Posted by: matheuscmss | February 22, 2015

First Bourbaki seminar of 2015 (I): Harari’s talk

About one month ago (on January 24, 2015), the first Bourbaki seminar of 2015 took place at Institut Henri Poincaré. As usual, this was an excellent opportunity to learn about recent advances in areas of Mathematics outside my field of expertise.

The first Bourbaki seminar of 2015 had the following four talks:

Today, I would like to discuss David Harari’s talk entitled “Zero cycles and rational points on fibrations in rationally connected varieties (after Harpaz and Wittenberg)”. Here, I will try to follow the first 38m50s of the video of Harari’s talk (in French) and sometimes his lecture notes (also in French). Of course, this goes without saying that any errors/mistakes are my full responsibility.

1. Introduction

One of the basic old problems in Number Theory is to determine whether a system of polynomial equations

\displaystyle P_i(x_1,\dots, x_n) = 0, \quad 1\leq i\leq r \ \ \ \ \ (1)

associated to homogeneous polynomials {P_i} with coefficients in a number field {k} has non-trivial solutions.

Equivalently, denoting by {X} the algebraic variety defined by the system (1), we want to know whether the set {X(k)} of points of {X} whose coordinates belong to {k} is not empty. In the literature, {X(k)} is called the set of {k}rational points of {X}.

It is not easy to answer this problem in general. Nevertheless, we have the following necessary condition: if {X(k)\neq\emptyset}, then {X(k_v)\neq\emptyset} for all completion {k_v} of {k} with respect to a place of {k} (i.e., {v} is an equivalence class of absolute values). In other words, we have that {X(k)=\emptyset} whenever there is a local obstruction in the sense that {X(k_v)=\emptyset} for some place {v} of {k}.

This necessary condition based on local obstructions is helpful because it is often easy to verify algorithmically that {X(k_v)\neq\emptyset}. For example, when {k=\mathbb{Q}}, its completions {k_v} are either {k_v=\mathbb{Q}_p} (for the place of {p}-adic absolute values, {p\in\mathbb{N}} prime) or {k_v=\mathbb{R}=\mathbb{Q}_{\infty}} (for the “place at infinity”), and, in this situation, we can check that {X(\mathbb{Q}_p)\neq\emptyset} with the help of Hensel’s lemma ({p}-adic analog of Newton’s method).

It is known that this necessary condition is sufficient in certain special cases. For instance, the classical Hasse-Minkowski theorem (from 1924) states that {X(k)\neq\emptyset} if and only if {X(k_v)\neq\emptyset} when {X} is a quadric, i.e., {X} is defined by just one polynomial equation of degree {2}.

Partly motivated by this, we introduce the following definition:

Definition 1 {X} satisfies Hasse’s principle (also called local-global principle) whenever {X(k)\neq\emptyset} if and only if {X(k_v)\neq\emptyset} for all places {v} of {k}.

As it turns out, Hasse’s principle is false in general: Swinnerton-Dyer constructed in 1962 some counterexamples among cubic surfaces, and Iskovskih constructed in 1970 a counterexample among the surfaces fibered in conics (given by intersections of two projective quadrics).

Of course, given that it is not hard to determine algorithmically when {X(k_v)\neq\emptyset} (with the help of Hensel lemma and/or Newton’s method), it is somewhat sad that Hasse’s principle fails in general.

In view of this state of affairs, we can try to generalize the problem of determining whether {X(k)\neq\emptyset} by replacing “rational points” by slightly more general objects (which then would be easier to find). In this direction, we have the following notion.

Definition 2 A zero-cycle {z} is a formal linear combination {z=\sum\limits_{x} n_x x} where:

  • {n_x\in\mathbb{Z}} vanishes for all but finitely many {x\in X}, and
  • if {n_x\neq 0}, then {x} is a closed point in the sense of Algebraic Geometry, i.e., {x} is a point defined over (its coordinates belong to) a finite extension {k(x)} of {k}.

The degree {deg(z)} of a zero-cycle {z} is {deg(z):=\sum\limits_{x} n_x [k(x):k]}.

Note that, by definition, a rational point {x\in X(k)} is a zero-cycle {z=x} of degree {deg(z) = 1}. Thus, we can ask the following more general question:

Does {X} possess a zero-cycle of degree {1} if {X} has such cycles over all {k_v}?

Remark 1 It follows from Bézout’s theorem that {X} has a zero-cycle of degree {1} if and only if {X} has points defined over finite extensions of {k} whose degrees are coprime.

