Intercity Number Theory SeminarMarch 1, Groningen. The first talk takes place in room 165 of the Bernoulliborg and the three other talks take place in room 253.
- Dino Festi Mainz, A method to compute the geometric Picard lattice of a K3-surface of degree 2
K3 surfaces are surfaces of intermediate type, i.e., they are in between surfaces whose arithmetic and geometry is fairly well understood (rational and ruled surfaces) and surfaces that are still largely mysterious (surfaces of general type). The Picard lattice of a K3 surface contains much information about the surface, both from a geometric and an arithmetic point of view. For example, it tells about the existence of elliptic fibrations on the surface; if the surface is over a number field, then by looking at the Picard lattice one can have information about the Brauer group, and the potential density of rational points. Although much effort, there is not yet a practical algorithm that, given an explicit K3 surface, returns the Picard lattice of the K3 surface. In this talk we are going to give an overview on how practically compute the geometric Picard lattice of a K3 surface of degree two over a field of characteristic zero.
- Florian Hess Oldenburg, Partially euclidean global fields
- Marc Paul Noordman Groningen, Algebraic first order differential equations
Autonomous algebraic first order differential equations (i.e. differential equations of the form P(u, u') = 0 for P a polynomial with constant coefficients) can be interpreted as rational differential forms on an algebraic curve. In this talk, based on joint work with Jaap Top and Marius van der Put, I will explain how this perspective clarifies the possible algebraic relations between solutions of such differential equations. In particular, we will see that there are almost no algebraic relations between distinct non-constant solutions of the same differential equation, unless that differential equation comes from a one-dimensional group variety.
- Jan Vonk Oxford, Singular moduli for real quadratic fields
The theory of complex multiplication describes finite abelian extensions of imaginary quadratic number fields using singular moduli, which are special values of modular functions at CM points. I will describe joint work with Henri Darmon in the setting of real quadratic fields, where we construct p-adic analogues of singular moduli through classes of rigid meromorphic cocycles. I will discuss p-adic counterparts for our proposed RM invariants of classical relations between singular moduli and the theory of weak harmonic Maass forms.
Intercity Number Theory SeminarMarch 15, Utrecht. The morning lecture takes place in KGB Atlas (Koningsbergergebouw Budapestlaan 4a-b, 3584 CD Utrecht) and in the afternoon we are in MIN 2.01 (Minnaertgebouw Leuvenlaan 4, 3584 CE Utrecht).
- Efrat Bank Technion, Israel, Primes in short intervals on curves over finite fields
We prove an analogue of the Prime Number Theorem for short intervals on a smooth proper curve of arbitrary genus over a finite field. Our main result gives a uniform asymptotic count of those rational functions, inside short intervals defined by a very ample effective divisor E, whose principal divisors are prime away from E. In this talk, I will discuss the setting and definitions we use in order to make sense of such count, and will give a rough sketch of the proof. This is a joint work with Tyler Foster.
- Francesca Balestrieri MPIM Bonn, Germany, Arithmetic of zero-cycles on products of Kummer varieties and K3 surfaces
The following is joint work with Rachel Newton. In the spirit of work by Yongqi Liang, we relate the arithmetic of rational points to that of zero-cycles for the class of Kummer varieties over number fields. In particular, if X is any Kummer variety over a number field k, we show that if the Brauer-Manin obstruction is the only obstruction to the existence of rational points on X over all finite extensions of k, then the Brauer-Manin obstruction is the only obstruction to the existence of a zero-cycle of any odd degree on X. Building on this result and on some other recent results by Ieronymou, Skorobogatov and Zarhin, we further prove a similar Liang-type result for products of Kummer varieties and K3 surfaces over k.
- Kęstutis Česnavičius Orsay, France, Macaulayfication of Noetherian schemes
To reduce to resolving Cohen-Macaulay singularities, Faltings initiated the program of "Macaulayfying" a given Noetherian scheme X. Under various assumptions Faltings, Brodmann, and Kawasaki built the sought Cohen-Macaulay modifications without preserving the locus where X is already Cohen-Macaulay. We will discuss an approach that overcomes this difficulty and hence completes Faltings' program.
- Judith Ludwig Heidelberg, Germany, Perfectoid Shimura varieties and applications
Given a tower of Shimura varieties (where the level at a fixed prime p grows) one may ask whether one can equip the inverse limit with a geometric structure. As I will explain in the talk, this is possible in many cases. The geometric structure is that of a perfectoid space. I will then show you the impact of this results by explaining some applications.
