Let A be an abelian variety defined over a number field K has the following property: Fix an integer m > 1 and for each good prime p, suppose the number of (F_p)-points of A mod p is divisible by m. Does there exist a K-isogenous abelian variety A' with rational torsion divisible by m?
This question was first posed by Lang and then taken up by Katz in the early 1980s where he showed the answer is `Yes' for elliptic curves and `No' for threefolds and higher. The case of abelian surfaces is more difficult and only some partial results are known. We will survey what is known along with some recent work on the problem. In particular, we will focus on the subtlety of abelian surfaces and present new results towards a complete answer to this question.
Some l-adic representations attached to low dimensional abelian varieties
The flow polytope associated to an acyclic graph is the set of all nonnegative flows on the edges of the graph with a fixed netflow at each vertex. We will examine flow polytopes arising from permutation matrices, alternating sign matrices and Tesler matrices. Our inspiration is the Chan-Robins-Yuen polytope (a face of the polytope of doubly-stochastic matrices), whose volume is equal to the product of the first n Catalan numbers (although there is no known combinatorial proof of this fact!). The volumes of the polytopes we study all have nice product formulas.
Product formulas for volumes of flow polytopes
Mar. 31, 2015 2pm (MATH 350)
Lie Theory
Lionel Levine (Cornell)
X
I will describe the role played by an Apollonian circle packing in the scaling limit of the abelian sandpile on the square grid Z^2. The sandpile solves a certain integer optimization problem. Associated to each circle in the packing is a locally optimal solution to that problem. Each locally optimal solution can be described by an infinite periodic pattern of sand, and the patterns associated to any four mutually tangent circles obey an analogue of the Descartes Circle Theorem. Joint work with Wesley Pegden and Charles Smart.