sextic in P Q (1, 1, 2, 3) is a dp1. The blow-up model: X is isomorphic to the blow-up of P 2 Q at 8 points in general position. In particular, Pic X = Z 9. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

25 Weak approximation fails for dp1s Theorem (V-A; 2008) Let p 5 be a rational prime number such that p 1 mod 12. Let X be the del Pezzo surface of degree 1 over Q given by w 2 = z 3 + p 3 x 6 + p 3 y 6 in P Q (1, 1, 2, 3). Then X is Q-minimal and there is a Brauer-Manin obstruction to weak approximation on X. Moreover, the obstruction arises from a quaternion algebra class in Br X / Br Q. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

26 Computing a Brauer-Manin obstruction To compute a Brauer-Manin obstruction, we need Br X / Br 0 (X ). For a del Pezzo surface, Br(X ) = Br 1 (X ), i.e., there are no transcendental classes in the Brauer group. Hence, the Hochschild-Serre spectral sequence gives an isomorphism Br X / Br 0 (X ) H 1( Gal(Q/Q), Pic X ), To compute the right hand side, we need the action of Gal(Q/Q) on Pic X explicitly. On a del Pezzo surface, Pic X is generated by the exceptional curves of X (C X with C 2 = K X C = 1). Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

27 Explicit exceptional curves on a dp1 Theorem (V-A; 2008) Let X be a del Pezzo surface of degree 1 over a field k, given as a smooth sextic hypersurface {f (x, y, z, w) = 0} in P k (1, 1, 2, 3). Let Γ : {z = Q(x, y), w = C(x, y)} P k (1, 1, 2, 3), where Q(x, y) and C(x, y) are homogenous forms of degrees 2 and 3, respectively, in k[x, y]. If Γ is a divisor on X k, then it is an exceptional curve of X. Conversely, every exceptional curve on X is a divisor of this form. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

28 Example: Exceptional curves on w 2 = z 3 + x 6 + y 6. Let Q(x, y) = ax 2 + bxy + cy 2, C(x, y) = rx 3 + sx 2 y + txy 2 + uy 3, Then the identity C(x, y) 2 = Q(x, y) 3 + x 6 + y 6 implies that u 2 c 3 1 = 0 2tu 3c 2 b = 0 2su + t 2 3ac 2 3cb 2 = 0 2ru + 2st 6acb b 3 = 0 2rt + s 2 3a 2 c 3ab 2 = 0 2rs 3a 2 b = 0 r 2 a 3 1 = 0 Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

29 Example: Exceptional curves on w 2 = z 3 + x 6 + y 6. Use Gröbner bases to solve this system of equations. Get 240 solutions, one for each exceptional curve of the surface. The Galois action can be read off from the coefficients of the equations of the exceptional curves. Sample exceptional curve: (s = 3 2, ζ = (1 + 3)/2) z = ( s 2 ζ + s 2 2s + 2ζ)x 2 + (2s 2 ζ 2s 2 + 3s 4ζ)xy + ( s 2 ζ + s 2 2s + 2ζ)y 2, w = (2s 2 ζ 4s 2 + 2sζ + 2s 6ζ + 3)x 3 + ( 5s 2 ζ + 10s 2 6sζ 6s + 16ζ 8)x 2 y + (5s 2 ζ 10s 2 + 6sζ + 6s 16ζ + 8)xy 2 + ( 2s 2 ζ + 4s 2 2sζ 2s + 6ζ 3)y 3. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

30 Brauer groups of diagonal dp1s Theorem (V-A; 2008) Let k be a field with char k 2, 3. Let X be a minimal del Pezzo surface of degree 1 over k of the form w 2 = z 3 + Ax 6 + By 6 for some A, B k. Then H 1 (Gal(k/k), Pic X ) is isomorphic to one of the following groups: {1}; (Z/2Z) i, i {1, 2, 3, 4, 6, 8}; (Z/3Z) j, j {1, 2, 3, 4}; (Z/6Z) k k {1, 2}; Z/2Z Z/6Z. Each group occurs for some field k. When k = Q only the following groups occur: {1}, Z/2Z, Z/2Z Z/2Z, Z/2Z Z/2Z Z/2Z, Z/3Z, Z/3Z Z/3Z, Z/6Z. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