Intercity Number Theory SeminarMarch 28, Leiden. This is a Thursday. All talks will be held in the Pieter de la Courtgebouw, room A5-47, which is walking distance from the train station. The PhD defense of Anna Somoza takes place in the Academiegebouw.
- Christophe Ritzenthaler Rennes, Reduction of plane quartics
Given a smooth plane quartic over a discrete valuation field K, we give a characterization of its reduction type (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over K in terms of the existence of a special plane quartic model and over the algebraic closure in terms of the valuations of the Dixmier-Ohno invariants of C. Joint work with Qing Liu, Elisa Lorenzo García and Reynald Lercier.
- Pınar Kılıçer Groningen, Modular invariants for genus-3 hyperelliptic curves
We discuss the connection between invariants of binary octics and Siegel modular forms of genus 3. Using this connection, we describe certain modular functions for hyperelliptic curves of genus 3 whose denominators are divisible by the primes of bad reduction for the associated hyperelliptic curves. We hope that this research will lead to an analogue of the Igusa invariants for hyperelliptic curves of genus 3. This is a joint work with Sorina Ionica, Kristin Lauter, Elisa Lorenzo Garcia, Maike Massierer, Adelina Manzateaunu and Christelle Vincent.
- Elisa Lorenzo-García Rennes, Modular expressions for Shioda invariants
Let C be a genus 3 hyperelliptic curve. It isomorphism class is determined by the so-called Shioda invariants J2,J3,...,J10. By using some previous results and alternative invariants of Tsuyumine, we give a modular expresion (in term of theta constants) for the products Ji*Di where D is the discriminant of the curve C. As as a consequence we present a set of absolute invariants of C as Siegel modular functions. In the special case in which C has CM, we give an easy computable criterium for determining the type of bad reduction of a prime dividing the denominator of any of these absolute invariants (by Kilicer's talk results we already know that such a prime is of bad reduction).
- Anna Somoza Leiden, PhD defense
Seminarium Computer Algebra Nederland / Intercity Number Theory SeminarApril 12, UvA Amsterdam. The first three lectures will be in room C0.110 in Science Park 904. Tim Dokchitser's inaugural lecture will be in the Aula.
- Steffen Löbrich UvA, On cycle integrals of meromorphic modular forms
In joint work in progress with Markus Schwagenscheidt, we study cycle integrals of meromorphic modular forms associated to quadratic forms of negative discriminant. In particular, we relate them to evaluations of locally harmonic Maass forms at CM-points. This allows us to generalize a recent theorem by Alfes-Neumann, Bringmann, and Schwagenscheidt on the rationality of these cycle integrals in several directions.
- Alex Bartel Glasgow, Torsion homology and regulators of isospectral manifolds
It is well known by now that two drums that sound the same need not look the same. But how different can they really look? There are two general constructions of such pairs of "drums", a representation theoretic one due to Sunada, and a number theoretic one due to Vigneras. After introducing the general setting and recalling the two constructions, I will discuss what can be said about the homology groups of such isospectral manifolds, e.g. for what primes p the p-torsion in the homology can differ. The answers in the two settings look very different... until one looks more closely! This is joint work with Aurel Page, and, fittingly for the occasion, it was originally inspired by the work of Tim and Vladimir Dokchitser on the Birch and Swinnerton-Dyer conjecture.
- Maarten Derickx MIT, Modularity of elliptic curves over totally real cubic fields
The key ingredient in Wiles' proof of Fermat's last theorem is his proof that all semistable elliptic curves over Q are modular. Wiles' techniques were later extended by Breuil, Conrad, Diamond and Taylor to prove modularity of all elliptic curves over Q. Using improvements in modularity lifting techniques Freitas, Le Hung and Siksek later proved that all elliptic curves over real quadratic fields are modular. In this talk I will discuss how the techniques for real quadratic fields together with new modularity lifting results due to Thorne and to Kalyanswamy can be used to prove modularity of elliptic curves over totally real cubic fields.
- Tim Dokchitser Bristol / UvA, Inaugural lecture
Intercity Number Theory SeminarMay 10, Leiden. The first talk will take place in room B3 of the Snellius building. The afternoon talks take place in room 412 and will be followed by the awarding of the Compositio Prize 2014-2016 to James Maynard, and a reception.
