## 2016

### Intercity Number Theory Seminar

January 22, Groningen. Bernoulliborg, room 105- 12:00–13:00
- Teresa Crespo Barcelona, Hopf Galois separable extensions
Hopf Galois theory generalizes classical Galois theory by replacing the action of the automorphism group by the action of a Hopf algebra. In the case of finite separable field extensions, Greither and Pareigis (1987) proved that the Hopf Galois property may be stated in terms of groups. In my talk I will survey Hopf Galois theory for finite separable field extensions and present some recent results on Hopf Galois structures obtained in collaboration with Anna Rio and Montserrat Vela.
- 13:30–14:30
- Ane Anema Groningen, Faltings method, Galois extensions of exponent four and abelian surfaces over Q
In 1983 Faltings introduced a method to decide if the Galois representation on the l-adic Tate module of two abelian varieties over a number field are isomorphic. In this talk we will recall an effective version of this method by Grenié (2007) and discuss Galois extensions of exponent four in order to apply the method to abelian surfaces over Q.
- 14:45–15:45
- Max Kronberg Groningen, Construction of Torsion Points on Jacobians of Curves using Hensel's Lemma
Given positive integers N and g, one can ask whether there exists a smooth curve of genus g defined over the rational numbers such that the jacobian has a rational N-torsion point. For g=1 this question was answered by Mazur in 1977. In the case g>1 not much is known. In this talk, we present a method based on Hensel's Lemma for the construction of curves with a rational p-torsion point on their jacobians.
- 16:00–17:00
- Zbigniew Hajto Kraków, Polynomial automorphisms and Picard-Vessiot theory
In 1973 L.A. Campbell published his landmark paper "A condition for a polynomial map to be invertible" characterizing polynomial automorphisms of affine spaces in terms of Galois extensions of function fields. I will present a differential version of the theorem of Campbell and some recent computational results on polynomial automorphisms obtained in collaboration with Elżbieta Adamus and Pawel Bogdan.

### Intercity Number Theory Seminar

February 5, Leiden. Snellius building, room 312- 11:15–12:15
- Martin Bright Leiden, Failures of weak approximation in families
Given a family of varieties over a number field, we investigate the variation of the Brauer-Manin obstruction within the family. We give sufficient conditions on a family of varieties over
for 100% of the family to have a Brauer-Manin obstruction to weak approximation. (Joint work with Tim Browning and Dan Loughran)**P**^{n} - 13:15–14:15
- Daniel Loughran Hannover, Fibrations with few rational points
In this talk, we study the problem of counting the number of varieties in fibrations over projective spaces which contain a rational point. We obtain geometric conditions that force very few of the varieties in the family to contain a rational point, in a precise quantitative sense. This generalises the special case of conic bundles, treated by Serre. This is joint work with Arne Smeets.
- 14:30–15:30
- Elisa Leiden, Twists of the Klein quartic: classification and modular interpretation
The Klein quartic is, up to isomorphism, the genus 3 curve with biggest automorphism group. We will compute all its twists over number fields, which provides a complete classification of twists of non-hyperelliptic genus 3 curves defined over number fields, already started by the speaker in her thesis. The Klein quartic is isomorphic to the modular curve X(7), which endows the twists with a modular interpretation. We use this interpretation to provide counterexamples to the Hasse principle.
- 15:45–16:45
- Rodolphe Richard Leiden, Witt vectors and p-adic differential equations

### Intercity Number Theory Seminar

March 4, UvA Amsterdam. The first three talks are in room F3.20 of the KdVI (entrance Nikhef), Science Park 107.Lenny Taelman's inaugural lecture will take place in the Oude Lutherse kerk (Singel 411, corner with Spui)

