# NTS/Abstracts Spring 2011

## Contents

## Anton Gershaschenko

Title: Moduli of Representations of Unipotent Groups |

Abstract: Representations of reductive groups are discretely parameterized, but unipotent groups can have non-trivial families of representations, so it makes sense try to construct and understand a moduli stack (or space) of representations of a given unipotent group. If you restrict to certain kinds of representations, it is possible to actually get your hands on the moduli stack and to construct a moduli space. I'll summarize the few things I know about the general case and then give you a tour of some interesting features that appear in small examples. |

## Keerthi Madapusi

Title: A rationality property of Hodge cycles on abelian varieties, with an application to arithmetic compactiﬁcations of Shimura varieties |

Abstract: Let A be an abelian variety over a number field E, and let v be a finite place of E where A has bad, split, semi-stable reduction. Then the toric part of the reduction of A at v gives rise to a partial integral structure on the (l-adic, p-adic, deRham, log crystalline) cohomology group H^1(A_{E_v} ), arising essentially from the character group of the torus. Let η be a Hodge cycle on A; then one can ask if η is rational (in a precise sense) with respect to this new integral structure on the cohomology. This question was first considered by Andre ́. Using the theory of Shimura varieties and the Faltings-Chai toroidal compactifications of the moduli of principally polarized abelian varieties, we convert this question into one of deciding if a certain sub-scheme of a torus embedding is again a torus embedding. In the situation where the Mumford-Tate group of A has a reductive model over Zp, for v|p (this is the unramified situation), we employ a generalization of the methods introduced by Faltings and Kisin–initially used to construct smooth integral models of Shimura varieties–to answer this question positively. A by-product of this rationality result is the construction of good toroidal compactifications of the integral models of Shimura varieties mentioned above. This was in fact the main motivation for considering, in the first place, the possibility of such a result. A formal consequence of the existence of these compactifications is the following result, which is a slightly weakened version of a conjecture of Yasuo Morita: Suppose the Mumford-Tate group G of A is anisotropic mod center, then, for any prime p such that G admits a reductive model over Zp, A has potentially good reduction at all finite places v|p of E. The first part of the talk will be expository: we will introduce Hodge cycles on abelian varieties and their properties, so that we can state the problem at hand. Then, we will switch track and talk about the question on toric embeddings referenced above. After this, we will focus on the case where the reduction of A at v is a split torus. In this case, the theory is more combinatorial, but the key ideas for the general case are already visible. We will quickly sketch the properties that we need of Shimura varieties and the Faltings-Chai compactification, and see how they can be used to reduce the problem to the one about toric embeddings. If time remains, we will say something about how to solve this latter problem using the Faltings-Kisin method. |

## Bei Zhang

Title: p-adic L-function of automorphic form of GL(2) |

Abstract: Modular symbol is used to construct p-adic L-functions associated to a modular form. In this talk, I will explain how to generalize this powerful tool to the construction of p-adic L-functions attached to an automorphic representation on GL_{2}(A) where A is the ring of adeles over a number field. This is a joint work with Matthew Emerton. |

## David Brown

Title: Explicit modular approaches to generalized Fermat equations |

Abstract: TBA |

## Tony Várilly-Alvarado

Title: TBA |

Abstract: TBA |

## Wei Ho

Title: TBA |

Abstract: TBA |

## Rob Rhoades

Title: TBA |

Abstract: TBA |

## TBA

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## Chris Davis

Title: TBA |

Abstract: TBA |

## Andrew Obus

Title: Cyclic Extensions and the Local Lifting Problem |

Abstract: The "local lifting problem" asks: given a G-Galois extension A/kt, where k is algebraically closed of characteristic p, does there exist a G-Galois extension A_R/Rt that reduces to A/kt, where R is a characteristic zero DVR with residue field k? (here a Galois extension is an extension of integrally closed rings that gives a Galois extension on fraction fields.) The Oort conjecture states that the local lifting problem should always have a solution for G cyclic. This is basic Kummer theory when p does not divide |G|, and has been proven when v_p(|G|) = 1 (Oort, Sekiguchi, Suwa) and when v_p(|G|) = 2 (Green, Matignon). We will first motivate the local lifting problem from geometry, and then we will show that it has a solution for a large family of cyclic extensions. This family includes all extensions where v_p(|G|) = 3 and many extensions where v_p(|G|) is arbitrarily high. This is joint work with Stefan Wewers. |

## Bianca Viray

Title: TBA |

Abstract: TBA |

## Frank Thorne

Title: TBA |

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## Rafe Jones

Title: Galois theory of iterated quadratic rational functions |

Abstract: I'll describe recent work investigating the arboreal Galoisrepresentation attached to a degree-2 rational function, focusing on the case where the function commuteswith a non-trivial Mobius transformation. In a sense this is a dynamical systems analogue to the p-adic Galois representation attached to an elliptic curve, with particular attention to the CM case. In joint work with Michelle Manes, we give criteria for the image of this representation to be large, and show thatthese criteria are often satisfied in the "CM" case. |

## Jonathan Blackhurst

Title: Polynomials of the Bifurcation Points of the Logistic Map |

Abstract: The logistic map f(r,x)=rx(1-x) was originally studied by ecologists modeling the population of a species from one generation to the next. Here, the population of the next generation f(r,x) depends on the population x of the current generation and a parameter r called the probiotic potential. To find the long-term behavior of the population for a fixed r, we look at iterates of the critical point c=1/2 under the map f. If 0<r<2 the iterates approach zero, so the population goes extinct. If 2<r<3, the iterates approach a single non-zero value, and the population is in equilibrium. If 3<r<1+sqrt(6), the iterates oscillate between two values (even iterates approach one value, while odd iterates approach another) in a boom-and-bust cycle. As r increases, the population begins oscillating between four values, then eight, then sixteen, and this period-doubling continues until an accumulation point (r=3.57, approximately) where the behavior of the population is no longer periodic. Even after the accumulation point, iterates of the critical point may begin again to exhibit periodic behavior. For example, at r=1+sqrt(8), they oscillate between three values. These values of r—2, 3, 1+sqrt(6), and 1+sqrt(8)—where qualitative behavior of the model changes are called bifurcation points. Little has been known about the polynomials these algebraic numbers satisfy. We find their degrees, show that their roots come in pairs whose mean is 1, put constraints on the size and prime factors of their constant coefficients, and record the number of real roots. |

## Liang Xiao

Title: TBA |

Abstract: There is an analogy among vector bundles with integrable connections, overconvergent F-isocrystals, and lisse l-adic sheaves. Given one of the objects, the property of being clean says that the ramification is controlled by the ramification along all generic points of the ramified divisors. In this case, one expects that the Euler characteristics may be expressed in terms of (subsidiary) Swan conductors; and (in first two cases) the log-characteristic cycles may be described in terms of refined Swan conductors. I will explain the proof of this in the vector bundle case and report on the recent progress on the overconvergent F-isocrystal case if time is permitted. |

## Winnie Li

Title: TBA |

Abstract: TBA |

## Organizer contact information

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