Bjorn Poonen

The Cassels-Tate pairing on polarized Abelian varieties.

Let (A, e) be a principally polarized abelian variety defined over a global field k, and let ..(A) be its Shafarevich-Tate group. Let ..(A)nd denote the quotient of ..(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing ..(A)nd . ..(A)nd > Q/Z. If A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on ..(A)nd. These criteria are expressed in terms of an element c . ..(A)nd that is canonically associated to the polarization e. In the case that A is the Jacobian of some curve, a down-to-earth version of the result allows us to determine effectively whether #..(A) (if finite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperelliptic curves of even genus g . 2 over Q have a Jacobian with nonsquare #.. (if finite). For example, it appears that this density is about 13% for curves of genus 2. The proof makes use of a general result relating global and local densities; this result can be applied in other situations.

Bertini theorems over finite fields

Let X be a smooth quasiprojective subscheme of Pn of dimension m iÝ 0 over Fq. Then there exist homogeneous polynomials f over Fq for which the intersection of X and the hypersurface f = 0 is smooth. In fact, the set of such f has a positive density, equal to ?AEX(m+1).1, where ?AEX(s) = ZX(q.s) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture.

Cycles of quadratic polynomials and rational points on a genus-

It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N=4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X

2

_1

curve

(16), whose rational points had been previously computed. We prove there are none with N=5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Galois-stable 5-cycles, and show that there exist Galois-stable N-cycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N=6.

The Grothendieck ring of varieties is not a domain

Let k be a field. Let K_0(V_k) denote the quotient of the free abelian group generated by the geometrically reduced varieties over k, modulo the relations of the form [X]=[X-Y]+[Y] whenever Y is a closed subvariety of X. Product of varieties makes K_0(V_k) into a ring. We prove that if the characteristic of k is zero, then K_0(V_k) is not a domain.

Union-closed families

Abstract A union-closed family F is a finite collection of sets not all empty, such that any union of elements of F is itself an element of F . Peter Frankl conjectured in 1979 that for any such family, there is an element in at least half of its sets. But the problem remains unsolved. We find a number of equivalent conjectures, and we prove the conjecture in special cases, including for example all families involving up to seven elements or having up to 28 sets, extending the previously known result for up to 18 sets. We also prove a general theorem stating exactly when a subfamily is enough to guarantee the existence of an element from the subfamily which is in half the sets of the whole family.

Random maximal isotropic subspaces and Selmer groups

Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over F_p. A random subspace chosen with respect to this measure is discrete with probability 1, and the dimension of its intersection with a fixed compact open maximal isotropic subspace is a certain nonnegative-integer-valued random variable. We then prove that the p-Selmer group of an elliptic curve is naturally the intersection of a discrete maximal isotropic subspace with a compact open maximal isotropic subspace in a locally compact quadratic space over F_p. By modeling the first subspace as being random, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar. Our model is compatible with Delaunays heuristics for p-torsion in Shafarevich-Tate groups, and predicts that the average rank of elliptic curves over a fixed number field is at most 1/2. Many of our results generalize to abelian varieties over global fields.

Squarefree values of multivariable polynomials

Given f in Z[x_1,...,x_n], we compute the density of x in Z^n such that f(x) is squarefree, assuming the abc conjecture. Given f,g in Z[x_1,...,x_n], we compute unconditionally the density of x in Z^n such that gcd(f(x),g(x))=1. Function field analogues of both results are proved unconditionally. Finally, assuming the abc conjecture, given f in Z[x], we estimate the size of the image of f({1,2,...,n}) in (Q^*/Q^*2) union {0}.

Lattice Polygons and the Number 12

1. PROLOGUE. In this article, we discuss a theorem about polygons in the plane, which involves in an intriguing manner the number 12. The statement of the theorem is completely elementary, but its proofs display a surprisingly rich variety of methods, and at least some of them suggest connections between branches of mathematics that on the surface appear to have little to do with one another. We describe four proofs of the main theorem, but we give full details only for proof 4, which uses modular forms. Proofs 2 and 3, and implicitly the theorem, appear in [4].