- 10:30–11:30
- Ronald van Luijk Leiden, Concurrent lines on del Pezzo surfaces of degree one
Let
*k*be a field and*k*an algebraic closure. A del Pezzo surface over*k*is a surface over*k*that is isomorphic over*k*to either×**P**^{1}(degree 8), or**P**^{1}blown up at**P**^{2}*r*<9 points in general position (degree 9-*r*). Famous examples are smooth cubic surfaces in(with**P**^{3}*r*=6 and degree 3), which over*k*contain 27 lines; at most three of these can be concurrent, that is, go through the same point. Analogously, we get 240 lines for*r*=8 and degree 1. Based on the graph on these lines, with edges between those that intersect, we get an upper bound of 16 for the number of concurrent lines. We show that this upper bound is only attained in characteristic 2, which makes the case*r*=8 different from all other cases. In most characteristics, including characteristic 0, the upper bound is 10. This is joint work with Rosa Winter. - 11:45–12:45
- Christophe Debry KU Leuven and UvA, Special values in positive characteristic
The setting of this talk is the one of global function fields and their special values. We introduce, associated to a Drinfeld module, a convergent Euler product analogous to the special values of interest in the class number formula and the BSD conjecture. We then state a conjectural formula relating this special value to the so-called class module and regulator of the Drinfeld module. After mentioning the many cases in which the formula is known to be true, we put this into a broader context of special values of zeta functions and Hecke characters.
- 13:45–14:45
- Ben Moonen Nijmegen, Can every K3 be dominated by a product of curves?
The question in the title is a well-known open problem (that I won't solve in the talk). I will consider it in the more general setting of surfaces with
*p*_{g}= 1. I will explain: (a) An 'easy' method that leads to interesting examples, (b) the consequences of having a positive answer, (c) why for general K3's we cannot hope to have a construction as simple as the one given in (a). - 16:00–17:00
- Lenny Taelman UvA, Onmogelijke verbanden

### Intercity Number Theory / Beeger Seminar

March 24, Leiden. This is a Thursday! Snellius building, room 312.- 11:00–12:00
- Rob Tijdeman Leiden, Finding well approximating lattices for a finite set of points.
The following question was asked by Andras Hajdu in view of applications in computer science: Given a finite set
*A*of points in inwhich do not fit in a hyperplane, how to find**R**^{n}*o*,*d*_{1}, ...,*d*_{n}∈such that the distance of**R**^{n}*a*-*o*to the lattice*d*is relatively small for every_{1}**Z**+ ... + d_{n}**Z***a*∈*A*? Using the LLL-algorithm his brother Lajos and I developed a method which works pretty well. We also derived some theoretical results, in particular in the one-dimensional case. - 13:15–14:15
- Florian Luca Johannesburg, Diversity in parametric families of number fields
Let
*f*(*x*,*t*)∈**Q**[*x,t*] be a polynomial with deg*f*_{x}=*D*≥ 2. For each positive integer*n*let*x*_{n}be a root of*f*(*x*,*n*). Dvornicich and Zannier proved thatlog [ **Q**[*x*_{1},...,*x*_{N}]:**Q**] ≫*N*log *N**F*= {_{f}(N)**Q**[*x*_{n}] : 1 ≤ n ≤ N}# *F*≫_{f}(N)*N*log *N*# *F*≫_{f}(N)*N*(log *N*)^{1-c}*c*:=*c*(*D*)>0 is a positive constant depending on*D*, but not on*f*. The main ingredient of the proof of the improvement consists on counting special divisors of values of polynomials with integer coefficients. (Coauthor: Yuri Bilu.) - 14:45–15:45
- Jan-Hendrik Evertse Leiden, Results and open problems related to Schmidt's Subspace Theorem
Schmidt's Subspace Theorem is a central theorem from Diophantine approximation with many applications. Among others, it implies Roth's Theorem on the approximation of algebraic numbers by rationals. We discuss the state of the art of Schmidt's Subspace Theorem, discuss some refinements, and pose some open problems.
- 16:00–17:00
- James Maynard Oxford, Polynomials representing primes
It is a famous conjecture that any one variable polynomial satisfying some simple conditions should take infinitely many prime values. Unfortunately, this isn't known in any case except for linear polynomials - the sparsity of values of higher degree polynomials causes substantial difficulties. If we look at polynomials in multiple variables, then there are a few polynomials known to represent infinitely many primes whilst still taking on `few' values; Friedlander-Iwaniec showed
*X*^{2}+*Y*^{4}is prime infinitely often, and Heath-Brown showed the same for*X*^{3}+2*Y*^{3}. We will describe recent work which gives a family of multivariate sparse polynomials all of which take infinitely many prime values.

