## 2018

### Intercity Number Theory Seminar

16 March, Utrecht. Location: Buys Ballotgebouw 161 (Princetonplein 5, 3584 CC Utrecht). Note: at the moment the main entrance of the BBG is closed and you will need to enter the Buys Ballotgebouw via the Koningsbergergebouw (Budapestlaan 4a-b, 3584 CD Utrecht)- 12:30–13:00
- tea break,
- 13:00–13:50
- Masha Vlasenko Warsaw, Dwork crystals and related congruences
In the talk I will describe a realization of the p-adic cohomology of an affine toric hypersurface which originates in Dwork's work and give an explicit description of the unit-root subcrystal based on certain congruences for the coeficients of powers of a Laurent polynomial. This is joint work with Frits Beukers.
- 14:00–14:50
- Diego Izquierdo Paris, On a conjecture of Kato and Kuzumaki
In 1986, Kato and Kuzumaki stated a set of conjectures which aimed at giving a Diophantine characterization of the cohomological dimension of fields in terms of Milnor K-theory and of points in projective hypersurfaces of small degree. The conjectures are known to be wrong in full generality, but they remain open for various fields that usually appear in number theory or in algebraic geometry. In this talk, I will present several results related to the conjectures of Kato and Kuzumaki for global fields and for some function fields.
- 15:00–15:30
- tea break,
- 15:30–16:20
- Sara Checolli Grenoble, On some arithmetic properties of Mahler functions
Mahler functions are power series
*f*(*x*) with complex coefficients for which there exist a natural number n and an integer*l*≥ 2 such that*f*(*x*),*f*(*x*^{l}),ldots,*f*(*x*^{ln-1}),*f*(*x*^{ln}) are linearly dependent over**C**(x). The study of these functions and of the transcendence of their values at algebraic points was initiated by Mahler around the 30's and then developed by many authors. In this talk we will investigate some arithmetic aspects of Mahler functions. In particular, when*f*(*x*) satisfies the equation*f*(*x*)=*p*(*x*)*f*(*x*^{l}) with*p*(*x*) a polynomial with integer coefficients, we will see how certain properties of*f*(*x*) mirrors on the polynomial*p*(*x*), also in connection with the theory of automatic sequences. If time allows, we will also discuss some analogies with*E*- and*G*- functions. This is a joint work with Julien Roques. - 16:30–17:20
- Jakub Byszewski Kraków, Sparse generalised polynomials and automatic sequences
We investigate generalised polynomials (i.e., polynomial-like expressions involving the use of the floor function) which take the value 0 on all integers except for a set of density 0. By a theorem of Bergelson-Leibman, generalised polynomials can be completely described in terms of dynamics on nilmanifolds. Our main result is that the set of integers where a sparse generalised polynomial takes non-zero value cannot be combinatorially rich (specifically, cannot contain a translate of an IP set). We study some explicit constructions and give some evidence for the claim that generalised polynomial sets of exponential growth have interesting arithmetic behaviour. In particular, we show that the characteristic functions of the Fibonacci and Tribonacci numbers are given by generalised polynomials. Finally, we show that any sufficiently sparse {0,1}-valued sequence is given by a generalised polynomial. We apply these results to a question on automatic sequences. This is joint work with Jakub Konieczny.

