Forthcoming Seminars › Seminar Series

Number Theory Seminar

Thu, 12/05/2011
16:00 		Daniel Bertrand (Paris) Number Theory Seminar Add to calendar L3
Ribet points on semi-abelian varieties : a nest for counterexamples
The points in question can be found on any semi-abelian surface over an elliptic curve with complex multiplication. We will show that they provide counter-examples to natural expectations in a variety of fields : Galois representations (following K. Ribet's initial study from the 80's), Lehmer's problem on heights, and more recently, the relative analogue of the Manin-Mumford conjecture. However, they do support Pink's general conjecture on special subvarieties of mixed Shimura varieties.
Thu, 12/05/2011
16:00 		Daniel Bertrand (Paris) Logic Seminar Add to calendar
Number Theory Seminar Add to calendar 		L3
" Ribet points on semi-abelian varieties : a nest for counterexamples"

The points in question can be found on  any semi-abelian surface over an elliptic curve with complex multiplication. We will show that they provide counter-examples to natural expectations in a variety of fields :  Galois representations (following K. Ribet's initial study from the 80's), Lehmer's problem on heights, and more recently, the relative  analogue of the Manin-Mumford conjecture. However, they do support Pink's general conjecture on special subvarieties of mixed Shimura varieties.

 
Thu, 26/05/2011
16:00 		David Loeffler (Warwick) Number Theory Seminar Add to calendar L3
Iwasawa theory for modular forms

he Iwasawa theory of elliptic curves over the rationals, and more
generally of modular forms, has mostly been studied with the
assumption that the form is "ordinary" at p -- i.e. that the Hecke
eigenvalue is a p-adic unit. When this is the case, the dual of the
p-Selmer group over the cyclotomic tower is a torsion module over the
Iwasawa algebra, and it is known in most cases (by work of Kato and
Skinner-Urban) that the characteristic ideal of this module is
generated by the p-adic L-function of the modular form.

I'll talk about the supersingular (good non-ordinary) case, where
things are slightly more complicated: the dual Selmer group has
positive rank, so its characteristic ideal is zero; and the p-adic
L-function is unbounded and hence doesn't lie in the Iwasawa algebra.
Under the rather restrictive hypothesis that the Hecke eigenvalue is
actually zero, Kobayashi, Pollack and Lei have shown how to decompose
the L-function as a linear combination of Iwasawa functions and
explicit "logarithm-like" series, and to modify the definition of the
Selmer group correspondingly, in order to formulate a main conjecture
(and prove one inclusion). I will describe joint work with Antonio Lei
and Sarah Zerbes where we extend this to general supersingular modular
forms, using methods from p-adic Hodge theory. Our work also gives
rise to new phenomena in the ordinary case: a somewhat mysterious
second Selmer group and L-function, which is related to the
"critical-slope L-function" studied by Pollack-Stevens and Bellaiche.

Thu, 02/06/2011
16:00 		Marco Streng (Warwick) Number Theory Seminar Add to calendar
Class invariants for quartic CM-fields
I show how invariants of curves of genus 2 can be used for explicitly constructing class fields of certain number fields of degree 4.
Thu, 09/06/2011
16:00 		David Masser (Basel) Number Theory Seminar Add to calendar L3
To Be Announced
Thu, 09/06/2011
16:00 		David Masser Logic Seminar Add to calendar
Number Theory Seminar Add to calendar 		L3
Unlikely intersections for algebraic curves.
In the last twelve years there has been much study of what happens when an algebraic curve in $ n $-space is intersected with two multiplicative relations $ x_1^{a_1} \cdots x_n^{a_n}~=~x_1^{b_1} \cdots x_n^{b_n}~=~1 \eqno(\times) $ for $ (a_1, \ldots ,a_n),(b_1,\ldots, b_n) $ linearly independent in $ {\bf Z}^n $. Usually the intersection with the union of all $ (\times) $ is at most finite, at least in zero characteristic. In Oxford nearly three years ago I could treat a special curve in positive characteristic. Since then there have been a number of advances, even for additive relations $ \alpha_1x_1+\cdots+\alpha_nx_n~=~\beta_1x_1+\cdots+\beta_nx_n~=~0 \eqno(+) $ provided some extra structure of Drinfeld type is supplied. After reviewing the zero characteristic situation, I will describe recent work, some with Dale Brownawell, for $ (\times) $ and for $ (+) $ with Frobenius Modules and Carlitz Modules.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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