Remark 2 A little curiosity about Bézout: as I discovered after moving from Paris to Avon, Bézout spent the last years of his life in Avon and the city gave his name to a street (not far from my appartment) in his honor.

Once more, the answer to this question is no: for example, it is known that there are counterexamples among surfaces fibered in conics.

Given this scenario, our goal is to explain how to refine the local-global principle with additional cohomological conditions (related to the so-called Brauer groups) introduced by Manin ensuring the existence of zero-cycles and/or rational points in certain situations.

2. Manin’s condition and Conjecture ({E_1})

From now on, let us assume for the sake of simplicity that the projective variety {X} is:

  • smooth (or non-singular), i.e., the Jacobian matrix associated to the polynomials {P_i} in (1) has maximal rank at all points of {X}, and
  • geometrically integral in the sense that if we pass from {k} to an algebraic closure {\overline{k}}, then {X(\overline{k})} does not break into several irreducible components.

In 1970, Manin had the idea of introducing a coupling

\displaystyle \prod\limits_v X(k_v)\times Br(X)\rightarrow \mathbb{Q}/\mathbb{Z} \ \ \ \ \ (2)

between the space {\prod\limits_{v} X(k_v)} of local points over all places of {k} and the Brauer group {Br(X)} of {X}. This (Brauer-Manin) coupling is defined as follows. We take a family {(p_v)\in \prod\limits_{v} X(k_v)} of local points and an element {\alpha\in Br(X)}, and we associated to them the following quantity:

\displaystyle ((p_v),\alpha)\mapsto \sum\alpha(p_v)

Of course, we have to explain what this means.

The Brauer group {Br(X)} is a (étale) cohomology group ({Br(X) = H^2(X,\mathbb{G}_m)} where {\mathbb{G}_m} is the multiplicative group) generalizing to {X} the notion of Brauer group of a field (whose elements are equivalence classes of central simple algebras of finite rank over the given field).

In general, {Br(X)} is a subgroup of the Brauer group {Br(k(X))} of the function field {k(X)} of X. Moreover, {Br(X)} is functorial: we can evaluate {\alpha(p_v)\in Br(k_v)} and, by the results from class field theory, we can embed {Br(k_v)} in {\mathbb{Q}/\mathbb{Z}} (and this embedding is actually an isomorphism when {v} is a finite place).

Therefore, we can consider the sum {\sum\limits_{v} \alpha(p_v)} of the elements {\alpha(p_v)\in\mathbb{Q}/\mathbb{Z}}. The fact that this sum has only finitely many non-trivial terms (i.e., {\alpha(p_v)=0} for all but finitely many {v}‘s) is a consequence of the projectivity of {X}.

At this point, it is natural to ask why Manin introduced this coupling and also what is its relevance for our purposes of studying rational points.

In order to explain this, let us setup some notations. The set of adelic points over {X} is {X(\mathbb{A}_k) := \prod\limits_{v} X(k_v)}. The kernel to the left of the coupling (2) is a subset of {X(\mathbb{A}_k)} denoted by {X(\mathbb{A}_k)^{Br(X)}}. In other terms, {(p_v)\in X(\mathbb{A}_k)^{Br(X)}} whenever {\sum\limits_{v} \alpha(p_v) = 0} for all {\alpha\in Br(X)}.

Using the global reciprocity law, one can show that {\overline{X(k)}\subset X(\mathbb{A}_k)^{Br(X)}} (where the closure of {X(k)} is taken with respect to the product topology of the {v}-adic topologies on {X(k_v)}.

This gives a new necessary condition called Manin’s condition for the existence of rational points in {X}: if {X(\mathbb{A}_k)^{Br(X)}=\emptyset}, then {X(k)=\emptyset}.

Of course, one of the main points of Manin’s condition is that, even though {X(\mathbb{A}_k)^{Br(X)}} seems a complicated, it can be computed in practice for many examples.

The perspective provided by Manin’s condition led Colliot-Thélène to make the following conjecture (previously formulated by Sansuc in the setting of rational surfaces):

Conjecture (Colliot-Thélène). If {X} is a rationally connected variety, then {\overline{X(k)} = X(\mathbb{A}_k)^{Br(X)}}.

Remark 3 It is known (since 2000) that, in general, {X(\mathbb{A}_k)^{Br(X)}\neq\emptyset} does not imply that {X(k)\neq\emptyset}. Thus, it is necessary to impose some geometrical conditions on {X} in the formulation of any conjecture in the spirit of Colliot-Thélène’s conjecture.

