NTS Fall 2011/Abstracts

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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 GL2 over CM fields

Abstract: We generalize the work of Harris–Soudry–Taylor, and constructs the compatible system of 2-dimensional p-adic Galois representations associated to a cuspidal automorphic representation of cohomological type on GL2 over a CM field, whose central character satisfies an invariance condition. A local-global compatibility statement, up to semi-simplification, can also be proved in this setting. This work relies crucially on Arthur's results on lifting from the group GSp4 to GL4.

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 L-functions.

September 29

Nigel Boston (Madison)
Title: Non-abelian Cohen-Lenstra heuristics.

Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian p-group A (p odd) arises as the p-class group of an imaginary quadratic field K is apparently proportional to 1/|Aut(A)|. The Galois group of the maximal unramified p-extension of K has abelianization A and one might then ask how frequently a given p-group 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 G2. In this talk, I will show how to use geometric Langlands theory to give a uniform construction of motives with motivic Galois groups E7, E8 and G2, 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, OK its ring of integers. Consider n + 1 Laurent polynomials fi in n variables with OK coefficients. We assume that they have support in given polytopes Δi. On one hand, we can associate a toric scheme X over OK to these polytopes, and consider the arithmetic intersection number of (f0, ...,fn)X in X. On the other hand, we have the mixed resultant Res(f0, ...,fn). When the associated scheme is projective and smooth at the generic fiber and we assume f0, ...,fn 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 K-admissible if there exists a Galois G-extension L/K such that L is a maximal subfield of a division algebra with center K (i.e. if there exists a G-crossed product K-division 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 2-Selmer rank r for any non-negative integer r. The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of Heath-Brown, Swinnerton-Dyer, 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 well-understood, 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 so-called Stark-Heegner 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 2-planes 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 L-function.

November 17

Robert Harron (Madison)
Title: L-invariants of symmetric powers of modular forms

Abstract: A fruitful way to study the arithmetic significance of special values of L-functions is via their interpolation by p-adic L-functions. In this talk, we will discuss the phenomenon of L-invariants, which arise when the interpolation property provides no immediate information. Specifically, the value of the p-adic L-function may vanish even when the value of the original L-function does not. Beginning with the work of Mazur–Tate–Teitelbaum on a p-adic Birch–Swinnerton-Dyer conjecture, it has been conjectured that the value of the derivative of the p-adic L-function should relate to the originalL-value, up to the introduction of a new factor: the L-invariant. We will discuss some of the known cases of this phenomenon, as well as ongoing work on the study of L-invariants of symmetric powers of modular forms.

December 1

Andrei Caldararu (Madison)
Title: The Hodge theorem as a derived self intersection, part 2

Abstract: The Hodge theorem on the decomposition of de Rham cohomology into Hp,q-pieces was phrased by Deligne–Illusie as the splitting of a complex in the derived category, which is then proved using positive characteristic methods.

A similar splitting result was obtained by Arinkin and myself for a certain complex associated to a closed embedding. In my earlier talk in the algebraic geometry seminar I explained how to recast the Deligne–Illusie so that it can be seen as a particular case of the Arinkin–Caldararu result.

In my current talk I shall quickly review this story, and then provide some of the details that were skipped in my earlier talk. This is joint work with Dima Arinkin and Marton Hablicsek.

December 8

Xinwen Zhu (Harvard)
Title: Local models of Shimura varieties

Abstract: I will report some recent progress in the study of local models of Shimura varieties, including the proof of the coherence conjecture of Pappas–Rapoport and the Kottwitz conjecture.

December 15

Shamgar Gurevich (Madison)
Title: Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation

Abstract: I will show how to encode in the language of the finite Weil representation, two basic results of number theory - the quadratic reciprocity, and the sign of the Gauss sum. This will enables us to use tools from group representation theory to give new proofs for these two results.

I will assume knowledge of basic linear algebra.

Joint work with Ronny Hadani (Austin), and Roger Howe (Yale).

Organizer contact information

Shamgar Gurevich

Robert Harron

Zev Klagsbrun

Melanie Matchett Wood

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