### The Big Picture, workshop on the occasion of Jan Stienstra's retirement

April 8, Utrecht. Unfortunately, the workshop will be moved to a later date, to be specified.See the website.

### 10th Belgian-Dutch Algebraic Geometry Day

April 22, Leuven. Kasteel van Arenberg, KU Leuven wetenschapscampus, Heverlee (there will be signs and there is parking next to the castle)- 13:45–14:45
- Sho Tanimoto Copenhagen, Towards a refinement of Manin's conjecture
Manin's conjecture predicts the generic distribution of rational points on Fano varieties, and it has the explicit asymptotic formula in terms of geometric invariants of the underlying variety. However the original version which predicts asymptotic formulae after removing proper closed sets is wrong due to covering families of subvarieties violating compatibility of Manin's conjecture, and a possible refinement, suggested by Peyre, removes thin sets instead of closed sets. One natural question is how to choose the exceptional thin set. In this talk, I would like to address this issue 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.
- 15:00–16:00
- John Christian Ottem Oslo, Effective cones of cycles on blow-ups of projective space.
While the cones of curves and effective divisors are important tools in algebraic geometry, the cones of subvarieties of intermediate codimension remain much more mysterious, mostly due to the lack of examples. In this talk, I will outline the two explicit computations of such cones; on blow-ups of projective space and on certain hyperkahler fourfolds. In the first case, we determine bounds on the number of points for which these cones are generated by the classes of linear cycles, and for which these cones are finitely generated. Surprisingly, we discover that in some cases, the higher codimension cones behave better than the cones of divisors. In the case of hyperkahler fourfolds, we show the cone of nef 2-cycles can be strictly larger than the cone of pseudoeffective 2-cycles, showing that the usual intuition from divisors do not generalize to higher codimension.
- 16:30–17:30
- Bruno Chiarellotto Padova, Monodromy action and special fiber for semistable schemes over a DVR
We plan to discuss some results on the action of the monodromy on the (unipotent) de Rham fundamental group of the generic fiber of some type of semistable schemes over a DVR in mixed characteristic. The action will be interpreted in terms of structure of the graph of the special fiber. This is joint work with Ch. Lazda.

### Intercity Number Theory Seminar

May 19, Eindhoven. PhD defense Emil Horobet. Thursday! The talks will be in MF 15; the defense in Aud 4.- 12:00–12:45
- Bernd Sturmfels UC Berkeley, Nearest Points on Toric Varieties
We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the A-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point. This is joint work with Martin Helmer.
- 13:30–14:15
- Bernard Mourrain Inria, Sparse decomposition from moments.
In 1795, G. de Prony proposed a method to decompose a function as a sum of exponentials from values at regularly spaced points. We show how this problem is naturaly connected to polynomials and Hankel matrices. We describe a multivariate extension of the method based of a generalization of Kronecker theorem for Hankels operators of finite rank, its relation with sparse decomposition and give some applications to tensor decomposition.
- 14:30–15:15
- Monique Laurent CWI and Tilburg University, Completely positive semidefinite matrices: some properties and applications
We consider the completely positive semidefinite cone CS
_{+}, which consists of all symmetric matrices admitting a Gram representation by positive semidefinite matrices (of any size). This cone permits to model some optimization problems in quantum information as conic optimization over affine sections of CS_{+}and its dual cone relates to tracial nonnegative quadratic polynomials. We will discuss hierarchies of inner and outer approximations for CS_{+}, and show classes of matrices in CS_{+}having exponentially large CS_{+}-rank (aka the smallest dimension of the psd matrices in a Gram representation). - 16:00–17:00
- Emil Horobet TU Eindhoven, PhD defense

