# UCSD Number Theory Seminar (Math 209)

### Thursday 2-3pm, AP&M 7421

#### Fall Quarter 2014

 September 11 Peter Schneider (University of Munster and MSRI) Rigid character groups, Lubin-Tate theory, and (phi,Gamma)-modules The talk will describe joint work with L. Berger and B. Xie in which we build, for a finite extension L of Q_p, a new theory of (phi,Gamma)-modules whose coefficient ring is the ring of holomorphic functions on the rigid character variety of the additive group o_L, resp. a "Robba" version of it. October 2 NO SEMINAR October 9 NO SEMINAR October 16 NO SEMINAR October 23 Michelle Manes (University of Hawai'i) Galois theory of quadratic rational functions Given a global field K and a rational function f(x) defined over K, one may take pre-images of 0 under successive iterates of f, and thus obtain an infinite tree by assigning edges according to the action of f. The absolute Galois group of K acts on the tree, giving a subgroup of the group of all tree automorphisms. Beginning in the 1980s with work of Odoni, and developing especially over the past decade, a significant body of work has emerged on the size and structure of this Galois representation. The analogy here is to Serre's finite index results for Galois representations arising from elliptic curves. I will discuss the contributions of several researchers, including Boston and Jones, along with my own work (joint with Jones) on these questions. October 30 Valentijn Karemaker (University of Utrecht) Hecke algebra isomorphisms and adelic points on algebraic groups Let G denote an algebraic group over Q and K and L two number fields. Assume that there is a group isomorphism of points on G over the adeles of K and L, respectively. We establish conditions on the group G, related to the structure and the splitting field of its Borel groups, under which K and L have isomorphic adele rings. Under these conditions, if K or L is a Galois extension of Q and G(A_K) and G(A_L) are isomorphic, then K and L are isomorphic as fields. As a corollary, we show that an isomorphism of Hecke algebras for GL(n) (for fixed n > 1), which is an isometry in the L^1 norm over two number fields K and L that are Galois over Q, implies that the fields K and L are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field if it is Galois over Q. November 6 Dorian Goldfeld (Columbia University) A standard zero free region for Rankin-Selberg L-functions on GL(n) For n > 1, let $\pi, \pi'$ be two irreducible cuspidal automorphic representations of GL(n, A) where A denotes the adeles over Q. Let $L(s, \pi \times \pi')$ be the Rankin-Selberg L-function. If one of $\pi$ or $\pi'$ is self dual then it was shown by Moreno and Sarnak that the Rankin-Selberg L-function does not vanish at s = c+it when 1-c is less than a positive fixed constant times a negative power of log(|t| +2). This is also called a standard zero free region. A standard zero free region for the Riemann zeta function was first obtained by de la Vallee Poussin (prime number theorem). Currently, the best known zero free region for Rankin Selberg L-functions on GL(n) (in the non self dual case) is due to Brumley who has proved 1-c is less than a fixed constant times a negative power of |t| +2. In joint work with Xiaoqing Li we obtain a standard zero free region in the non self dual case. November 13 Kim Laine (Berkeley) Security in genus 3 The security of genus 3 curves in public key cryptography has long been somewhat unclear. For non-hyperelliptic genus 3 curves Claus Diem found a way to exploit the geometry of the curve to speed up index calculus on the Jacobian, achieving an impressive running time of Õ(q). Unfortunately the algorithm suffers from massive memory requirements. We have our own variation of non-hyperelliptic genus 3 index calculus, which improves Diem's approach in several ways. We study both the computational complexity and the memory cost of our method in great detail and make the results completely explicit. Combining this with some techniques to alleviate the memory cost, we get a very clear understanding of the security and show that for certain field sizes of practical interest the non-hyperelliptic genus 3 index calculus is a threat worth taking into account. The so-called isogeny attacks make genus 3 hyperelliptic curves equally vulnerable. November 20 TBA TBA TBA November 27 THANKSGIVING BREAK December 4 TBA TBA TBA December 11 TBA TBA TBA