### Intercity Number Theory Seminar

20 April, Groningen. Bernoulliborg, room 165, start at 11:45.- 11:45–12:45
- Maarten Derickx Groningen, Torsion subgroups of rational elliptic curves over infinite extensions of Q.
Recently there has been much interest in studying the torsion subgroups of elliptic curves base-extended to infinite extensions of Q. In this talk, given a finite group G, we will study what happens with the torsion of an elliptic curve E over Q when changing base to the compositum of all number fields with Galois group G. This is done by studying a group theoretic condition called generalized G-type, which is a necessary condition for a number field with Galois group H to be contained in that compositum. In general, group theory allows one to reduce the original problem to the question of finding rational points on finitely many modular curves. This method is illustrated by completely determining which torsion structures occur for elliptic curves defined over Q and base-changed to the compositum of all fields whose Galois group is of generalized A4-type.
- 13:30–14:30
- Pınar Kılıçer Oldenburg, On primes dividing the invariants of Picard curves
The j-invariants of elliptic curves with complex multiplication (CM) are algebraic integers. For invariants of genus g = 2 or 3, this is not the case, though suitably chosen invariants do have smooth denominators in many cases. Bounds on the primes in these denominators have been given for g=2 (Goren-Lauter) and some cases of g=3. For Picard curves of genus 3, we give a new approach based not on bad reduction of curves but on a very explicit type of good reduction. This approach simultaneously yields much sharper bounds and a simplification of the proof. This is joint work with Marco Streng and Elisa Lorenzo García.
- 14:45–15:45
- Harm Voskuil Amsterdam, Mumford curves in positive characteristic.
Mumford curves can be defined as quotients of an open analytical subspace of the projective line over a complete non-archimedean field by the action of a free discontinuous group. We consider Mumford curves in positive characteristic that have many automorphisms.
Then the free group Δ defining the Mumford curve

*X*is contained as a normal subgroup in a finite amalgam Γ of finite groups. The automorphism group*Aut*(*X*) is the quotient Γ/*Delta*.We describe the amalgams Γ that occur and obtain a formula for the upper bound of the order of the automorphism group in terms of the genus of the Mumford curve (assuming that the genus is >1). Furthermore, all the Mumford curves that realise this upper bound are described in terms of the corresponding amalgam Γ and the automorphism group Γ/Δ is determined.

(This is joint work with Marius van der Put).

- 16:00–17:00
- Florian Hess Oldenburg, Explicit isomorphisms and fields of definition and moduli of curves
We define some isomorphism invariants of pointed curves and discuss a constructive approach to compute fields of definition and fields of moduli of curves. These invariants generalise the well known j-invariant of elliptic curves in a different direction than for example the Igusa invariants of hyperelliptic curves.

### Intercity Number Theory Seminar

22 June, Leiden. This is the last day of the workshop Effective Methods for Diophantine Problems. All talks will be in the Havingazaal of the Gorlaeus Building, located on the first floor of what's indicated by LMUY on this map and the detailed map on the last page of that document. Note that you can not get there through the main entrance of the Gorlaeus Lecture Hall, indicated by (4) on the map.### Belgian-Dutch Algebraic Geometry seminar

20 September, Nijmegen. This is a two-day event: September 20 and 21. See the website, also for (free) required registration.### Aachen-Bonn-Koeln-Lille-Siegen seminar on automorphic forms

23 November, Utrecht. Location: Minnaert Building, room 201- 13:15–14:15
- Peter Bruin Leiden, On explicit computations with modular Galois representations
I will explain a compact way of encoding representations of the absolute Galois group of a field
*K*on finite Abelian groups as*dual pairs of finite*. These are in principle equivalent to finite commutative group schemes or Hopf algebras, but are easier to compute and to store. I will show how to compute these objects and to work with them, with a focus on representations attached to modular forms over finite fields. Work is ongoing to include such representations in the L-Functions and Modular Forms Database.*K*-algebras - 14:15–15:15
- Alexandru Ciolan Köln, Asymptotics and inequalities for partitions into squares
In this talk we show that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further prove that, for
*n*large enough, the two quantities are different and which of the two is bigger depends on the parity of*n*. This solves a recent conjecture formulated by Bringmann and Mahlburg (2012). - 15:45–16:45
- Jan-Willem van Ittersum Utrecht, A symmetric Bloch-Okounkov theorem
Bloch and Okounkov showed that the generating series associated to a wide class of functions on partitions of integers are quasimodular forms. We consider a different class of functions on partitions. In this class the functions are symmetric in the parts and multiplicities of the parts of the partitions. We show that the associated generating series are quasimodular forms as well.
- 16:45–17:45
- Annalena Wernz Aachen, On Hermitian modular forms - Theta series and Maass spaces
It is well known (Cohen, Resnikoff 1978 and Hentschel, Nebe 2009) that Hermitian theta series of an even unimodular theta lattice of rank
*k*belong to [*U*(2,2;*O*_{K}),*k*] where*U*(2,2;*O*_{K}) is the full Hermitian modular group of degree 2. In this talk, we consider the normalizer*U*^{*}(2,2;*O*_{K}) of*U*(2,2;*O*_{K}) in the special unitary group*SU*(2,2;**C**) and examine the behavior of Hermitian theta series under its action. Furthermore, we consider the Hermitian Maaß spaces*S*(*k*,*O*_{K}) and*M*(*k*,*O*_{K}) introduced by Sugano (1985) and Krieg (1991) respectively. In an approach which is similar to the paramodular case (Heim, Krieg 2018), we prove that a Maaß form in the Sugano sense is an element of Krieg's Maaß space if and only if it is a modular form with respect to*U*^{*}(2,2;*O*_{K}).

