Oihana Garaialde (University of the Basque Country) Cohomology of p-groups of maximal class abstract
14 September 2015
no seminar
21 September 2015 2 talks
14:30-15:30
René Schoof (Università di Roma “Tor Vergata”) The discrete logarithm problem
16:00-17:00
Alberto Bellardini (Leuven) Picard functor and logarithmic geometry abstract
5 October 2015
16:00-17:00
Daniel Tubbenhauer (Bonn) Web calculi in representation theory abstract
19 October 2015
no seminar
26 October 2015
16:00-17:00
Hang Xue (MPIM Bonn) Gan--Gross--Prasad conjecture for unitary groups
We explain a classical formula of Waldspurger and its generalization to
higher rank unitary groups. We will also discuss the relative trace formula
approach of Jacquet--Rallis.
2 November 2015
no seminar
9 November 2015
no seminar
16 November 2015
16:00-17:00
Sho Tanimoto (Copenhagen) Towards a refinement of Manin's conjecture
We explain our attempt to formulate a refinement of Manin's conjecture using
birational geometry, e.g., the minimal model program and the boundedness of log
Fano varieties. This is joint work with Brian Lehmann and Yuri Tschinkel.
23 November 2015
16:00-17:00
Léo Ducas
(CWI Amsterdam) Recovering Short Generators of Principal Ideals in Cyclotomic Rings
Joint work with Ronald Cramer, Chris Peikert and Oded Regev
A handful of recent cryptographic proposals rely on the conjectured hardness of
the following problem in the ring of integers of a cyclotomic number field:
given a basis of an ideal that is guaranteed to have a ``rather short''
generator, find such a generator. Recently, Bernstein and
Campbell-Groves-Shepherd have sketched potential attacks against this problem.
Most notably, the latter authors claimed a *quantum polynomial-time* algorithm.
(Alternatively, replacing the quantum component with an algorithm of Biasse and
Fieker would yield a classical subexponential-time algorithm.) A key claim of
Campbell etal. is that one step of their algorithm---namely, decoding the
*log-unit* lattice of the ring to recover a short generator from an arbitrary
one---is efficient (whereas the standard approach on general lattices takes
exponential time). However, very few convincing details were provided to
substantiate this claim.
In this work, we clarify the situation by giving a rigorous proof that the
log-unit lattice is indeed efficiently decodable, in cyclotomics of
prime-power index. The proof consists of two main technical contributions: the
first is a geometrical analysis, using tools from analytic number theory, of the
standard generators of the group of cyclotomic units. The second shows that for
a wide class of typical distributions of the short generator, a standard
lattice-decoding algorithm can recover it, given any generator.
30 November 2015
16:00-17:00
Roland van der Veen (Leiden) Algebraic curves from knots
To any knot one can assign an algebraic curve together with a distinguished
point. Roughly speaking it parametrizes the representations of the fundamental
group into SL(2). I will explain how to find the curves in two different ways
and how to extract interesting numbers from it. For example the figure eight
knot gives rise to 15A8 in the Cremona table.
7 December 2015
16:00-17:00
TBA TBA
Spring 2015
9 February 2015
16:00-17:00
Arno Kret The Arthur-Selberg trace formula and the geometry of Shimura varieties modulo p
We show how the Arthur-Selberg trace formula can be used to study the Newton stratification of
Shimura varieties. In the first half of the talk we will recall the definition of the Newton
stratification, and give some examples. In the second half of the talk we explain the (idea of)
the Kottwitz point counting formula, and how it can be compared with the Arthur-Selberg trace
formula. This comparison can then be used to answer geometric questions, such as non-emptiness
and dimensions of Newton strata.
16 February 2015
16:00-17:00
David Holmes (Leiden) Families of elliptic curves over bases of various dimensions
Elliptic curves over fields have been studied in various guises for thousands of
years. Around 1965, André Néron studied how 1-parameter families of elliptic
curves degenerate, and constructed canonical (or Néron) models for them. These
models have many wonderful properties, some of which we will outline in the
first part of the talk.
We will then investigate what happens in families with more parameters (more
formally, over base-schemes of higher dimensions). In this situation things
become considerably more complicated. We will give a reasonably complete
classification of when Néron models do and do not exist, and describe various
applications.
