The Tate conjecture

The American Institute of Mathematics

The following compilation of participant contributions is only intended as a lead-in to the

ARCC workshop “The Tate conjecture.” This material is not for public distribution. Corrections and new material are welcomed and can be sent to workshops@aimath.org

Version: Fri Jul 20 20:44:49 2007

Table of Contents

A. Participant Contributions . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Blasius, Don

2. Ellenberg, Jordan

3. Geisser, Thomas

4. Getz, Jayce

5. Hall, Chris

6. Iyer, Jaya

7. Knightly, Andrew

8. Lichtenbaum, Stephen 9. Lyons, Christopher 10. Milne, James

11. Pacheco, Amilcar

12. Ramakrishnan, Dinakar

13. Raskind, Wayne

14. Ribet, Kenneth

15. Schoen, Chad

16. Silverberg, Alice

17. Suh, Junecue

18. Ulmer, Douglas

19. Virdol, Cristian

20. Wambach, Eric

21. Zarhin, Yuri

Chapter A: Participant Contributions

A.1 Blasius, Don

In the late 1980’s I proved, with Jon Rogawski, the Tate Conjecture for H^{2} of the Shimura varieties attached to forms of U(3). In the early 1990’s I considered the same problem for all Shimura varieties and reduced the general case to several key ones, notably the case of Shimura surfaces attached to quaternion algebras. As this case was under study by Murty and Ramakrishnan at the time, who were optimistic that they could get a general result, I did not pursue the matter further. Now 15 years later the automorphic theory is much more developed, and it would seem to be a good test of the state of theory to revisit this question in the key cases, and hopefully to get closer to a general result.

Unfortunately, in my opinion, this is primarily an interesting exercise in the theory of Shimura varieties and automorphic forms, and is very unlikely to shed any light on either the general case of H^{2} of a variety or even the cases of H^{2j} of a Shimura variety (j > 1).

A.2 Ellenberg, Jordan

I’m very involved (in collaboration with Venkatesh) in trying to understand the way that statements about varieties over function fields over finite fields can be used to develop conjectures and even suggest methods of proving theorems about varieties over number fields. In a recent paper, “Selmer groups and Mordell-Weil groups of elliptic curves over towers of function fields” I explore some Iwasawa-flavored questions about the growth of Selmer groups in certain non-abelian towers. These results “deconditionalized” results of Silverman which were conditional on Tate’s conjectures. It would be quite interesting (at least for me!) to understand the extent to which those results match up with recent work of the non-abelian Iwasawa theorists about growth of ranks of elliptic curves over non-abelian towers of number fields, and to what extent one can really get at the difference between Selmer and Mordell-Weil as we go up such a tower (this being the Tate conjecture part.) Of course the L-functions form the natural bridge – on the function field side, careful consideration of the way the Lfunctions vary in the tower is the key to Ulmer’s results on unboundedness of Mordell-Weil ranks of elliptic curves.

A.3 Geisser, Thomas

I am interested in Tate’s conjecture for schemes of finite fields. It follows from work of Kahn, Lichtenbaum and myself, that Tate’s conjecture together with Beilinson’s conjecture that rational and numerical equivalence agree up to torsion, is equivalent to the finite generation of Weil-etale (or arithmetic) cohomology groups H^{i}(X_{W} , Z(n)). Morever, the above mentioned conjectures imply many other good properties of these cohomology groups, for example Poincare duality. Conversely, if one is able to establish sufficiently many good properties of these cohomology, then finite generation, hence Tate’s conjecture follows.

A.4 Getz, Jayce

Let π be an automorphic representation of a reductive group G, and assume that π contributes to the cohomology of a finite-level Shimura variety Sh_{K}(G, X) of dimension n. The following is a recurring theme in the literature:

• π is a lifting from a smaller group if and only if there is a π_{f}-isotypical algebraic cycle contributing to H_{et}^{n}(Sh_{K}(G, X) × \overline{ℚ}, ℚ_{ℓ})

(some of the registrants at the conference have contributed to this theme). I would like to learn more about the techniques that have been applied to prove statements of this type.

