Intercity Number Theory Seminar


Belgian-Dutch Algebraic Geometry seminar

February 1, Utrecht. See website.

Intercity Number Theory Seminar

March 1, Groningen. The first talk takes place in room 165 of the Bernoulliborg and the three other talks take place in room 253.

12:00-13:00Dino Festi (Mainz), A method to compute the geometric Picard lattice of a K3-surface of degree 2
Abstract. 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.
13:45-14:45Florian Hess (Oldenburg), Partially euclidean global fields
15:00-16:00Marc Paul Noordman (Groningen), Algebraic first order differential equations
Abstract. 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.
16:15-17:15Jan Vonk (Oxford), Singular moduli for real quadratic fields
Abstract. 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 Seminar

March 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).

11:00-12:00Efrat Bank (Technion, Israel), Primes in short intervals on curves over finite fields
Abstract. 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.
13:15-14:15 Francesca Balestrieri (MPIM Bonn, Germany), Arithmetic of zero-cycles on products of Kummer varieties and K3 surfaces
Abstract. 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.
14:15-15:15Kęstutis Česnavičius (Orsay, France), Macaulayfication of Noetherian schemes
Abstract. 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.
15:30-16:30Judith Ludwig (Heidelberg, Germany), Perfectoid Shimura varieties and applications
Abstract. 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 Seminar

March 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.

11:00-12:00Christophe Ritzenthaler (Rennes), Reduction of plane quartics
Abstract. 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.
12:45-13:45Pınar Kılıçer (Groningen), Modular invariants for genus-3 hyperelliptic curves
Abstract. 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.
14:00-15:00Elisa Lorenzo-García (Rennes), Modular expressions for Shioda invariants
Abstract. 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).

16:15-17:15Anna Somoza (Leiden), PhD defense

Seminarium Computer Algebra Nederland / Intercity Number Theory Seminar

April 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.

10:15-11:15Steffen Löbrich (UvA), On cycle integrals of meromorphic modular forms
Abstract. 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.
11:30-12:30Alex Bartel (Glasgow), Torsion homology and regulators of isospectral manifolds
Abstract. 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.
13:30-14:30Maarten Derickx (MIT), Modularity of elliptic curves over totally real cubic fields
Abstract. 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.
16:00-17:00Tim Dokchitser (Bristol / UvA), Inaugural lecture

Intercity Number Theory Seminar

May 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.

11:00-12:00Guido Lido (Leiden, Roma), Computations in the Poincaré torsor and the quadratic Chabauty method
Abstract. 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.
13:15-14:15Damaris Schindler (Utrecht), On prime values of binary quadratic forms with a thin variable
Abstract. 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.
14:30-15:30Peter Koymans (Leiden), The spin of prime ideals and applications
Abstract. 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.

15:45-16:45James Maynard (Oxford), Dense clusters of primes in subsets
Abstract. 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.

Belgian-Dutch Algebraic Geometry Day

June 21, UvA Amsterdam. See the website.

Intercity Number Theory Seminar

September 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.

11:00-12:00Riccardo Pengo (Copenhagen), Mahler’s measure and elliptic curves with complex multiplication
Abstract. 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.
12:15-13:15Alina Ostafe (Sydney), On some unlikely intersections for values and orbits of rational functions
Abstract. 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.

14:00-15:15 , informal discussions at math department
15:20-16:20Berend Ringeling (Utrecht), Zeros of modular forms and congruences
Abstract. 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 Seminar

October 18, Utrecht.

Intercity Number Theory

November 1, Leiden.

Belgian Dutch Algebraic Geometry Day

November 8, Antwerpen.

DIAMANT Symposium

November 29, De Bilt. The is part of a two-day event, November 28-29.