31 Hardest step: inverting Br X / Br 0 (X ) H 1( Gal(k/k), Pic X ) Br X / Br 0 (X ) H 1( Gal(k/k), Pic X ) Br k(x )/ Br 0 (X ) H 1( Gal(K/k), Pic X K ) inf inf H 1( Gal(L/k), Pic X L ) Br cyc (X, L) ψ ker N L/k / im Br cyc (X, L) := { } classes [(L/k, f )] in the image of the map Br X / Br 0 (X ) Br k(x )/ Br 0 (X ) Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

32 Weak approximation fails for dp1s Theorem (V-A; 2008) Let p 5 be a rational prime number such that p 1 mod 12. Let X be the del Pezzo surface of degree 1 over Q given by w 2 = z 3 + p 3 x 6 + p 3 y 6 in P Q (1, 1, 2, 3). Then X is Q-minimal and there is a Brauer-Manin obstruction to weak approximation on X. Moreover, the obstruction arises from a quaternion algebra class in Br X / Br Q. For the surfaces X of the theorem, we have Br(X )/ Br 0 (X ) = Z/2Z Z/2Z. One of the nontrivial classes gives the quaternion algebra (p, f /g), where... Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

33 Weak approximation fails for dp1s f = 12z 6 72pz 5 y 2 192pz 5 yx 48pz 5 x p 2 z 4 y p 2 z 4 y 3 x + 576p 2 z 4 y 2 x p 2 z 4 yx p 2 z 4 x 4 288p 3 z 3 y 6 720p 3 z 3 y 5 x 888p 3 z 3 y 4 x 2 768p 3 z 3 y 3 x 3 756p 3 z 3 y 2 x 4 264p 3 z 3 yx 5 204p 3 z 3 x p 4 z 2 y p 4 z 2 y 7 x p 4 z 2 y 6 x p 4 z 2 y 5 x p 4 z 2 y 4 x p 4 z 2 y 3 x p 4 z 2 y 2 x p 4 z 2 yx 7 48p 4 z 2 x p 5 zy 10 48p 5 zy 9 x 720p 5 zy 8 x p 5 zy 7 x 3 600p 5 zy 6 x 4 216p 5 zy 5 x 5 240p 5 zy 4 x 6 480p 5 zy 3 x 7 504p 5 zy 2 x 8 24p 5 zyx p 5 zx p 6 y p 6 y 11 x + 192p 6 y 10 x p 6 y 9 x p 6 y 8 x p 6 y 7 x 5 192p 6 y 6 x 6 288p 6 y 5 x p 6 y 4 x p 6 y 3 x 9 48p 6 yx 11. g = z 6 6pz 5 y 2 24pz 5 yx 6pz 5 x p 2 z 4 y p 2 z 4 y 3 x + 132p 2 z 4 y 2 x p 2 z 4 yx p 2 z 4 x 4 + 8p 3 z 3 y 6 60p 3 z 3 y 5 x 168p 3 z 3 y 4 x 2 276p 3 z 3 y 3 x 3 168p 3 z 3 y 2 x 4 60p 3 z 3 yx 5 + 8p 3 z 3 x 6 24p 4 z 2 y 8 24p 4 z 2 y 7 x + 156p 4 z 2 y 6 x p 4 z 2 y 5 x p 4 z 2 y 4 x p 4 z 2 y 3 x p 4 z 2 y 2 x 6 24p 4 z 2 yx 7 24p 4 z 2 x p 5 zy 9 x + 24p 5 zy 8 x 2 120p 5 zy 7 x 3 324p 5 zy 6 x 4 432p 5 zy 5 x 5 324p 5 zy 4 x 6 120p 5 zy 3 x p 5 zy 2 x p 5 zyx p 6 y p 6 y 11 x + 48p 6 y 10 x p 6 y 9 x p 6 y 8 x p 6 y 7 x p 6 y 6 x p 6 y 5 x p 6 y 4 x p 6 y 3 x p 6 y 2 x p 6 yx p 6 x 12. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

34 Transcendental Brauer classes For curves we have Br(X ) = Br 1 (X ), i.e., curves carry have no transcendental Brauer classes. Theorem (Harari; 1996) There exist infinitely many explicit conic bundles V over P 2 with a transcendental Brauer-Manin obstruction to the Hasse principle. Question Are there nice algebraic surfaces that fail to satisfy the Hasse principle on account of a transcendental Brauer-Manin obstruction? Can we write down an example? Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

35 Transcendental Brauer classes For surfaces X of negative Kodaira dimension we also have Br(X ) = Br 1 (X ), i.e., these varieties carry have no transcendental Brauer classes. We start searching for transcendental classes by looking at surfaces of Kodaira dimension 0. Within this class, we consider K3 surfaces. Definition A K3 surface is a nice surface with trivial canonical bundle and h 1( X, O X ) = 0. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