### Intercity Seminar Number Theory

7 December, Eindhoven. In Metaforum 14 (6e verdieping); PhD defense Guus Bollen in Senaatszaal of Auditorium.- 11:45–12:30
- Wieb Bosma Nijmegen, Enumerating self-complementary graphs
Graphs that are isomorphic to their complement are among the objects to which Polya's counting method has been most successfully applied. We will discuss methods for counting and enumerating self-complementary graphs (both ordinary and directed, with and without particular structure) on small numbers of points, and interaction between the two. On the way we will also encounter some interesting properties and problems of these graphs.
- 13:15–14:00
- Winfried Hochstättler Hagen, The Varchenko Determinant of an Oriented Matroid
The Varchenko matrix
*M*of a hyperplane arrangement is a symmetric square matrix indexed by the full dimensional regions of the arrangement, where*M*_{ij}equals the product of the hyperplanes seperating the cells*i*and*j*. Varchenko proved 1993 that the determinant of this matrix has a nice factorization. Using a proof strategy suggested by Denham and Henlon in 1999 we show that the same factorization works in the abstract setting of oriented matroids. For that purpose we show that every*T*-convex region of the set of topes, considered as a subcomplex of the Edelman poset, has a contractible order complex, which might be of independent interest. - 14:15–15:00
- Dustin Cartwright Tennessee, One-dimensional groups and algebraic matroids
I will talk about a construction of algebraic matroids from modules over endomorphism rings of 1-dimensional algebraic groups, which generalizes both linear and monomial realizations of matroids. The algebraic matroid constructed in this way coincides with a linear matroid over the endomorphism ring. I will explain how this relationship also extends to the Lindström valuation and the Frobenius flock of the algebraic matroid. This is joint work with Jan Draisma and Guus Bollen.
- 16:00–17:00
- Guus Bollen Eindhoven, PhD defense

### Intercity Number Theory Seminar

18 December, Leiden. In Snellius 412; PhD defense Erik Visse in the Academiegebouw in the center of Leiden.- 10:00–11:00
- Erik Visse Leiden, PhD defense
- 13:45–14:45
- Rachel Newton Reading, Number fields with prescribed norms
Let
*G*be a finite abelian group, let*k*be a number field, and let*x*be an element of*k*. We count Galois extensions*K*/*k*with Galois group*G*such that*x*is a norm from*K*/*k*. In particular, we show that such extensions always exist. This is joint work with Christopher Frei and Daniel Loughran. - 15:00–16:00
- Daniel Loughran Manchester, Integral points on Markoff surfaces
Integral solutions to Markoff-type equations of the form
*x*^{2}+*y*^{2}+*z*^{2}-*xyz*=*m*were studied by Ghosh and Sarnak. In this talk we explain how to re-interpret their work using the Brauer-Manin obstruction, and quantify the number of such surfaces which fail the integral Hasse principle. This is joint work with Vlad Mitankin. - 16:15–17:15
- Efthymios Sofos Bonn, The size of the primes
*p*for which a Diophantine equation is not soluble modulo*p*The set of the primes*p*for which a variety over the rational numbers has no*p*-adic point plays a fundamental role in arithmetic geometry. This set is deterministic, however, we prove that if we choose a typical variety from a family then the set has random behaviour. We furthermore prove that this behaviour is modelled by a random walk in Brownian motion. This has several consequences, the main one being the description of the finer properties of the distribution of the primes in this set via the Feynman-Kac formula.