- Guido Lido Leiden, Roma, Computations in the Poincaré torsor and the quadratic Chabauty method
Joint work with Bas Edixhoven. Faltings's theorem states that a curve C of genus g>1 defined over the rationals has only finitely many rational points. In practice there is no general procedure to provably compute the set C(Q). When the rank of the Mordell-Weil group J(Q) (with J the Jacobian of C) is smaller than g we can use Chabauty's method, i.e. we can embed C in J and, after choosing a prime p, we can view C(Q) as a subset of the intersection of C(Qp) and the closure of J(Q) inside the p-adic manifold J(Qp); this intersection is finite and computable up to finite precision. Minhyong Kim has generalized this method inspecting (possibly non-abelian) quotients of the fundamental group of C. His ideas have been made effective in some new cases by Balakrishnan, Dogra, Muller, Tuitman and Vonk: their "quadratic Chabauty method" works when the rank of the Mordell-Weil group is strictly less than g+s-1 (with s the rank of the Neron-Severi group of J). In this lecture we will give a reinterpretation of the quadratic Chabauty method, only using the Poincaré torsor of J and a little of formal geometry, and we will show how to make it effective.
- Damaris Schindler Utrecht, On prime values of binary quadratic forms with a thin variable
A result of Fouvry and Iwaniec states that there are infinitely many primes of the form x^2+y^2 where y is a prime number. In this talk we will see a generalization of this theorem to the situation of an arbitrary primitive positive definite binary quadratic form. This is joint work with Peter Cho-Ho Lam and Stanley Xiao.
- Peter Koymans Leiden, The spin of prime ideals and applications
Let K be a cyclic, totally real extension of Q of degree at least 3, and let σ be a generator of Gal(K/Q). We further assume that the totally positive units are exactly the squares of units. In this case, Friedlander, Iwaniec, Mazur and Rubin define the spin of an odd principal ideal a to be spin(σ, a) = (α/σ(α))K, where α is a totally positive generator of a and (*/*) is the quadratic residue symbol in K. Friedlander, Iwaniec, Mazur and Rubin prove equidistribution of spin(σ, p) as p varies over the odd principal prime ideals of K. In this talk I will show how to extend their work to more general fields. I will then give various arithmetic applications.
This is joint work with Djordjo Milovic.
- James Maynard Oxford, Dense clusters of primes in subsets
We discuss how one can generalize weak versions of the prime k-tuples conjecture to apply to arbitrary well-distributed sets of integers and primes, with good uniformity in the results. This has several consequences for large gaps between primes, strings of congruent primes, and many primes in short intervals.
Intercity Number Theory SeminarSeptember 6, Nijmegen. The morning talks will be held in HG00.307, the informal discussions in leg HG03.07 (3rd floor), and the afternoon talk in room HG00.062.
- Riccardo Pengo Copenhagen, Mahler’s measure and elliptic curves with complex multiplication
Elliptic curves with complex multiplication have historically formed a fertile test ground for many conjectures on the arithmetic of elliptic curves. In this talk, we will explore one instance of this phenomenon by looking at the conjectures relating special values of L-functions to the Mahler measure of polynomials in multiple variables. These conjectures, initiated by the work of Boyd on Lehmer’s problem, can be approached for elliptic curves with complex multiplication using the proof of the conjectures of Bloch and Kato for Hecke characters, due to Deninger and Kings.
- Alina Ostafe Sydney, On some unlikely intersections for values and orbits of rational functions
For given rational functions f1,dots,fs defined over a number field K, Bombieri, Masser and Zannier (1999) proved that the algebraic numbers α for which the values f1(α),dots,fs(α) are multiplicatively dependent are of bounded height (unless this is false for an obvious reason).
Motivated by this, we present recent finiteness results on multiplicative relations of values of rational functions at arguments restricted to the maximal abelian extension of K. We go even further and discuss the presence of multiplicative relations modulo finitely generated groups, posing some open questions. If time allows, we will present some finiteness results regarding the presence of powers of S-integers in orbits of polynomial dynamical systems.