### Intercity Number Theory Seminar

June 10, Leuven. Departement wiskunde, lokaal 01.07, Celestijnenlaan 200B, 3001 Heverlee, België- 13:00–13:45
- Jennifer Balakrishnan Oxford, Coleman integration for hyperelliptic curves
In the 1980s, Coleman formulated a p-adic theory of line integration and used it to re-interpret the method of Chabauty to find rational points on higher genus curves. I will give an overview of Coleman integration and explain how Kedlaya's algorithm for computing zeta functions of hyperelliptic curves (using p-adic cohomology) provides the essential input for an algorithm for computing Coleman integrals. I will also give some examples of single and double Coleman integrals on hyperelliptic curves which play a role in the study of rational points.
- 14:00–14:45
- Jan Tuitman Leuven, Coleman integration for general curves
Over the past years I have developed an extension of Kedlaya's algorithm for computing zeta functions of hyperelliptic curves over finite fields to general curves. More recently, in joint work with Jennifer Balakrishnan, we have modified this algorithm to compute (both single and iterated) Coleman integrals on general curves over the rationals as well. I will give an overview of this work and show some examples computed with our Magma implementation.
- 15:15–16:15
- Giulia Battiston Heidelberg, A theory of Galois descent for finite inseparable extensions
The well known Galois descent allows to descend algebraic objects along a field extension L/K when the latter is Galois, in particular separable. Using the theory of the automorphism group scheme and of descent along torsors, I will present a generalization of the classical Galois descent to the case where L/K is a possibly inseparable field extension, and give some examples of applications.
- 16:30–17:30
- Margaret Bilu ENS Paris, Motivic Euler products
The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties over k. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. I will focus on Manin's conjecture and on its motivic analogue: the latter predicts the behaviour of moduli spaces of curves of given degree on some algebraic varieties, and may be formulated in terms of the generating series of their classes in the Grothendieck ring, the motivic height zeta function. This will lead me to explain how some power series with coefficients in the Grothendieck ring can be endowed with an Euler product decomposition and to give an overview of the techniques used to solve Manin's problem for equivariant compactifications of vector groups.

### “Hecke meets Galois” - a minisymposium

June 13, Utrecht. PhD defense Valentijn Karemaker, academiegebouw- 09:30–10:15
- Maarten Solleveld Radboud U., Instances of the local Langlands correspondence, based on class field theory
- 10:20–11:05
- Elena Mantovan Caltech, Differential operators and families of automorphic forms on unitary groups of arbitrary signature
- 11:30–12:15
- Jeroen Sijsling U. Ulm, Reconstructing plane quartics from their invariants
- 14:30–15:30
- Valentijn Karemaker U. Utrecht, PhD defense: Hecke algebras, Galois representations, and abelian varieties

### Intercity Number Theory Seminar

August 26, Leiden. Havinga zaal, Gorlaeus building. (Enter through the main entrance of the Gorlaeus building. Walk up the stairs, cross the long corridor and pass through the doors at the end of the corridor. Directly after passing those doors, turn to the right and you will find the Havingazaal 50 meters down the hall on the right hand side.)This is the last day of the workshop Mathematical Structures for Cryptography.

- 09:45–10:45
- Nadia Heninger University of Pennsylvania, Cryptographic applications of capacity theory
- 11:15–12:15
- Florian Hess Carl von Ossietzky Universität Oldenburg, Asymptotically fast arithmetic in Jacobians of curves of large genus
- 13:20–13:50
- Andreas Enge INRIA Bordeaux-Sud-Ouest, Short addition sequences for theta functions
- 14:00–15:00
- Chaoping Xing Nanyang Technological University, Singapore, Codex and applications to local decoding of Reed-Muller codes