A.5 Hall, Chris

Let F be a number field and K be the function field F (t). Given an elliptic curve E/K with non-constant j-invariant we consider the one-parameter family of quadratic twists of E by the polynomial t − c. We can show that, up to replacing E by a quadratic twist of ‘sufficiently large degree’, for almost all values of c ∈ F, the corresponding twist has rank zero. More precisely, if we regard each twist as an elliptic surface over F, then we can show that for almost all c ∈ F the space of vanishing cycles — the interesting piece of middle cohomology — is an irreducible G_{F} -module.

We can also show how to apply similar ideas to Lefschetz fibrations of (smooth) complete intersections. More precisely, if X/F is a complete intersection together with a general Lefschetz fibration by hypersurfaces of sufficiently high degree, then we can show that, for almost all F -fibers, the space of vanishing cycles is an irreducible G_{F} -module and the Tate conjectures are true.

A.6 Iyer, Jaya

The rational Chow groups of a smooth projective variety is expected to have a finite descending filtration satisfying certain compatibility conditions. J. Murre has proposed a filtration in terms of a Chow-Kuenneth decomposition of the variety. A Chow-Kuenneth decomposition is known to exist for few examples; curves, surfaces, abelian varieties and certain threefolds. We have also provided some examples recently. Jannsen has recently given some examples which satisfy Murre’s conjecture which are defined over the rational or elliptic function field over a finite field. Assuming Tate’s conjecture he derives some interesting consequences on the motivic cohomology. We would like to discuss his paper with other participants and hopefully it may lead to other examples.

A.7 Knightly, Andrew

I am interested in the use of automorphic forms to understand cases of Tate’s conjecture. My background is stronger on the automorphic/analytic side, so I am hoping that the workshop will improve my understanding of the geometric part of the picture. I echo the call for a clear, detailed exposition of the case of a product of two modular curves, without using Faltings.

A.8 Lichtenbaum, Stephen

I am particularly interested in formulas which express special values of zeta and Lfunctions in terms of cohomological Euler characteristics. Typically, any proof of such formulas relies on knowing the Tate conjecture.

I have recently developed a cohomology theory (Weil-etale cohomology) for varieties over finite fields, where conjecturally all the cohomology groups of motivic complexes should be finitely-generated abelian groups. This conjecture would imply the Tate conjecture, and conceivably might provide a new method of attack, although I must admit I don’t see how at the moment.

A.9 Lyons, Christopher

I’m interested in the Tate Conjecture for modular surfaces over finite fields, and am currently considering the case of a product of two modular curves, with a view towards proving this case without the use of Faltings’ theorem.

A.10 Milne, James

It is known that the Hodge conjecture for abelian varieties of CM-type implies the Tate conjecture for the same varieties (Pohlmann) and also for abelian varieties over finite fields (Milne). Thus the Hodge conjecture for abelian varieties of CM-type provides a key test for both conjectures. It is discouraging that, although we know what the Hodge cycles are on such varieties, there seems to be no immediate hope of proving that they are algebraic.

An alternative is to attempt to prove that the cohomology classes predicted by the Hodge and Tate conjectures to be algebraic have some of the properties that this would entail. Deligne did this for abelian varieties in characteristic zero. Another very interesting case to study is abelian varieties over finite fields. As with Deligne’s result, this will having important implications for Shimura varieties.

A.11 Pacheco, Amilcar

Let f : Y → X be a proper flat morphism of relative dimension d of smooth projective irreducible varieties Y and X defined over a number field k of dimensions n and m. Let S be a finite set of places of k such that for every place v ∉ S the reduction f_{v} of f modulo v is still a proper flat morphism of varieties over the residue field κ_{v} of v and the reductions Y_{v} and X_{v} of both varieties are smooth. For each x ∈ X_{v}(κ_{v}) let a_{v}(Y_{v}), resp. b_{v}(Y_{v}) be the trace of Frobenius of κ_{v} on H_{et}^{1}(Y_{x,v} × \overline{κ}_{v}), resp. H_{et}^{2}(Y_{x,v} × \overline{κ}_{v}). Let q_{v} := #κ_{v}. Consider the average trace of Frobenius

A_{v}(Y) := \frac{1}{q_{v}^{m}} ∑_{x∈X (κ_{v})} a_{v}(Y_{x,v}), B_{v}(Y) := \frac{1}{q_{v}^{m}} ∑_{x∈X (κ_{v})} b_{v}(Y_{x,v}).

