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 Events
Alexander Molnar
SFU
Thu 3 Mar 2011, 3:00pm
Number Theory Seminar
Room ASB 10900 (IRMACS - SFU Campus)
Affine minimal rational functions
Room ASB 10900 (IRMACS - SFU Campus)
Thu 3 Mar 2011, 3:00pm-3:50pm

Abstract

Many arithmetic geometric results have an arithmetic dynamic analogue. For instance, Siegel's theorem that an elliptic curve has only finitely many integer points is analogous to the fact that any orbit under a rational function whose second iterate has non-constant denominator has only finitely many distinct integer values.
 
A conjecture of Lang states that the number of integer points on a minimal Weierstrass model of an elliptic curve is uniformly bounded. In order to translate this conjecture, one needs a dynamic concept of minimality. We present one such notion, affine minimality, an algorithm to compute affine minimal forms of rational functions and some recent results pertaining to the dynamical analogue of Lang's conjecture.

Note for Attendees

Refreshments will be served between the two talks.
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UBC
Thu 3 Mar 2011, 4:10pm
Number Theory Seminar
Room ASB 10900 (IRMACS - SFU Campus)
Friable values of polynomials
Room ASB 10900 (IRMACS - SFU Campus)
Thu 3 Mar 2011, 4:10pm-5:00pm

Abstract

We summarize the current meager state of knowledge concerning how often values of polynomials have only small prime factors (that is, the values are "friable" or "smooth"). We also present some evidence, in the form of a theorem conditional upon a suitably explicit hypothesis on prime values of polynomials, to support a conjectured asymptotic formula for the number of friable values of any polynomial.
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UBC
Thu 17 Mar 2011, 3:00pm
Number Theory Seminar
Room WMAX 216 (PIMS - UBC Campus)
Fourier coefficients and Siegel theta series
Room WMAX 216 (PIMS - UBC Campus)
Thu 17 Mar 2011, 3:00pm-3:50pm

Abstract

Thanks to the work of Howe, Freitag, J.-S. Li, and most others, we can identify the holomorphic Siegel modular forms of degree n that arise as theta liftings associated with quadratic forms in at most n variables in terms of their Fourier coefficients. We shall review this work (using the language of theta correspondence) and discuss some ideas for studying the case when the number of variables in the quadratic form is greater than n.

Note for Attendees

Refreshments will be served between the two talks.
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UBC
Thu 17 Mar 2011, 4:10pm
Number Theory Seminar
Room WMAX 216 (PIMS - UBC Campus)
Constructible exponential functions, integrability, and characters of p-adic groups
Room WMAX 216 (PIMS - UBC Campus)
Thu 17 Mar 2011, 4:10pm-5:00pm

Abstract

I will talk about a class of functions on local fields that can be defined by means of logic (the so-called constructible exponential functions). These functions are in a sense built from charactersitic functions of balls, and additive characters of the field. It turns out that they have unexpectedly simple "integration theory". It also turns out that Harish-Chandra characters of representations of p-adic groups near the identity element belong to this class of functions. This allows to transfer Harish-Chandra's theorem about local integrability of characters that was known in general only for local fields of characteristic zero, to large positive characteristic. All of the notions mentioned in this abstract will be defined in the talk.
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Guillaume Maurin
UBC
Thu 31 Mar 2011, 3:00pm
Number Theory Seminar
Room ASB 10900 (IRMACS - SFU Campus)
Polynomial equations with constant coefficients over function fields
Room ASB 10900 (IRMACS - SFU Campus)
Thu 31 Mar 2011, 3:00pm-3:50pm

Abstract

We will present new conjectures on polynomial equations with constant coefficients over a function field of arbitrary characteristic (joint work with Ghioca). These statements are inspired by previous conjectures from Zilber, Pink and Bombieri, Masser and Zannier. We will try to explain how known results on the latter may give some information on the former in the case of characteristic zero.

Note for Attendees

Refreshments will be served between the two talks.
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UBC
Thu 31 Mar 2011, 4:10pm
Number Theory Seminar
Room ASB 10900 (IRMACS - SFU Campus)
On additive combinatorics in higher degree systems
Room ASB 10900 (IRMACS - SFU Campus)
Thu 31 Mar 2011, 4:10pm-5:00pm

Abstract

We consider a system of k diagonal polynomials of degrees 1, 2,..., k. Using methods developed by W.T. Gowers and refined by Green and Tao to obtain bounds in the 4-term case of Szemeredi's Theorem on long arithmetic progressions, we show that if a subset A of the natural numbers up to N of size d_N*N exhibits sufficiently small local polynomial bias, then it furnishes roughly the expected number of solutions to the given system.  If A furnishes no non-trivial solutions to the system, then we show via an energy incrementing argument that there is a concentration in a Bohr set of pure degree k, and consequently in a long arithmetic progression.  We show that this leads to a bound on the density d_N of the set A of the form d_N << exp(-c*sqrt(log log N)), where c>0 is a constant dependent at most on k.
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