Two examples where this conjecture is known to be true are:

  • intersections of two quadrics in the projective space {\mathbb{P}^n} if {n\geq 7} (by the results of Colliot-Thélène, Sansuc and Swinnerton-Dyer from 1987);
  • smooth compactifications of homogenous spaces of algebraic linear groups with connected stabilizers (by the results of Borovoi).

Remark 4 At this point, Serre asked Harari whether the particular choice of smooth compactification was important in the second item above. Harari replied that, even though this is not obvious, our whole discussion so far is birationally invariant, and this implies that it is not important what smooth compactification was taken in the statement of Borovoi’s theorem.

Logically, one can expect that this discussion of rational points has a counterpart for zero-cycles. In fact, the Brauer-Manin coupling (2) extends by linearity to Chow’s groups of zero-cycles modulo rational equivalence, so that we have a coupling

\displaystyle \prod\limits_{v} CH_0(X_{k_v})\times Br(X)\rightarrow \mathbb{Q}/\mathbb{Z} \ \ \ \ \ (3)

where {X_{k_v}:=X\times_k k_v}. By analogy with Colliot-Thélène’s conjecture, this leads us to suspect that there might be a relation between the kernel (to the left) of this coupling and the zero-cycles of X over {k}. In this direction, we have the following conjecture:

Conjecture ({E_1}) (Colliot-Thélène, Kato, Saito). If there exists a family {(z_v)} of (local) zero-cycles orthogonal to {Br(X)} (with respect to the coupling (3)) with {deg(z_v)=1}, then {X} has a (global) zero-cycle of degree {1} over {k}.

Remark 5 Note that this time we made no geometric assumption on {X}.

Remark 6 There is a refined version of this conjecture (called Conjecture {(E)}) where one describes the image of {\prod\limits_{v} CH_0(X_{k_v})} in {Hom(Br(X), \mathbb{Q}/\mathbb{Z})} under the natural application induced by the Brauer-Manin coupling (3).

Remark 7 Concerning the nomenclature, Harari told that “Conjecture ({E_1})” probably means “Conjecture of Existence of zero-cycles of degree one”, and, then, when this conjecture was refined, the subscript {1} was removed leading to “Conjecture (E)”.

Again, let us give some examples where the conjecture ({E_1}) is known to be true:

  • curves {X} whose Tate-Shafarevich group of its Jacobian is finite (by the results of Saito in 1989);
  • surfaces fibered in conics over {\mathbb{P}^1} (by the results of Salberger in 1988);
  • smooth compactifications of homogenous spaces of algebraic linear groups with connected stabilizers (by the results of Liang in 2013).

These examples indicate that one can prove non-trivial results about the existence of rational points and/or zero-cycles when {X} has an extra structure. As we are going to see now, Harpaz and Wittenberg obtained important results in this direction when {X} is fibered over a curve.

3. Statement of Harpaz-Wittenberg theorems

Suppose that {X} is fibered over a curve {C} (say {f:X\rightarrow C}) whose Tate-Shafarevich group is finite or simply take {C=\mathbb{P}^1_k}. Assume that the generic fiber {X_{\eta}=f^{-1}(\eta)}, {\eta\in C}, is rationally connected.

Before stating the results of Harpaz-Wittenberg, we need the following notion introduced by Skorobogatov:

Definition 3 A {k}-variety {Y} is split if it contains an irreducible component of multiplicity {1} which is geometrically integral.

Remark 8 For {k=\mathbb{Q}}, {x_1^2-x_2^2=0} is split: it decomposes as {(x_1-x_2)(x_1+x_2)=0}, and both {x_1-x_2=0} and {x_1+x_2=0} are defined over {k=\mathbb{Q}}. On the other hand, {x_1^2+x_2^2=0} is not split.

Coming back to {f:X\rightarrow C} as above, it is worth to mention that, from the point of view of the so-called fibration method of construction of rational points and zero-cycles, the non-split fibers of {f:X\rightarrow C} are the bad fibers. The reason for this fact is that if {Y} is split, then one can show (with the aid of the Lang-Weil estimate and Hensel’s lemma) that {Y(k_v)\neq\emptyset} for almost all places {v}.

In this setting, the question that we want to discuss is the following. Can we prove conjectures above for {X} assuming its validity for the fibers of {f:X\rightarrow C}?