Let K := k(X). Let P_{Y} be the Picard variety of the generic fiber Y of f and (τ, B) the K/k-trace of P_{Y}. Define a_{v}(B) as above and let A_{v}^{∗}(Y) := A_{v}(Y) − a_{v}(B).

Denote by conjecture N the following statement: the function

∑_{v∉S} −A_{v}^{∗}(Y)\frac{log(q_{v})}{q_{v}^{s}} + ∑_{v∉S} B_{v}(Y)\frac{log(q_{v})}{q_{v}^{s+1}}

which is a priori defined analytically over ℜ(s) > 1 extends to the whole plane with a simple zero at s = 1 with residue

rank (\frac{P_{Y} (K)}{τB(k)}) + rank(NS(Y/K)).

In a previous paper with M. Hindry and R. Wazir we proved that any two of the following statements imply the third: (1) Tate’s conjecture (the divisor case) for Y; (2) Tate’s conjecture for X; (3) conjecture N. In fact, conjecture N can be thought of as “generic BSD + generic Tate”. We can associate L-functions L_{i}(Y/K, s) for i = 1, 2 to the fibration f, so that conjecture N can be rewritten as

ord_{s=1} (\frac{L_{1}(Y/K, m + s)}{L_{1}(B/k, s)L_{2}(Y/K, m + 1 + s)}) = rank (\frac{P_{Y} (K)}{τB(k)}) + rank(NS(Y/K)).

Question 1. What happens if we replace H^{1} by H^{2m−1} and H^{2} by H^{2m} obtaining a

new conjecture N′ = “generic Bloch” + “generic Tate” for codimension m cycles. Do we get a similar result with respect to the corresponding Tate’s conjecture for codimension m cycles in X and Y?

We have actually proved an unconditional result, namely that the function

∑_{v∉S} −A_{v}^{∗}(Y)\frac{log(q_{v})}{q_{v}^{s}} + ∑_{v∉S} B_{v}(Y)\frac{log(q_{v})}{q_{v}^{s+1}} − ∑_{v∉S} b_{v}(Y)\frac{log(q_{v})}{q_{v}^{s+1}} + ∑_{v∉S} \frac{log(q_{v})}{q_{v}^{s+1}}

that is a priori analytic over ℜ(s) > 1 can be extended to the half-plane ℜ(s) ≥ 1 with a simple pole at s = 1 with residue equal to

r := rank (\frac{P_{Y} (K)}{τB(k)}) + rank(NS(Y/K)) − rank(NS(Y/k)) + rank(NS(X /k)).

This is in fact the order at s = 1 of the rational function

M(Y/X , s) := \frac{L_{1}(Y/K, m + s)L_{2}(Y, s + 1)}{L_{1}(B/k, s)L_{2}(Y/K, m + s + 1)L_{2}(X , s + 1)}.

Question 2. What can be said about the “special value”

lim_{s→1}(s − 1)^{−r}M(Y/X , s) =^{?} \frac{R_{1}(Y/K)R_{2}(Y/k)}{R_{1}(B/k)R_{2}(Y/K)R_{2}(X /k)},

where R_{i}(·) denotes the hypothetical special value of the corresponding L-function?