### PhD defense Maarten Derickx

September 21, Leiden. room C102 Sterrewacht, Sterrenwachtlaan 11, 2311 GP, Leiden- 11:15–12:15
- Maarten Derickx Leiden, PhD defence (Academiegebouw, Rapenburg 73)
- 14:00–15:00
- Pierre Parent Université de Bordeaux, Rational points of modular curves: an arakelovian point of view
General methods from diophantine geometry have been very successful in proving finiteness results for points of algebraic curves with values in number fields. Those results however are in general not effective, for deep reasons, and this prevents from proving triviality (and not only finiteness) of relevant sets of rational points. In this talk I will explain how the situation can be much better in the case of modular curves, by using specific arakelovian methods.
- 15:15–16:15
- Marusia Rebolledo Université Blaise Pascal Clermont-Ferrand 2, Survey of Yuan-Zhang proof of averaged Colmez conjecture.
In 1993, Colmez proposed a conjectural formula for the Faltings height of a CM abelian variety in terms of logarithmic derivatives of Artin L functions. In 2015, Yuan and Zhang and independently Andreatta, Goren, Howard and Madapusi-Pera proved an average version of this formula. In this talk I will give a survey of this result and Yuan-Zhang proof.
- 16:30–17:30
- Maarten Derickx Universiteit Leiden, Torsion points on elliptic curves over number fields of degree > 6
If
*E*is an elliptic curve overthen the torsion subgroup of**Q***E(*is either isomorphic to**Q**)for**Z**/N**Z***N < 13*and*N*not equal to 11, or isomorphic tofor**Z**/2**Z**x**Z**/2N**Z***N < 5*as famously proven by Mazur. Actually Merel generalized this to number fields of bounded degree: For every integer*d*, the set*Φ(d)*of isomorphism classes of groups that one encounters as torsion subgroup of an elliptic curve over a number field of degree*d*is finite. A first step in determining*Φ(d)*is determining*S(d)*, the set of primes that can divide the order of a group in*Φ(d)*. Before I wrote my thesis the sets*S(1)*,*S(2)*,*S(3)*and*S(4)*were already known, and jointly with Stoll, Stein and Kamienny I determined*S(5)*and*S(6)*. In this talk I will sketch a strategy I think that, when successfully implemented on a computer, should allow the determination of*S(7)*and maybe even more*S(d)*with*d>6*. I will also explain what one needs to prove if one hopes to use this strategy theoretically, in order to bound the largest element in*S(d)*by some polynomial in*d*.

### Intercity Number Theory Seminar

October 7, Leiden. Snellius Building, room B03- 11:15–12:15
- Gabriel Zalamansky Universiteit Leiden, The complexity of a flat Groupoid
We introduce the notion of complexity of a flat groupoid scheme, as a measure of its defect to be an equivalence relation. We will then prove a descent theorem for quotients of groupoids of complexity less or equal than 1, allowing descent along certain ramified coverings and generalizing descent along torsors in this setting.
- 13:30–14:30
- Jinbi Jin Universiteit Leiden - MPIM Bonn, Computability of the étale Euler-Poincaré characteristic with finite coefficients
The classical point-counting algorithms by Schoof and Pila for curves compute the number #X(
*F*) of_{q}*F*-points on a fixed smooth projective curve_{q}*X*in time polynomial in log*q*. A motivating question for this talk is whether such efficient algorithms exist for more general varieties*X*. Via the Lefschetz trace formula, it suffices to be able to efficiently compute the so-called étale cohomology groups for suitable coefficients; the methods of Schoof and Pila can also be reinterpreted to be using this observation.Computability of the étale cohomology groups is known already, first in 2015 by Poonen, Testa, and van Luijk in characteristic 0, and later that year by Madore and Orgogozo in general. However, their algorithms aren't known to be primitive recursive (loosely speaking, primitive recursive algorithms are those that only use finite loops of precomputed length), as at some point in their computation, they enumerate along an infinite set. In this talk, I will describe the results obtained in my upcoming thesis (under the supervision of Bas Edixhoven and Lenny Taelman), which are partial results towards a primitive recursive algorithm computing étale cohomology, using a different approach.