5 March 2015 Thursday, room 407
10:00 - 10:45
Yuri Bilu (Bordeaux) Special points on straight lines and hyperbolas
11:00 - 11:45
Pascal Autissier (Bordeaux) Abelian varieties and the Minkowski-Hlawka theorem
13:45
Academiegebouw, Rapenburg Leiden, PhD defense of Junjiang Liu p-adic decomposable form inequalities
9 March 2015
16:00-17:00
José Ignacio Burgos Gil (Madrid) Equidistribution of small points on toric varieties
As the culmination of work of many mathematicians, Yuan has obtained a very
general equidistribution result for small points in arithmetic varieties.
Roughly speaking Yuan's theorem states that given a « very » small generic
sequence of points, with respect to a positive hermitian line bundle, the
associated sequence of measures converges weakly to the measure associated
to the hermitian line bundle. Here very small means that the height of the
points converges to the lower bound of the essential minimum given by Zhang
inequalities. The existence of a very small generic sequence is a strong
condition on the arithmetic variety because it implies that the essential
minimum attains its lower bound. We will say that a sequence is small if the
height of the points converges to the essential minimum. By definition every
arithmetic variety contains small generic sequences.
We show that for toric line bundles on toric varieties arithmetic Yuan's
theorem can be splitted in two parts.
A) Given a small generic sequence of points, with respect to a positive
hermitian line bundle, the associated sequence of measures converges weakly
to a measure.
B) If the sequence is very small, the limit measure agrees with the measure
associated to the hermitian line bundle.
23 March 2015
16:00-17:00
Christopher Daw (IHES) Heights of pre-special points of Shimura varieties
In this talk we discuss a bound for the height of the pre-image of a special
point on a Shimura variety in a fundamental set of the associated Hermitian
symmetric domain. This bound is polynomial in terms of standard invariants
associated with the corresponding Mumford-Tate torus and generalises an earlier
result of Pila and Tsimerman for the moduli space of principally polarised
abelian varieties of dimension g. The result constitutes the final ingredient
needed to complete a new proof of the André-Oort conjecture under the
Generalised Riemann Hypothesis, using a strategy of Pila and Zannier. This is
joint work with Martin Orr (University College London).
30 March 2015
16:00-17:00
Julian Lyczak (Leiden) Nodal curves on surfaces, an application of algebraic cobordism
Severi published in 1921 a result on nodal curves on the projective plane. He
showed that the set of nodal curves of degree d with exactly k nodes on the
projective plane has an obvious embedding in a projective space. He also stated
that the projective scheme obtained by taking the Zariski closure is non-empty
and of codimension k. His proofs of the first and third statement were correct,
but it was not until 1985 that a correct proof for the irreducibility was found.
The result on the dimension shows that given enough points on a surface, the
number of nodal curves through these points does not depend on the choice of
points. These number, depending on d and k, were computed recursively by
Caporaso and Harris in 1998. Eventually it were Fomin and Mikhalkin in 2009 who
showed that these Severi degrees are polynomials in d, for k small in comparison
to d.
Even before this proof, mathematicians have been trying to generalize these
results and conjectures to arbitrary smooth projective surfaces. One will need
to replace degree d curves by a line bundle and in this way one can still define
the Severi variety and degree. Göttsche conjectured in 1997 that under an
assumption on this line bundle, generalizing ampleness, the Severi degree should
be a polynomial in topological invariants of the surface and the line bundle.
Tzeng was able to prove this theorem by using algebraic cobordism: an
equivalence relation on classes of surfaces with a line bundle. The crux of the
proof lies in the fact that any equivalence class of algebraic cobordism is
uniquely determined by the topological invariants, occurring in Göttsche's
conjecture.
9 April 2015 Thursday, room B3
different day, time and place
A popular subject in Diophantine geometry is the existence of rational
points in varieties, a subject which includes topics such as the Hasse
principle and the Brauer-Manin obstruction. On the opposite end of the
spectrum there are varieties whose rational points are Zariski dense and
where one is interested in more refined properties of their distribution.