36 Examples of K3 surfaces Double covers of P 2 ramified along a smooth sextic plane curve: {w 2 f (x 0, x 1, x 2 ) = 0} P(1, 1, 1, 3) = Proj Q[x 0, x 1, x 2, w], where f (x 0, x 1, x 2 ) Q[x 0, x 1, x 2 ] 6. Smooth quartic surfaces in P 3. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

37 Previous work Many authors have constructed explicit transcendental Brauer classes, including Artin Mumford (1969), Colliot-Thélène Ojanguren (1989), Harari (1996), Wittenberg (2004), Harari Skorobogatov (2005), Skorobogatov Swinnerton-Dyer (2005), Ieronymou (2009), Ieronymou Skorobogatov Zarhin (2009), Preu (2010). This body of work includes examples of transcendental Brauer-Manin obstructions to weak approximation on K3 surfaces. In all cases, the K3 surfaces considered are endowed with an elliptic fibration, which is used in an essential way to construct transcendental Brauer classes. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

38 Transcendental obstructions to the Hasse principle Theorem (Hassett, V-A; 2011) Let X be a K3 surface of degree 2 over a number field k, given as a sextic in P(1, 1, 1, 3) = Proj k[x 0, x 1, x 2, w] of the form ( ) w 2 = 1 2A B C 2 det B 2D E, (1) C E 2F where A,..., F k[x 0, x 1, x 2 ] are homogeneous quadratic polynomials. Then the class A of the quaternion algebra (B 2 4AD, A) in Br(k(X )) extends to an element of Br(X ). When k = Q, there exist particular polynomials A,..., F Z[x 0, x 1, x 2 ] such that X has geometric Picard rank 1 and A gives rise to a transcendental Brauer-Manin obstruction to the Hasse principle on X. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

39 Transcendental obstructions to the Hasse principle For the second part of the theorem, we can take A := 7x0 2 16x 0 x x 0 x 2 24x x 1 x 2 16x2 2, B := 3x x 0 x 2 + 2x1 2 4x 1 x 2 + 4x2 2, C := 10x x 0 x 1 + 4x 0 x 2 + 4x1 2 2x 1 x 2 + x2 2, D := 16x x 0 x 1 23x x 1 x 2 40x2 2, E := 4x0 2 4x 0 x x1 2 4x 1 x 2 + 6x2 2, F := 40x x 0 x 1 40x1 2 8x 1 x 2 23x2 2. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

40 Hodge theoretic motivation How did we know where to look for these examples? Let X be a complex projective K3 surface. Let T X := NS(X ) H 2 (X, Z) be the transcendental lattice of X. There is a one-to-one correspondence {α Br X of exact order n} 1 1 {surjections T X Z/nZ} Hence, to α as above, we may associate T α T X : T α = ker(α: T X Z/nZ). Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

41 Hodge theoretic motivation Theorem (van Geemen; 2005) Let X be a complex projective K3 surface of degree 2 with Pic X = Z, and let α (Br X )[2]. Then one of the following three things must happen: 1 There is a unique primitive embedding T α Λ K3. This gives a degree 8 K3 surface Y associated to the pair (X, α). 2 T α ( 1) = h 2, P H 4 (Z, Z), where Z is a cubic fourfold with a plane P (h is the hyperplane class). 3 T α ( 1) = h 2 1, h 1h 2, h 2 2 H 4 (Y, Z), where Y is a double cover of P 2 P 2 ramified along a type (2, 2) divisor (h 1, h 2 are the pullbacks to Y of the hyperplane classes of P 2 under the two projections P 2 P 2 P 2 ). Idea: go backwards and work over any field k of characteristic 2. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

42 Difficulties No elliptic fibrations: Our construction going backwards doesn t necessarily yield K3 surfaces for which Pic X = Z. We use recent work of Elsenhans and Jahnel certify this (requires intensive point counts over finite fields). Computing local invariants for Brauer-Manin sets: We show that if the singular locus at a place of bad reduction for X consists of at most 7 ordinary double points, then the local invariants of A are constant at this prime. We evaluate them by looking at a single p-adic point. Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39

43 Difficulties Primes of bad reduction of X : A Groebner basis computation over Z shows these primes divide (346 digits!) Surprisingly, we use algebraic geometry to factor this integer! Anthony Várilly-Alvarado (Rice) Explicit Arithmetic on Algebraic Surfaces January 30th, / 39