A.12 Ramakrishnan, Dinakar

I have worked on, and continue to be interested in, a few directions concerning the Tate conjecture. The first is the case of divisors on varieties X over number fields. By repeatedly taking hyperplane sections and applying Lefschetz, one can reduce this case to (smooth projective) surfaces. There is as of yet no general idea to approach this situation for arbitrary surfaces, though the analogue for the Hodge conjecture has been known for long, due to the classical work of Lefschetz. It is of interest to try to establish this for different classes of interesting surfaces, like the various modular surfaces. (For abelian surfaces -even for divisors on abelian varieties, this is known by Faltings, and for K3 surfaces one can reduce to the abelian variety situation by using the Kuga-Satake-Deligne construction.) For the Hilbert modular case, the Tate conjecture was established by Harder, Langlands and Rapoport over abelian fields, then extended for general number fields by the independent and simultaneous works of Klingenberg on the one hand and of mine with Kumar Murty on the other. For Picard modular surfaces, which form another key class of surfaces, this was later done by Blasius and Rogawski. Another crucial class consists of quaternionic Shimura surfaces, defined by a quaternion division algebra D over a totally real number field F with B split at exactly two infinite places w, w’, say. I have some unpublished work with Kumar Murty concerning this situation (from over ten years ago). The difficulty comes only when F is non-Galois and/or there is no involution permuting w, w’. There was one essential case in which we had a circuitous way to approach it, but now, this seems amenable to solution by using in addition the recent array of results on Galois representations and modularity. Our primary method is to use period relations, involving a refinement of Waldspurger, and reduce to the Hodge conjecture on a related variety. In sum, the Tate conjecture is known for divisors on a few (classes of) surfaces, but in each class, not just the modular ones, the argument involves special techniques. One of the goals of the workshop, which I am coorganizing with Raskind, is to bring together people with different backgrounds, and try to see if a cross-fertilization of ideas can be useful. There is no guarantee of success, but may be a worthwhile thing to do. In any case, there are explicit problems which people can work on, and they are exciting.

The second direction of interest to me involves finding interesting examples of higher dimensional varieties, in particular fourfolds (like products of modular curves and Hilbert fourfolds), still over number fields, where there are Tate classes represented by codimension 2 algebraic cycles which are not intersections of divisors. Delicate new problems arise depending on the type of the underlying totally real number field, and there are some problems here which some young participants might want to try their hands on.

The third direction of interest is the situation for divisors on surfaces over finite fields, and a student of mine is beginning to work on this for certain modular surfaces. The interesting situation is when the reduction mod p of known divisors in characteristic 0 and their Frobenius transforms do not exhaust all the Tate classes. This is also crucial for making progress on a related conjecture of Beilinson concerning a Bloch Chow group, and also an interesting case of the Bloch-Kato conjecture. In fact, the relationship to the Beilison conjecture at the near-central point of an even weight motive is of abiding interest to me.

Finally, I am curious about the case of divisors on surfaces X of geometric genus 1, which are not K3 surfaces. When one can put such an X in a family, one can use the period surjectivity and an analogue of Kuga-Satake.

A.13 Raskind, Wayne

Anyone who does research in the theory of algebraic cycles and etale cohomology is bound to be interested in the Tate conjecture. A few years ago, I stumbled on a local version of the Tate conjecture for certain types of varieties over p-adic fields which we call varieties with totally degenerate reduction. I then formulated an analogue of the Tate conjecture and the generalized Hodge conjecture that predicts that the coniveau filtration on the p-adic etale cohomology can be given by the successive kernels of what we call enriched monodromy operators. These operators are a finer version of the monodromy operators on the graded pieces of the p-adic etale cohomology for the monodromy filtration. They take values in odd, continuous Quillen algebraic K-theory and generalize the period pairing for abelian varieties with totally toric reduction. For varieties over number fields that have this type of reduction at some finite place dividing p, this local Tate conjecture implies the usual Tate conjecture. I expect that these methods will help us understand and prove the Tate conjecture in some cases.

A.14 Ribet, Kenneth

Quite a long time ago, I contributed to the subject by proving the Tate conjecture for H^{1} of a modular curve. The method was to show, essentially, that the Galois action on H^{1} is so rich that there is room for no more potentially algebraic cycles than the ones that we can account for by correspondences. This method is generic and flexible enough that it makes sense to apply it in other contexts. In particular, to the extent that we can show for other motives that the Galois action is as rich as we expect it to be, we can prove the Tate conjecture by inspection.

I am looking forward very much to this workshop.

A.15 Schoen, Chad

I would be interested in learning about the insights of the Langlands program into the cohomology of Shimura varieties. I don’t understand much about how the Tate conjecture might relate to some of the more recent conjectures about motives. Has work on motives led to insights concerning the Tate conjecture?