Manin's conjecture attempts to describe the distribution of points in
specific subvarieties via the use of heights. It has never been established
for a single smooth cubic surface although it is known for particular del
Pezzo surfaces of any degree d>3. In this talk we will prove the one sided
estimate predicted by Manin's conjecture for a family of smooth cubic
surfaces containing a rational line.
13 April 2015
16:00-17:00
Barinder Banwait (Essen) Tetrahedral Elliptic Curves and the local-global principle for isogenies
If an elliptic curve over a number field admits a rational p-isogeny over every
completion, does it necessarily admit one over the number field itself? As
Sutherland has shown, the answer is 'No' in general. I will discuss joint work
with John Cremona going further into this question, giving some interesting
counterexamples. There'll also be an (ultimately failed though nevertheless
worthwhile) attempt, with Alex Bartel, at proving isomorphisms between Jacobians
of modular curves of genus 3 and level 13.
20 April 2015
16:00-17:00
Arne Smeets On a monodromy trace formula for varieties over discretely valued fields
Let R be a local ring which is complete for a discrete valuation, with fraction
field K and algebraically closed residue field k. Let X be a smooth, proper
K-variety. Nicaise conjectured that the rational volume of X is equal to the
trace of the tame monodromy operator on the l-adic cohomology of X, if X
satisfies a cohomological tameness condition. We prove his conjecture for a
large class of varieties satisfying this condition: those with logarithmic good
reduction.
29 April 2015 Wednesday
17:00-18:00
Gabriel Zalamansky Ramification theory for inseparable coverings abstract
30 April 2015 Thursday
14:45-15:45 room 401
Jeanine van Order
(MPIM Bonn) Dirichlet twists of GL(n)-automorphic L-functions
Let π be a cuspidal automorphic representation of GL(n) over a number field F.
Let p be a rational prime, and let X be the set of all finite-order Hecke
characters of F obtained by composition with the norm homomorphism from F to ℚ
with some Dirichlet character of p-power conductor. I will explain why one
expects (and sometimes knows) that if π is cohomological, then for all but
finitely many characters ξ in the set X, the central value L(1/2,π×ξ) does not
vanish. I will also present some preliminary results, as well as explain how the
general conjecture can be reduced to certain estimates with hyper-Kloosterman
sums of p-power modulus.
4 May 2015
16:00-17:00
Daniele Turchetti Lifting Galois covers of curves: a non-Archimedean analytic approach
The problem of lifting Galois covers of curves to characteristic zero asks
roughly which Galois covers of smooth projective curves defined over a field of
positive characteristic arise as reduction of Galois covers of the same group
in characteristic zero. This question has been studied in several fashions
since the '60s and is still a source of open problems and ongoing research.
The last big breakthrough was the proof of the so-called "Oort conjecture",
achieved in 2012 by Pop, and built on the work of Obus-Wewers. The aim of this
talk is to explain the techniques used in proving main results on the topic, to
discuss new open problems and to introduce a new viewpoint on the subject,
related to Berkovich theory of non-Archimedean analytic spaces.
11 May 2015
16:00-17:00
Manolis Tzortzakis Dessins d' enfants
In 1979, G.Belyi proved his theorem characterizing, analytically, all compact
Riemann surfaces which can be defined over a number field. Using that as a
springboard, A.Grothendieck defined dessins d'enfants (children's drawings) as
certain graphs embedded on Riemann surfaces which essentially are a
combinatorial description of Belyi's characterization. As such they admit an
action of the absolute Galois group ie they are an instance of the so called
geometric Galois actions. Dessins were used to study Riemann surfaces and to
provide combinatorial invariants of the Galois action. In this talk we will
outline Belyi's thoerem, introduce dessins and some of their properties along
with some connections with different areas.
26 May 2015 Tuesday 2 talks, early start
14:45-15:45 room 312
Michiel Kosters The last fall degree and an application to HFE
We first briefly discuss the standard method for solving
polynomial systems, using Groebner bases. Then we will discuss the
first fall degree conjecture, which predicts the complexity of such
computations. We present some evidence in favor and some evidence
against this conjecture. We will propose a new concept, the last fall
degree, which can be used to give complexity of Groebner basis
computations as well, without any heuristics. Finally, we discuss how
one can prove that HFE (hidden field equations) is insecure by
bounding the complexity of Groebner basis algorithms using the last
fall degree.
