NTS Fall 2011/Abstracts
Contents
September 8
Alexander Fish (Madison) 
Title: Solvability of Diophantine equations within dynamically defined subsets of N 
Abstract: Given a dynamical system, i.e. a compact metric space X, a homeomorphism T (or just a continuous map) and a Borel probability measure on X which is preserved under the action of T, the dynamically defined subset associated to a point x in X and an open set U in X is {n  T^{ n}(x) is in U} which we call the set of return times of x in U. We study combinatorial properties of sets of return times for certain types of dynamical systems for generic points x in X. Among examples of such sets are normal sets which correspond to the system X = [0,1], T(x) = 2x mod 1, Lebesgue measure, U = [0, 1/2]. We give a complete classification of linear Diophantine systems solvable within every normal set. The methods combine the probabilistic method together with the use of van der Corput's lemma. At the end of the talk we will discuss open problems 
September 15
Chung Pang Mok (McMaster) 
Title: Galois representation associated to cusp forms on GL_{2} over CM fields 
Abstract: We generalize the work of Harris–Soudry–Taylor, and constructs the compatible system of 2dimensional padic Galois representations associated to a cuspidal automorphic representation of cohomological type on GL_{2} over a CM field, whose central character satisfies an invariance condition. A localglobal compatibility statement, up to semisimplification, can also be proved in this setting. This work relies crucially on Arthur's results on lifting from the group GSp_{4} to GL_{4}. 
September 22
Yifeng Liu (Columbia) 
Title: Arithmetic inner product formula 
Abstract: I will introduce an arithmetic version of the classical Rallis' inner product for unitary groups, which generalizes the previous work by Kudla, Kudla–Rapoport–Yang and Bruinier–Yang. The arithmetic inner product formula, which is still a conjecture for higher rank, relates the canonical height of special cycles on certain Shimura varieties and the central derivatives of Lfunctions. 
September 29
Nigel Boston (Madison) 
Title: Nonabelian CohenLenstra heuristics. 
Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian pgroup A (p odd) arises as the pclass group of an imaginary quadratic field K is apparently proportional to 1/Aut(A). The Galois group of the maximal unramified pextension of K has abelianization A and one might then ask how frequently a given pgroup G arises. We develop a theory wherein this frequency is inversely proportional to the size of its automorphism group in a new category and then test this against computations. If time permits, I shall describe progress on the real quadratic case. This is joint work with Michael Bush and Farshid Hajir. 
October 6
Zhiwei Yun (MIT) 
Title: Exceptional Lie groups as motivic Galois groups 
Abstract: More than two decades ago, Serre asked the following question: can exceptional Lie groups be realized as the motivic Galois group of some motive over a number field? The question has been open for exceptional groups other than G_{2}. In this talk, I will show how to use geometric Langlands theory to give a uniform construction of motives with motivic Galois groups E_{7}, E_{8} and G_{2}, hence giving an affirmative answer to Serre's question in these cases. 
October 13
Melanie Matchett Wood (Madison) 
Title: The probability that a curve over a finite field is smooth 
Abstract: Given a fixed variety over a finite field, we ask what proportion of hypersurfaces (effective divisors) are smooth. Poonen's work on Bertini theorems over finite fields answers this question when one considers effective divisors linearly equivalent to a multiple of a fixed ample divisor, which corresponds to choosing an ample ray through the origin in the Picard group of the variety. In this case the probability of smoothness is predicted by a simple heuristic assuming smoothness is independent at different points in the ambient space. In joint work with Erman, we consider this question for effective divisors along nef rays in certain surfaces. Here the simple heuristic of independence fails, but the answer can still be determined and follows from a richer heuristic that predicts at which points smoothness is independent and at which points it is dependent. 
October 20
Jie Ling (Madison) 
Title: Arithmetic intersection on Toric schemes and resultants 
Abstract: Let K be a number field, O_{K} its ring of integers. Consider n + 1 Laurent polynomials f_{i} in n variables with O_{K} coefficients. We assume that they have support in given polytopes Δ_{i}. On one hand, we can associate a toric scheme X over O_{K} to these polytopes, and consider the arithmetic intersection number of (f_{0}, ...,f_{n})_{X} in X. On the other hand, we have the mixed resultant Res(f_{0}, ...,f_{n}). When the associated scheme is projective and smooth at the generic fiber and we assume f_{0}, ...,f_{n} intersect properly, the arithmetic intersection number is given by the norm of the mixed resultant. This could be thought of as an arithmetic analog of Beinstein's theorem. 
October 27
Danny Neftin (Michigan) 
Title: Arithmetic field relations and crossed product division algebras 
Abstract: A finite group G is called Kadmissible if there exists a Galois Gextension L/K such that L is a maximal subfield of a division algebra with center K (i.e. if there exists a Gcrossed product Kdivision algebra). Motivated by the works of Neukirch, Uchida, and Iwasawa, Sonn defined an arithmetic equivalence of number fields by admissibility and posed (1985) the following open problem: Do two number fields with the same admissible groups necessarily have the same degree over Q? Up until recently only positive results were known. We shall discuss equivalence by admissibility and construct two number fields that have the same odd order admissible groups, providing evidence to a negative answer to Sonn's problem. 
November 3
Zev Klagsburn (Madison) 
Title: Selmer ranks of quadratic twists of elliptic curves 
Abstract: Given an elliptic curve E defined over a number field K, we can ask what proportion of quadratic twists of E have 2Selmer rank r for any nonnegative integer r. The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of HeathBrown, SwinnertonDyer, and Kane for elliptic curves over the rationals with E(Q)[2] = Z/2 × Z/2. We present new results for elliptic curves with E(K)[2] = 0 and with E(K)[2] = Z/2. I will present joint work of Mazur, Rubin, and myself supporting the conjecture for curves with E(K)[2] = 0. Additionally, I will present some new results of my own for curves with E(K)[2] = Z/2, including some surprising results that conflict with the conjecture.

November 8
Christelle Vincent (Madison) 
Title: On the construction of rational points on elliptic curves 
Abstract: If E is an elliptic curve defined over a number field K, Mordell's Theorem asserts that the group E(K) of points of E defined over the field K is abelian and finitely generated. While the torsion part of this group is considered to be wellunderstood, the rank of the infinite part is very difficult to compute in general. In an effort to understand this quantity better, Darmon has proposed a conjectural construction of socalled StarkHeegner points. We will begin by presenting a prototypical example of such a construction found in the work of Gross and Zagier, which will be accessible to undergraduate students. Building on this framework, we will explain how this can be generalized to construct points on a larger class of elliptic curves. 
November 10
Luanlei Zhao (Madison) 
Title: Integral of Borcherds forms of orthogonal type 
Abstract: Following S. Kudla, J. Bruinier and T. Yang, we compute the integral of an automorphic Green functions coming from vector valued harmonic Maass forms for the dual pair (O(n); Sp(1)) over the negative 2planes with signature (r, 2) for 0 < r < n, which is of interest in Arakelov geometry. We connect this integral of Borcherds form with the derivative of a Rankin–Selberg Lfunction. 
November 17
Robert Harron (Madison) 
Title: Linvariants of symmetric powers of modular forms 
Abstract: tba 
December 1
Andrei Calderaru (Madison) 
Title: tba 
Abstract: tba 
December 8
Xinwen Zhu (Harvard) 
Title: tba 
Abstract: tba 
December 15
Shamgar Gurevich (Madison) 
Title: Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation 
Abstract: tba 
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