- , informal discussions at math department
- Berend Ringeling Utrecht, Zeros of modular forms and congruences
For p a prime larger than 7, the Eisenstein series of weight p-1 has some remarkable congruence properties modulo p. Those imply, for example, that the j-invariants of its zeros (which are known to be real algebraic numbers in the interval [0,1728]), are at most quadratic over the field with p elements and are congruent modulo p to the zeros of a certain truncated hypergeometric series. In my thesis, I introduce the "theta modular form" of weight k as the unique modular form of weight k, for which the first dimMk Fourier coefficients are identical to those of the Jacobi theta series. Theta modular forms modulo p relate to the average weight enumerators in coding theory. In my talk, I will show that theta modular forms of weight (p+1)/2 behave in many ways like Eisenstein series: the j-invariants of their zeros all belong to the interval [0,1728], are modulo p all in the ground field with p elements, and are congruent modulo p to the zeros of a truncated hypergeometric sum.
Intercity Number Theory SeminarOctober 18, Utrecht. The morning lecture is in Minnaert 201, the first two afternoon lectures In Koningsberger Atlas, and the final lecture in Buys Ballot 001 (all these buildings are connected internally)
- Andrew Schopieray Sydney/Berkeley, Quadratic d-numbers
In the context of conformal field theory, Moore and Seiberg claimed the study of modular tensor categories "should be viewed as a generalization of group theory". In this analogy the order of the group is the category's dimension, now taking values outside the natural numbers, which begs the question: what is the set of possible dimensions? This question is almost entirely open. In this talk we focus on the number-theoretic condition that dimensions generate Galois-invariant ideals in the ring of algebraic integers (so-called "algebraic d-numbers"). We will show how the simplicity of the unit groups of quadratic extensions of the rational numbers allows us to constructively classify all quadratic d-numbers, and provide a discrete list of potential dimensions. Discussion of higher extensions and other generalizations will follow with a hope to spark interest in these rare and valuable algebraic integers.
- Andrew Bridy Yale, The arboreal finite index problem
Let K be a global field, f in K[x], and b in K. Let Kn be the splitting field of fn(x)−b, where fn denotes n-fold composition. The projective limit of the groups Gal(Kn/K) embeds into the automorphism group of an infinite rooted tree. A major problem in arithmetic dynamics, motivated by Serre's open image theorem, is to determine when the index is finite. I discuss the solution of the problem for K a function field and f a low degree polynomial by classifying all obstructions to finite index. When K is a number field the solutions are conditional on deep conjectures from diophantine geometry.
- Jakub Byszewski Krakow, Automatic sequences: structure and randomness
Automatic sequences (generated by finite automata) constitute one of the basic complexity classes and arise naturally in various contexts in mathematics and computer science. In the talk we will study their uniformity. We show that any automatic sequence can be separated into a structured part and a pseudorandom part. The structured part is described explicitly (in particular, for sequences produced by strongly connected automata, it is rational almost periodic); the pseudorandom part has exponentially decaying Gowers uniformity norms of all orders, and so, roughly speaking, does not correlate with nilsequences. This provides explicit examples of Gowers-uniform sequences. As an application, we obtain asymptotic counts of the number of solutions of linear equations in automatic sets. The talk is based on joint work with Jakub Konieczny (Jerusalem) and Clemens Müllner (Vienna).
- Vandita Patel Manchester, A Galois property of even degree Bernoulli polynomials
Let k be an even integer such that k is at least 2. We give a (natural) density result to show that for almost all d at least 2, the equation (x+1)k + (x+2)k + ... + (x+d)k = yn with n at least 2, has no integer solutions (x,y,n). The proof relies upon some Galois theory and group theory, whereby we deduce some interesting properties of the Bernoulli polynomials. This is joint work with Samir Siksek (University of Warwick).
Intercity Number TheoryNovember 1, Leiden. Room 312 of the Snellius building
- Martin Bright Leiden, A walk on the wild side: p-torsion in the Brauer group
The Brauer group of a variety over a number field is a powerful tool for studying failures of the Hasse principle. To apply this, one needs to understand the function obtained by evaluating a given element of the Brauer group at the p-adic points of the variety. For elements of order coprime to p, this evaluation map factors through reduction modulo p. For elements of order p, the situation is much more intricate: one may need to look at the points to higher p-adic precision. We relate the resulting filtration on the Brauer group to one defined by Bloch–Kato, and show that Kato's refined Swan conductor controls the local behaviour of the evaluation map. This is joint work with Rachel Newton.
- Leila Schneps Jussieu, Paris, Grothendieck-Teichmüller theory, a crossroads between geometry and number theory.