16:00-17:00 room 312
Everett Howe The number of genus-2 curves with nonsimple Jacobians
Up to isomorphism, there are approximately $2 q^3$ curves of genus 2 over the
finite field ${\bf F}_q$. How many of these curves have nonsimple Jacobian
varieties? Jeff Achter (Colorado State University) and I have recently shown
that up to explicit powers of $\log q$, the number of genus-2 curves with
nonsimple Jacobians is $q^{2.5}$, so that roughly speaking there is one chance
out of $\sqrt q$ that a randomly-chosen curve has nonsimple Jacobian. I will
outline part of the proof of this result, and say something about the problem
that motivated Achter to ask about this question.
Lars Kuehne (MPIM Bonn) Effective and uniform results of André-Oort type
The André-Oort Conjecture (AOC) states that the irreducible components of the
Zariski closure of a set of special points in a Shimura variety are special
subvarieties. Here, a special variety means an irreducible component of the
image of a sub-Shimura variety by a Hecke correspondence. The AOC is an analogue
of the classical Manin-Mumford conjecture on the distribution of torsion points
in abelian varieties. In fact, both conjectures are special instances of the
far-reaching Zilber-Pink conjecture(s).
I will present a rarely known approach to the AOC that goes back to Yves André
himself: Before the recent model-theoretic proofs of the AOC in certain cases by
the Pila-Wilkie-Zannier approach, André presented in 1998 the first
unconditional (i.e., not depending on GRH) proof of the AOC in a non-trivial
case, namely, a product of two modular curves. In my talk, I discuss several
results in the style of André's method, allowing to compute all special points
on a non-special curve in a product of two modular curves.
These results are effective - in stark contrast to those obtainable by the
Pila-Wilkie-Zannier approach - and have sometimes the additional advantage of
being uniform both in the degrees of the curve and its definition field. For
example, this allows to show that there are actually no two singular moduli
(= j-invariants associated with complex CM-elliptic curves) x and y satisfying
x+y=1.
15 June 2015 2 talks
15:15-16:00
Frauke Bleher (Iowa) Ordering modules using orbit closures
It is a fundamental problem in representation theory to classify the
finitely generated modules for a given finite dimensional algebra
over a field. One approach to this problem is to give conditions for
one module to be "simpler" than another in some sense. For example,
because of the Jordan-Hölder Theorem, one can take the
semi-simplification of a module. In this talk I will introduce several
partial orders that can be put on the isomorphism classes of finitely
generated modules. I will then concentrate on one particular partial
order coming from taking closures of orbits under the action of an
algebraic group. I will discuss two different settings in which this occurs,
one affine and the other projective. At the end I will raise some questions
and give some answers in a low-dimensional setting. This is joint work
with T. Chinburg and B. Huisgen-Zimmermann.
16:15-17:00
Ted Chinburg (Pennsylvania) Chern classes in Iwasawa theory
The classical case of Iwasawa theory has to do
with how quickly the p-parts of the ideal class groups of
number fields grow in certain towers of number fields.
I will discuss the connection of these growth rates to Chern
classes. By the end of the talk I'll describe some joint work
over imaginary quadratic fields with F. Bleher, R. Greenberg, M. Kakde,
G. Pappas, R Sharifi and M. Taylor.
16 June 2015 Tuesday PhD defence + 3 talks
10:00-11:00 Academiegebouw Rapenburg 73
PhD defence of Santosh Nadimpalli
13:30-14:30 Snellius 412
Santosh Nadimpalli (Orsay and Leiden) Classification
of typical representations for GL_4(F)
Let F be a non-discreet non-Archimedean local field. In this talk we
will be concerned with irreducible smooth representations of GL_4(O_F)
which determine the inertial support of irreducible smooth
representations of GL_4(F). There representations are called typical
representations. We will obtain a complete classification of typical
representations in terms of Bushnell-Kutzko theory of types for the
case where the cardinality of the residue field is greater than 3. The
case of GL_4(F) will serve as an example for showing techniques used
for the general case.