The first result of Harpaz-Wittenberg provides an affirmative answer for this question in the context of the Conjecture ({E_1}):

Theorem 4 (Harpaz-Wittenberg (2014)) Let {f:X\rightarrow C} as above. Suppose that all smooth fibers satisfy Conjecture ({E_1}). Then, Conjecture ({E_1}) holds for {X}.

Remark 9 One can replace “all smooth fibers” by “many smooth fibers” in the previous statement (where “most” means a non-empty Zariski open subset of fibers, for example). Also, we can replace “Conjecture ({E_1})” by “Conjecture (E)” in this statement.

The next two examples illustrate the range of applicability of this theorem:

  • If {f:X\rightarrow\mathbb{P}^1} has a generic fiber birational to a homogenous space of algebraic linear group with connected stabilizer, then Conjecture ({E_1}) holds for {X}.
  • Consider the equation {N_{k'|k}(x_1\omega_1+\dots x_r\omega_r)=P(t)}, where {k'} is a finite extension of {k}, {N_{k'|k}} is the corresponding norm, {\omega_1,\dots, \omega_r} is a basis of {k'|k}, {x_1,\dots, x_r} are the variables, and {P(t)} is a non-zero polynomial in {t\in\mathbb{P}^1_k}. Let {X} be a projective smooth model associated to this “normic equation” (such {X} always exists by Hironaka’s theorem of resolution of singularities). Then, the generic fibers of {X} are birational to a homogenous space of algebraic linear group with connected stabilizer, so that Conjecture ({E_1}) holds for {X}.

Let us now compare this result with other theorems previously known in the literature.

Colliot-Thélène, Skorobogatov and Swinnerton-Dyer (in 1998) and Liang (in 2013) proved some cases of Harpaz-Wittenberg theorem under much more restrictive hypothesis on the Brauer group {Br(X_{\eta})} of the generic fiber and/or on the bad fibers of {f:X\rightarrow C}. Indeed, Colliot-Thélène-Skorobogatov-Swinnerton-Dyer imposed in their work that {Br(X_{\eta})} is trivial and the bad fibers are split over a finite extension of {k} with Abelian Galois group, while Liang makes no assumption on {Br(X_{\eta})} but he imposes that there exists at most one bad fiber.

In particular, one of the great advantages of Harpaz-Wittenberg theorem is that there is no need to make any assumptions on {Br(X_{\eta})} or on the bad fibers, so that it can be applied to a whole new class of examples.

Concerning Colliot-Thélène’s conjecture on rational points, there are analogs of Harpaz-Wittenberg theorem (with “Conjecture ({E_1})” replaced by Colliot-Thélène’s conjecture) under certain restrictive hypothesis. For example, in the same work mentioned above, Colliot-Thélène-Skorobogatov-Swinnerton-Dyer proved such an analog assuming the validity of the so-called Schnizel’s hypothesis (a broad conjectural generalization of Dirichlet’s theorem on arithmetic progressions), while Harari (in 1997) proved such an analog under the assumption that there is at most one bad fiber.

Harpaz and Wittenberg have an analog of their theorem in the context of Colliot-Thélène’s conjecture, but, as it turns out, they were not able to completely removed the restrictive hypothesis mentioned in the previous paragraph. More precisely, they showed the following result.

Theorem 5 (Harpaz-Wittenberg) Suppose that {f:X\rightarrow\mathbb{P}^1_{\mathbb{Q}}} (as above) has all of its bad (non-split) fibers over {\mathbb{Q}}-points. Then, the validity of Colliot-Thélène’s conjecture for the fibers of {f:X\rightarrow\mathbb{P}^1_{\mathbb{Q}}} implies that {X} satisfies Colliot-Thélène’s conjecture.

Once more, a basic example illustrating Harpaz-Wittenberg comes from normic equations {N_{k'|\mathbb{Q}}(x_1\omega_1 + \dots + x_r\omega_r) = P(t)} such that the polynomial {P} splits over {\mathbb{Q}}. In fact, the statement of Harpaz-Wittenberg theorem in the case of these examples was previously established by Browning and Matthiesen (in 2013) using methods from Analytic Number Theory (inspired from the works of Green-Tao-Ziegler).

At this point, Harari started the discussion of the elements of the proofs of the theorems of Harpaz-Wittenberg. However, I will not pursue this discussion here as I do not feel confident to comment on this part of Harari’s talk.

Instead, we will close this post here: the curious reader can consult the lecture notes of Harari and/or the video of Harari’s talk (from 38h50s on) for more details.

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