Let K/k be a finite extension of number fields. Both k and K can be embedded in their idèle groups A_k and A_K , and the norm map N_{K/k} : K^* → k^* extends to a map N_{A_K /A_k} on these groups. The Hasse Norm Theorem states that, if K/k is Galois with cyclic Galois group, every element of k^* that, after embedding in A^*_k, lands in the image of N_{A_K /A_k} , is also in the image of N_{K/k}. In other words, ever element of k^* that is a norm everywhere locally is a norm globally. Field extensions with this property are said to satisfy the Hasse norm principle (HNP).
There are several families of extensions for which the HNP is known to hold. On the other hand, Hasse already showed that not all extensions satisfy the HNP. An interesting question is therefore, how often does the HNP fail?
In this talk I will give an introduction to the Hasse norm principle and different ways that people have tried to answer the aforementioned question. I will then report on joint work with Anand Deopurkar, Rachel Newton, and Vaidehee Thatte, where we study the failure of the Hasse norm principle in families of extensions of global function fields.
Arithmetic dynamics studies the number-theoretic properties of algebraic numbers under iteration by a polynomial or rational function. A central object in the theory is the canonical height introduced by Call and Silverman. Roughly speaking, this function measures the arithmetic complexity of dynamical orbits. The Dynamical Lehmer Conjecture predicts an optimal lower bound for the canonical height of a wandering algebraic number in terms of its degree, but it remains open in general.
In this talk, I will present ongoing work on lower bounds for the canonical height associated with quadratic post-critically finite polynomials defined over a number field and wandering points lying in its cyclotomic closure.
The values of the Riemann zeta function at odd positive integers greater than 1 are conjectured to be transcendental, yet even their irrationality remains a mostly open question. Recently, new inputs to irrationality proofs have come from geometric methods, especially in connection with periods of hyperplane arrangements. In this talk, we outline the connection between zeta values and moduli spaces of curves of genus zero and explain how this geometric perspective can shed some light on irrationality questions. Then, we describe recent work on a special class of hyperplane arrangements, whose periods are generated by special values of multiple polylogarithms.