15:00-16:00 Snellius 412
Shaun Stevens (UEA, Norwich) On modular
representations of p-adic classical groups
I will discuss some work in progress, with Rob Kurinczuk, attempting
to understand the category of smooth l-modular representations of
p-adic classical groups, in particular towards a classification of
cuspidal/supercuspidal representations, the notion of "endo-class" and
implications for decompositions of the category.
16:15-17:15 Snellius 412
Guy Henniart (Orsay) Higher ramification theory
and the local Langlands correspondence for GL(n)
This is a report on joint work in progress with C. J. Bushnell. Let F
be a non-Archimedean locally compact field, F^sep a separable
algebraic closure of, G_F its Galois goup over F. The Langlands
correspondence for GL(n,F) associates to a smooth irreducible
representation r of G_F, of dimension n, a smooth irreducible cuspidal
representation pi(r) of GL(n,F). When n=1 that is given by class field
theory, and the higher ramification theorem of class field theory says
that for two characters r_1 and r_2 of G_F, the characters pi(r_1) and
pi(r_2) of GL(1,F)=F* have the same restriction to the higher unit
subgroup U_F^i of F* if and only if r_1 and r_2 have the same
restriction to the higher ramification subgroup G_F^i of G_F. We
determine what happens if n>1; that introduces Herbrand functions of a
new type.
23 June 2015 Tuesday Algant graduation talks room 412
10:30 - 11:30
Alessandro Pezzoni On the complexity of irrational algebraic numbers
At the beginning of the XX century Borel introduced the notion of normal number
and showed that almost every real number is normal. Later he conjectured that
every irrational algebraic number is normal, but this result is still out of
reach. After giving the necessary background we will use the p-adic Subspace
Theorem to prove a transcendence criterion for numbers with a special kind of
b-ary expansion. Time permitting, we will then use this criterion to show that
the complexity function of the b-ary expansion of every irrational algebraic
number grows more than linearly, and that every irrational automatic number is
transcendental.
11:30 - 12:30
Edoardo Lanari Compatibility of homotopy colimits and homotopy pullbacks of simplicial presheaves
This work is basically the analysis of the problem of under which conditions
pullbacks commute with colimits, but from an homotopical point of view. Hence
the role of ordinary pullback and colimit functor in the classical theory is
played by homotopy pullback and homotopy colimit functors. The framework is that
of a general category of simplicial presheaves, i.e. one of the form sSet^I, for
some small category I.
13:30 - 14:30
Marco Vergura A Giraud-type theorem for model topoi abstract
14:30 - 15:30
Stefano Nicotra Finite dimensional pure motives abstract
16:00 - 17:00
Stefan van der Lugt Arakelov intersection theory applied to torsors of semi-stable elliptic curves
Arakelov intersection theory arises as a way to generalize the intersection
theory on smooth projective surfaces over a field to an intersection theory on
arithmetic surfaces. In 1987 Paul Hriljac gave, for each element in the
Tate-Shafarevich group of a semi-stable elliptic curve over a number field, an
upper bound for the discriminant of a splitting field of this element in terms
of its order. We will look at Hriljac's proof for this upper bound using
Arakelov intersection theory, and generalize the result of Hriljac to include
all torsors of this elliptic curve.
17:00 - 18:00
Sergio Zarabara An étale view of Galois Cohomology abstract
Fall 2014
15 September 2014
16:00-17:00
René Schoof (Università di Roma “Tor Vergata”) Serre’s Uniformity Conjecture
Serre’s Uniformity Conjecture is a statement concerning the Galois action on
torsion points of elliptic curves defined over number fields. In this talk,
we formulate the conjecture and explain some of the difficulties in proving it.
29 September 2014
16:00-17:00
Rodolphe Richard (Universiteit Leiden)
6 October 2014
15:30 - 16:30
Elisa Lorenzo (Universiteit Leiden)
Some arithmetic properties of twists of the Klein quartic
(abstract)
13 October 2014
16:00-17:00
Carlo Pagano (Universiteit Leiden) There are no non-zero abelian varieties defined over Z
with good reduction at every prime
In this talk we'll present the overall strategy of Fontaine's proof of the
statement in the title. In particular we'll explain how the problem reduces to
the following: given G a commutative finite flat group scheme over O_K, the DVR
of a complete field K of charateristic 0 and residue field of positive
charateristic, give a good upper-bound of the ramification of the extension of K
obtained extending K with the point of G in the algebraic closure of K. So we'll
show how to obtain such an estimation.