A.16 Silverberg, Alice

I am interested in abelian varieties and Shimura varieties. For this workshop, I am especially interested in connections between the Tate Conjecture and ranks of Mordell-Weil groups of abelian varieties, as in the work of Ulmer. Since I haven’t thought about the Tate Conjecture for a number of years, I hope that the workshop will include several introductory lectures.

A.17 Suh, Junecue

One cannot have a Weil cohomology theory for projective smooth varieties over finite fields: As pointed out by Serre, if one did, the (quaternion) endomorphism algebra of a supersingular elliptic curve would act on a 2-dimensional rational vector space.

As one is not directly led to a contradiction with an *ordinary* elliptic curve, one may still wonder whether there could be such a theory, applicable only for ordinary projective smooth varieties over finite fields.

In part of my thesis, I proved that if one assumes Tate’s conjecture for (i) projective smooth varieties over finite fields and (ii) CM abelian varieties over number fields, then one does have a neutral fibre functor on the Tannakian subcategory of the whole motive category over finite fields, consisting of motives with integral Newton slopes of Frobenius. The proof of a key ingredient closely follows the method of Pohlmann (1968).

I am interested in the behaviour of algebraic/absolute Hodge cycles on CM abelian varieties under reduction modulo p.

A.18 Ulmer, Douglas

I am interested in ideas around the well-known but under-exploited fact that a variety which is dominated by a product of curves satisfies the Tate conjecture relating divisors and classes in H^{2}.

Let k be a perfect field, K = k(t). If X is a curve over K, let X → P^{1} be the corresponding fibered surface over k.

A funny-looking result in my recent work on ranks is this: if k is finite and X is a curve over K which has a plane model defined by g(t, x, y) = 0 with g the sum of exactly four non-zero monomials satisfying mild conditions on the exponents, then the Jacobian of X satisfies the BSD conjecture. The key point is that the 4-monomials hypothesis guarantees that X is dominated by a Fermat surface and so by a product of curves. Thus 4-monomials is just a stand-in for DPC. It has the virtue of obviously being preserved in towers, i.e., when t is replaced by u^{d}.

Other methods can be used to produce high analytic ranks in towers k(u) = k(t^{1/d}) for k finite. Combined with the above, we get: for all p and all g > 0 there are Jacobians over F_{p}(u) which satisfy BSD and which have arbitrarily large rank. This motivates me to try to construct other families of curves X such that the fiber product of X → P^{1} and P^{1} → P^{1} (u^{d} ← t) is again DPC. This can be done (Berger, Arizona thesis 2007) and it yields lots more Jacobians over F_{p}(u) with BSD and high rank.

My current interest is in exploiting the geometry of Berger’s construction for other fields k. When k is finitely generated, we get lots of surfaces X which are DPC and so satisfy Tate. For any k, we get a supply of curves X over K so that the rank of Jac_{X}(K) is calculable in terms of simple geometry and isogenies of Jacobians of curves over k. This seems to be a significant reduction in complexity with interesting applications.

A.19 Virdol, Cristian

The second part of the Tate conjecture asserts that the for a smooth projective variety X defined over a number field F, the L-function has a pole at s = i + 1 of order equal to dimℚ̄_{l}(H_{et}^{2i}(X, ℚ̄_{l})(i))^{Γ_{E}}, where E is an arbitrary extension of F. We are able to show this result when X is a quaternionic Shimura surface or (under certain assumptions) a quaternionic Shimura fourfold and E is an arbitrary solvable extension of a totally real field that contains the canonical field of definition of X.

A.20 Wambach, Eric

My main interest is in the Tate conjecture for Shimura varieties and for Abelian varieties. For Shimura varieties associated to the quasisplit U(3), Tate conjecture for H^{2} is known due to Blasius and Rogawski. However not all algebraic cycles have been constructed explicitly. I am wondering wether the embedding of SO(2, 1) inside U(3) gives rise to nontrivial algebraic cycles which contribute to H^{2}. Possible methods could be Theta-lift and Siegel-Weil formula, or the relative trace formula. Either one may also work in other situations.

A.21 Zarhin, Yuri

I hope to learn how to construct exotic algebraic classes, using say, modular forms and/or the geometry of algebraic varieties with automorphisms.