- 14:45–15:45
- Erik Visse Univeriteit Leiden, Explicit bounds for the transcendental Brauer group of Kummer surfaces over a number field
Kummer surfaces are K3 surfaces that arise from abelian surfaces. For any K3 surface over a finitely generated field, Skorobogatov and Zarhin have proved that the Brauer group is finite. Their proof is not explicit, but it uses abelian varieties in a way that the proof can be made explicit for Kummer surfaces by studying bounds for the degrees of certain isogenies. For Kummer surfaces over a number field, we find an explicit bound on the number of elements of the transcendental Brauer group, solely in terms of the degree of the number field and the Faltings height of the associated abelian surface, which we assume to be principally polarized. Furthermore, we give an algorithm that (in theory) calculates the algebraic Brauer group that for Kummer surfaces with minimal Picard rank is also practical. This is joint work with Victoria Cantoral-Farfán, Yunqing Tang and Sho Tanimoto.
- 16:00–17:00
- Richard Griffon Universiteit Leiden, Analogue of the Brauer-Siegel for the Legendre elliptic curves over function fields
The classical Brauer-Siegel theorem gives upper and lower bounds on the product of the regulator and the class-number of a number field in terms of its discriminant. These bounds can be seen as a measure of the "arithmetic complexity" of number fields.
Now, consider an elliptic curve

*E*defined over a global field*K*: assuming that its Tate-Shafarevich group is finite, one can form the product of the order of this group by the Néron-Tate regulator of*E*. Heuristically, this product measures the complexity of computing the Mordell-Weil group of*E*over*K*. This prompts the question of bounding this quantity in terms of simpler invariants of*E*, e.g. its height. In other words, one would like to prove an analogue of the Brauer-Siegel theorem for elliptic curves.In some explicit examples, it's possible to prove that a perfect analogue actually holds. In this talk, I'll start by describing in more details this problem and its motivations (mostly concentrating on the case where

*K*is a function field of characteristic*p*) and I will describe my results about the family of Legendre elliptic curves.

### Intercity Number Theory Seminar

November 11, UvA and VU Amsterdam. At the VU: before lunch in room WN-F619 and after lunch in room WN-M639 (both in the W&N building).- 11:00–12:00
- Martin Orr Imperial College London, Finiteness theorem for K3 surfaces with complex multiplication
I will discuss the following theorem, which is joint work with Alexei Skorobogatov: For any integer n, there are finitely many Qbar-isomorphism classes of K3 surfaces with complex multiplication which can be defined over number fields of degree n. This is derived from Tsimerman's Galois bound for CM abelian varieties by Shimura variety constructions.
- 13:15–14:15
- James Lewis University of Alberta, A Going-up Theorem in K-Theory and a Question of Bruno Kahn
Over an afternoon at the Confederation Lounge in Edmonton, Bruno Kahn suggested that there should be a connection between the Griffiths group of a smooth projective threefold and the group of
*K*_{1}indecomposables of a smooth surface related to such a threefold. Although unclear at that time, we now present, via well-known conjectures, some concrete evidence in support of this. - 14:30–15:30
- Chun Yin Hui VU Amsterdam, Invariant dimensions, semisimplicity, and maximality of geometric monodromy action
Let
*X*be a smooth separated geometrically connected variety over,**F**_{q}*f*:*Y*→*X*a smooth projective morphism, and*w*an integer. We compare the invariant dimensions of sufficiently many*l*-adic and mod*l*representations of the geometric étale fundamental group*π*arising from the higher direct image sheaves_{1}:=π_{1}^{ét}(X_{Fq})*R*and^{w}f_{*}**Q**_{l}*R*respectively. These dimension data are used to prove mod^{w}f_{*}**F**_{l}*l*semisimplicity and*l*-adic maximality of*π*-action when_{1}*l*is sufficiently large. This is a joint work with Anna Cadoret and Akio Tamagawa. - 16:00–17:00
- Wadim Zudilin University of Newcastle and RU Nijmegen, Short random walks and Mahler measures
An
*n*-step uniform random walk is a walk that starts at the origin and consists of*n*steps of length 1 each taken into a uniformly random direction. It is particularly interesting for*n*=2,3,4,5 because of its beautiful links to modular and hypergeometric functions. The Mahler measure of an*n*-variable polynomial is its geometric mean over the*n*-dimensional torus. There are many cases when*n*-variate Mahler measures are known or conjectured to be linked to hypergeometric functions and noncritical*L*-values, for small*n*as well. In the talk I will outline the links above and indicate some new interconnections between the short random walks and Mahler measures. These novel results are from joint work in progress with Armin Straub.