27 October 2014
16:00-17:00
Pavel Solomatin (Universiteit Leiden) Curves with many points over finite fields: the classfield theory approach.
The problem of constructing curves with many points over finite fields has
received considerable attention in the recent years. Using the classfield theory
approach, we construct new examples of curves ameliorating some of the known
bounds. More precisely, we improve the lower bounds on the maximal number of
points $N_q(g)$ for many values of the genus $g$ and of the cardinality $q$ of
the finite field $\mathbb F_q$.
3 November 2014
16:00-17:00
Andrea Ferraguti (Universität Zürich) Q-curves, modularity and L-functions
Q-curves are elliptic curves over \bar{Q} which are isogenous to all
their Galois conjugates. For example, all elliptic curves with CM are
Q-curves, while not all elliptic curves without CM are Q-curves. In the
first part of the talk we will outline the proof of the following
theorem, proposed by Ribet in 1992 and validated by Khare and
Winterberger's work on Serre's conjecture in 2006: Q-curves are exactly
modular elliptic curves, namely quotients over \bar{Q} of modular curves
for \Gamma_1(N). This theorem provides a deep connection between
L-functions of Q-curves and certain products of L-functions of newforms
of level \Gamma_1(N). In the second part of the talk, we will see how it
is possible, under the generalized Manin conjecture, to exploit this
connection to provide a lower bound on |L(C,1)|, where L(C,s) is the
L-function attached to a Q-curve C completely defined over a quadratic
field.
10 November 2014
16:00-17:00
Miguel Grados (HU Berlin) Estimates of canonical Green’s functions
(abstract)
17 November 2014
16:00-17:00
Eva Martínez (FU-Berlin) The tensor product problem in Arakelov geometry
The tensor product problem for vector bundles over smooth projective curves
defined over a field of characteristic 0 is well-known. It states that tensor
products of semistable vector bundles are semistable. Analogously, in 1997
J.B. Bost conjectured it in the arithmetic context, for the so called Arakelov
vector bundles over arithmetic curves. In this seminar we will explore this
problem and show some examples where the strict parallelism with the geometric
case breaks down.
24 November 2014
11:15-12:10 room B1
(note the time
and place)
Ziyang Gao (Universiteit Leiden) The mixed Ax-Lindemann theorem and its applications to the Zilber-Pink conjecture (followed by the PhD defense of the speaker at 13:45 in the city centre)
The Zilber-Pink conjecture is a diophantine conjecture concerning unlikely
intersections in mixed Shimura varieties. It is a common generalization of the
Mordell-Lang conjecture and the Andre-Oort conjecture. Following the
Pila-Zannier method, recent development has been made about this conjecture by
Pila, Pila-Tsimerman, Ullmo-Yafaev, Klingler-Ullmo-Yafaev, Gao in the direction
of Andre-Oort conjecture and by Habegger-Pila, Orr in the direction of unlikely
intersections beyond Andre-Oort. In all the proofs, a key point is to prove the
Ax-Lindemann theorem (or more generally the Ax-Schanuel theorem), which is a
generalization of the classical Lindemann-Weierstrass theorem in the functional
case. In this talk, I will explain how it is a natural generalization of the
classical Lindemann-Weierstrass theorem and give a sketch of its proof. Then I
will discuss about its application to some special cases of the Zilber-Pink
conjecture, in particular the Andre-Oort conjecture and Andre-Pink-Zannier
conjecture. I will focus on the universal family of abelian varieties in my
talk.
8 December 2014
16:00-17:00
Misja Steinmetz (Cambridge) Asymptotic Behaviour of the Conjugacy Probability of the Symmetric and
Alternating Groups (last talk of the semester)
For a group G, we let kappa(G) be the probability that two elements of the group
are conjugate, when chosen uniformly independently at random. We will have a
look at kappa(Sn) and study the natural question: how does kappa(Sn) behave as n
goes to infinity? With remarkably little effort we will then use these results
to get similar answers for kappa(An).