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Abstracts for Number theory lunch seminar

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Quaternionic Darmon points and arithmetic applications

Posted in
Speaker: 
Matteo Longo
Date: 
Wed, 2011-09-28 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

In 2001 H. Darmon propossed a systematic construction of local
points on rational elliptic curves using p-adic integration combined
with the theory of modular symbols. Conjecturally, these points should
actually be defined over abelian extensions of real quadratic fields and
their behaviour in the Mordell-Weil group is governed by L-functions, as
much as Heegner points in the case of real imaginary fields. The aim of
the talk is to present a Darmon-style construction in the context of
Shimura curves. I will also give some arithmetic applications of this

CM Values of Higher Green's Functions

Posted in
Speaker: 
Maryna Viazovska
Affiliation: 
MPI
Date: 
Wed, 2011-10-05 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Higher Green functions are real-valued functions of two variables on
the upper half plane which are bi-invariant under the action of a congruence
subgroup, have logarithmic singularity along the diagonal, and satisfy the equation
Delta f =k(1 - k)f, where Delta is a hyperbolic Laplace operator and k is a
positive integer. Such functions were introduced in the paper of
Gross and Zagier  "Heegner points and derivatives of $L-series"(1986).
Also it was conjectured in this paper that higher Green's  functions have

Modular-type functions attached to mirror quintic Calabi-Yau threefolds

Posted in
Speaker: 
Hossein Movasati
Date: 
Wed, 2011-10-12 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

In this talk we introduce a differential algebra  of modular-type
functions attached to the periods of a one parameter family
of Calabi-Yau varieties which is mirror dual to the universal family of
quintic threefolds. Such an algebra is generated by seven functions
satisfying functional and differential equations in parallel to  the
modular functional equations of classical Eisenstein series and the
Ramanujan differential equation. Our result is the first
example of automorphic-type functions attached to varieties whose period

Multiplicities of discriminants

Posted in
Speaker: 
Dennis Eriksson
Date: 
Wed, 2011-10-19 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

The discriminant of a homogenous polynomial F is another homogenous
polynomial  in the coefficients of the polynomial, which is zero if and
only if the  corresponding hypersurface F = 0 is singular. In case the
coefficients are in a  discrete valuation ring, the order of the
discriminant (if non-zero) measures  the bad reduction. We give some new
results on this order, and in particular tie  it to Bloch's conjecture/the
Kato-T.Saito formula on equality of localized Chern  classes and Artin

Measure theoretic rigidity for Mumford curves

Posted in
Speaker: 
Janne Kool
Affiliation: 
Universiteit Utrecht
Date: 
Wed, 2011-10-26 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Measure theoretic rigidity (in the style of Mostow) states
that two compact hyperbolic Riemann surfaces of the same genus are
isomorphic if and only if the "boundary map" associated to their
uniformizations is absolutely continuous. In this talk, we will
formulate the analog of this result for "p-adic Riemann surfaces",
i.e., for Mumford curves. In this case, the mere absolute continuity
of the boundary map implies only isomorphism of the special fibers of
the Mumford curves, and needs to be enhanced by a finite list of

On fields of algebraic numbers with bounded local degrees

Posted in
Speaker: 
Sara Checcoli
Affiliation: 
Basel
Date: 
Wed, 2011-11-02 11:15 - 12:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

We provide a characterization of in finite algebraic Galois extensions of
the rationals with uniformly bounded local degrees.

In particular we show that for an infi nite Galois extension of the
rationals the following three properties are equivalent: having uniformly
bounded local degrees at every prime; having uniformly bounded local
degrees at almost every prime; having Galois group of finite exponent.
The proof of this result enlightens interesting connections with
Zelmanov's work on the Restricted Burnside Problem. We give a formula to

Cyclic Extensions and the Local Lifting Problem

Posted in
Speaker: 
Andrew Obus
Affiliation: 
MPI
Date: 
Wed, 2011-11-09 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

The lifting problem we will consider roughly asks: given a
smooth, proper, geometrically connected curve X in characteristic p
with an action of a finite group G, does there exist a smooth, proper
curve X' with G-action in characteristic zero such that X' (with
G-action) lifts X (with G-action)?  It turns out that solving this
lifting problem reduces to solving a local lifting problem in a formal
neighborhood of each point of X where G acts with non-trivial inertia.
  The Oort conjecture states that this local lifting problem should be

Stability of gamma factors, I

Posted in
Speaker: 
Tung-Lin Tsai
Affiliation: 
MPI
Date: 
Wed, 2011-11-16 14:15 - 15:10
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Gamma-factors are crucial in defining root numbers satisfying 
functional equations. Stability of gamma factors plays an important 
role in proving Langlands principle of functoriality in certain cases, 
and it provides a proof of the equality of root numbers defined from 
Langlands-Shahidi method with others including those of Artin. In the 
first part of the talk, I will briefly explain why stability of gamma 
factors leads to Langlands principle of functoriality. I will also 

Stability of gamma factors, II

Posted in
Speaker: 
Tung-Lin Tsai
Affiliation: 
MPI
Date: 
Wed, 2011-11-16 15:20 - 16:00
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Variations on Artin's primitive root conjecture

Posted in
Speaker: 
Adam Felix
Affiliation: 
Queen's University Kinsgston/MPI
Date: 
Wed, 2011-11-23 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Fix an integer b.  In 1927, Artin made a conjecture about the density of primes p for
which b generates the group of residues modulo p.  We will discuss this conjecture
and variations of it.
 

 

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