# Bjorn Poonen's work, Fall 2016

Basic setup: every week one of the graduate students picks a paper by Bjorn and explains what is happening in the paper. Please visit Bjorn Poonen's website for the list of papers.

Organizational:

1. Please email me if you want to be on the associated mailing list.
2. The talks will be 2x45 minutes with a short break.
3. Time and place: Room 407, Fridays 10:30 -- 12:00 AM.
4. First meeting: September 9. Everybody interested please attend.

List of lectures:

1. September 9.
• Speaker: Remy van Dobben de Bruyn
• Title: The Hodge ring of varieties in positive characteristic
• Abstract: Which Hodge diamonds can occur as a sum of Hodge diamonds of varieties? Are there any linear relations between the Hodge numbers of a variety in characteristic p, besides the ones coming from Serre duality? Which linear combinations of Hodge numbers are birational invariants? These questions were answered over the complex numbers by D. Kotschick and S. Schreieder. In this talk, we will discuss the version in characteristic p, where the answers are different due to the failure of Hodge symmetry.
• This is the unique lecture not about a paper of Poonen.
2. September 16.
• Speaker: Shuai Wang
• Title: An explicit algebraic family of genus-one curves violating the Hasse principle
• Abstract: We prove that for any t in Q, the curve 5 x^3 + 9 y^3 + 10 z^3 + 12((t^12-t^4-1)/(t^12-t^8-1))^3 (x+y+z)^3 = 0 in P^2 is a genus 1 curve violating the Hasse principle. An explicit Weierstrass model for its Jacobian E_t is given. The Shafarevich-Tate group of each E_t contains a subgroup isomorphic to Z/3 x Z/3.
3. September 23.
• Speaker: Joseph Gunther
• Title: Most odd degree hyperelliptic curves have only one rational point
• Abstract: Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g tends to infinity. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves. The method can be summarized as follows: using p-adic analysis and an idea of McCallum, we develop a reformulation of Chabauty's method that shows that certain computable conditions imply #C(Q)=1; on the other hand, using further p-adic analysis, the theory of arithmetic surfaces, a new result on torsion points on hyperelliptic curves, and crucially the Bhargava-Gross equidistribution theorem for nonzero 2-Selmer group elements, we prove that these conditions are often satisfied for p=2.
4. September 30.
• Speaker: Dmitrii Pirozhkov
• Title: Bertini theorems over finite fields
• Abstract: Let X be a smooth quasiprojective subscheme of P^n of dimension m >= 0 over F_q. Then there exist homogeneous polynomials f over F_q 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 zeta_X(m+1)^{-1}, where zeta_X(s)=Z_X(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.
5. October 7.
• Speaker: Carl Lian
• Title: Bertini irreducibility theorems over finite fields
• Abstract: Given a geometrically irreducible subscheme X in P^n over F_q of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that the intersection of H and X is geometrically irreducible tends to 1 as d tends to infinity. We also prove variants in which X is over an extension of F_q, and in which the immersion of X in P^n is replaced by a more general morphism.
6. October 7.
• Speaker: Keaton Naff
• Title: Using elliptic curves of rank one towards the undecidability of Hilbert's Tenth Problem over rings of algebraic integers
• Abstract: Let F and K be number fields, with F contained in K. and let O_F and O_K be their rings of integers. If there exists an elliptic curve E over F such that E(F) and E(K) have rank 1, then there exists a diophantine definition of O_F over O_K.
7. October 21.
• Speaker: Monica Marinescu
• Title: Multiples of subvarieties in algebraic groups over finite fields
• Abstract: Let X be a subvariety of a commutative algebraic group G over Fq such that X generates G. Then any closed point of G is the image of a closed point of X under an endomorphism of G. If G is semiabelian, you only need to use the multiplication by n endomorphisms of G, and there is a density-1 set of primes S such that X(\bar Fq) projects surjectively onto the S-primary part of G(\bar Fq). These results build on work of Bogomolov and Tschinkel.
8. October 28.
• Speaker: Shizhang Li
• Title: Néron-Severi groups under specialization
• Abstract: Andr\'e used Hodge-theoretic methods to show that in a smooth proper family X to B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different approach to Andr\'e's theorem, which also proves the following refinement: in a family of varieties with good reduction at p, the locus on the base where the Picard number jumps is nowhere p-adically dense. Our proof uses the p-adic Lefschetz (1,1) theorem'' of Berthelot and Ogus, combined with an analysis of p-adic power series. We prove analogous statements for cycles of higher codimension, assuming a p-adic analogue of the variational Hodge conjecture, and prove that this analogue implies the usual variational Hodge conjecture. Applications are given to abelian schemes and to proper families of projective varieties.
9. November 4. AGNES. No lecture
10. November 11.
• Speaker: Qixiao Ma
• Title: The Cassels-Tate pairing on polarized abelian varieties
• Abstract: Let (A,\lambda) be a principally polarized abelian variety defined over a global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing \Sha(A)_\nd \times \Sha(A)_\nd \rightarrow \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 \Sha(A)_\nd. These criteria are expressed in terms of an element c \in \Sha(A)_\nd that is canonically associated to the polarization \lambda. 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 \#\Sha(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 \ge 2 over \Q have a Jacobian with nonsquare \#\Sha (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.
11. November 18. No lecture
12. November 25. Holiday. No lecture
13. December 2.
• Speaker: Monica Marinescu
• Title: Multivariable polynomial injections on rational numbers
• Abstract: For each number field k, the Bombieri-Lang conjecture for k-rational points on surfaces of general type implies the existence of a polynomial f(x,y) in k[x,y] inducing an injection k x k --> k.
14. December 9.
• Speaker: Johan de Jong
• Title: TBA
• Abstract: TBA