Grothendieck-Teichmüller theory was originated by Alexander Grothendieck as a way to study the absolute Galois group of the rationals by considering its action on fundamental groups of varieties, in particular of moduli spaces of curves with marked points: the special properties of the Galois action with respect to inertia generators and the fact of respecting the relations in the fundamental group gave rise to the definition of the group GT which contains GQ.
The group GT is profinite, but its defining relations can also be used to give a pro-unipotent avatar, and an associated graded Lie algebra grt. The study of the Lie algebra grt reveals many unexpected relations with number theory that are completely invisible in the profinite situation. We will show how Bernoulli numbers, cusp forms on SL2(Z) and multiple zeta values arise in the Lie algebra context.
- Pierre Lochak Jussieu, Paris, A topological version of Grothendieck-Teichmüller theory.
After recalling some basic features of G.-T. theory, mainly with a view to emphasize its "nonlinear" versus "linear" aspects, I will take the topological (and nonlinear) path, introducing in particular the profinite completions of various simplicial complexes (curve an arc complexes mainly, as well as some closely related graphs) which feature far-reaching generalizations of the so-called "dessins d'enfants" and enable one to build a topological version of G.-T. (valid in every genus) which may (or may not) be rather close to the blueprint appearing in Grothendieck's famous Esquisse d'un programme.
- Igor Shparlinski Sydney, Integers of prescribed arithmetic structure in residue classes
We give an overview of recent results about the distribution some special integers in residues classes modulo a large integer q. Questions of this type were introduced by Erdos, Odlyzko and Sarkozy (1987), who considered products of two primes as a relaxation of the classical question about the distribution of primes in residue classes. Since that time, numerous variations have appeared for different sequences of integers. The types of numbers we discuss include smooth, square-free, square-full and almost primes integers. We also expose the wealth of different techniques behind these results: sieve methods, bounds of short Kloosterman sums, bounds of short character sums and many others.
Belgian Dutch Algebraic Geometry DayNovember 8, Antwerpen.
DIAMANT SymposiumNovember 29, De Bilt. The is part of a two-day event, November 28-29.
Intercity Number Theory SeminarDecember 13, UvA and VU Amsterdam. At the VU: before lunch in room WN–S623 and after lunch in room WN–P647 (both in the W&N building).
- Ilke Canakci VU, Cluster algebras and continued fractions
I will present a combinatorial realisation of continued fractions given as quotients of cardinalities of sets. These sets are given in terms of perfect matchings of certain graphs, snake graphs, which appear in the theory of cluster algebras. Cluster algebras are widely studied worldwide since their introduction in 2002 due to their link to various areas including quiver representations, Teichmuller theory, integrable systems, and knot theory. This realisation of continued fractions recovers classical results in elementary number theory and gives rise to applications to cluster algebras.
- David Hansen MPIM Bonn, Completed cohomology and the p-adic Langlands program
Completed cohomology, as defined by Calegari and Emerton, gives a natural candidate for general spaces of p-adic automorphic forms. I’ll give some motivated introduction to completed cohomology and its role in the p-adic Langlands program. In the latter part of the talk, I’ll explain some new vanishing theorems for completed cohomology. The proofs involve a fun combination of perfectoid methods and more classical Shimura variety techniques, and I’ll try to highlight the main ingredients. This is joint work with Christian Johansson.
- Joey van Langen VU, Automating the modular method for Frey Q-curves
The modular method has effectively been applied to solve a variety of exponential Diophantine equations. Although the procedure is in essence very similar for every case, only implementations for particular Diophantine equations can be found in the literature. In this talk I will describe the mathematical obstacles to generalizing such approaches. The talk will focus on a general algorithm that can be applied to problems for which a Frey Q-curve exists. An implementation of this work in Sage provides a simple way to obtain the results of the modular method for such problems. As an illustration we will use this approach on the Diophantine equation (x - y)4 + x4 + (x + y)4 = zl.
- Lars Kühne University of Basel, The Equidistribution Conjecture for semiabelian varieties
The (Strong) Equidistribution Conjecture for semiabelian varieties yields substantial information on the points of small height on those varieties, including the Manin-Mumford and the Bogomolov Conjecture. Chambert-Loir has settled this conjecture affirmatively in the case of almost split semiabelian varieties. The general case, however, has remained intractable because the canonical height of a semiabelian variety is negative unless it is almost split. In fact, this places the conjecture outside the scope of Yuan's Equidistribution Theorem on algebraic dynamical systems. In my talk, I will outline my recent proof of the (Strong) Equidistribution Conjecture for general semiabelian varieties.