Intercity Number Theory Seminar


Intercity number theory seminar

17 March, Utrecht. Pangea lecture room, Koningsbergergebouw
Alisa Sedunova MPIM/Université de Montréal, Binary quadratic forms in prime variables
The number of solutions to a2 + b2 = c2 + d2 < x in integers is a well-known result, while if one restricts all the variables to primes Erdős showed that only the diagonal solutions, namely, the ones with {a, b} = {c, d} contribute to the main term, hence there is a paucity of the off-diagonal solutions. Daniel considered the case of a, c being prime and proved that the main term has both the diagonal and the non-diagonal contributions. We survey the main results in this direction and investigate the remaining cases.
Harald Helfgott CNRS/University of Göttingen, Expansion, divisibility and parity
Expander graphs can be defined in any of several equivalent ways: in terms of boundaries of sets of vertices, or eigenvalues of the adjacency problem, or random walks. Expander graphs have become a central object of study in discrete mathematics (and theoretical computer science). They appear in group theory, combinatorics and also number theory. We will discuss a recent application: one can prove (with difficulty!) that a graph that encodes the divisibility properties of integers is an expander of sorts (a “strong local expander almost everywhere”) – and that fact has several strong, immediate consequences on the statistics of the factorization of integers into primes.
Simon Rydin Myerson University of Warwick, Conic fibrations over elliptic curves
A theorem of Serre states that almost all plane conics over ℚ have no rational point. We prove an analogue of this for a family of conics parametrised by certain elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve. Another way to think about this result is: we show that 0% of points on elliptic curves have a denominator which is a sum of two squares.
Vandita Patel University of Manchester, Values of the Ramanujan tau-function
The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey–Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).

Arithmétique en Plat Pays / Getaltheorie in het Vlakkeland

20 March, Leuven. See website for details.

Belgian/Dutch Algebraic Geometry Seminar

24 March, Nijmegen. See website for details. Please register by 17 March.

Inaugural lecture of Assia Mahboubi

26 April, VU Amsterdam. Mini-symposium in room HG-08A33, followed by the inaugural lecture in the Aula.
Antoine Chambert-Loir Université Paris Cité, Simplicity — Group actions and proof formalization
The theorem that the alternating group in at least 5 letters is a simple group is a cornerstone of many group theory courses, sometimes in connection with Abel’s theorem that the general equation of degree at least n is not solvable by radicals. Willing to join the game of “proof formalization”, I recently wrote a proof of that theorem within the proof assistant Lean and its mathematical library mathlib. The principle of that proof, which is certainly well known to specialists of simple groups theory — but apparently not otherwise — combines a classic criterion of Iwasawa (1941) which is usually only used in geometric contexts and even more classic results of Jordan (1872). I will try to share various aspects of that experiment.
Benedikt Ahrens Delft, Univalent foundations for the formalization of (higher) category theory

Category theory is a mathematical theory of structures and their interactions. It was originally developed to axiomatize algebraic topology; in the meantime, it has proved useful as a language and infrastructure for organizing knowledge in many areas of mathematics, computer science, data science, biology, and more.

The formulation of category theory in set-theoretic foundations has been seen as problematic: the notion of “sameness” between mathematical objects provided by set theory does not coincide with the fundamental notion of isomorphism in category theory. This divergence is exacerbated when considering higher-categorical structures, such as categories themselves.

In this talk, we explain how univalent foundations provides exactly the right setting in which to formalize category theory. In particular, equivalence of categories coincides with “sameness” of categories in the sense of univalent foundations.

Assia Mahboubi VU/Inria, Computer assisted mathematics
Doors open from 15:30; lecture starts at 15:45 sharp. Followed by a reception.

Intercity Number Theory Seminar

9 June, UvA and VU Amsterdam. All lectures will be in the NU building at the VU Campus, room NU–05A47
Casper Putz VU, A case study of generalised Fermat equations
We will discuss a method for attacking generalised Fermat equations. The focus of the talk is its applications to generalised Fermat equations with signature (2,5,7). We give an overview of the method, which involves computation of p-adic étale algebras, Hunter searching, and rational points on algebraic curves. We go more in depth on the computation of rational points, specifically the use of elliptic curve Chabauty.
Eugenia Rosu Leiden, A higher degree Weierstrass function
The Weierstrass P-function plays a great role in the classic theory of complex elliptic curves. A related function, the Weierstrass sigma-function, is used by Guerzhoy to construct preimages under the xi-operator of newforms of weight 2, corresponding to elliptic curves. In this talk, I will discuss a generalization of the Weierstrass sigma-function and an application to harmonic Maass forms. More precisely, I will describe a construction of a preimage of the xi-operator of a newform of weight k for k>2. This is based on joint work with C. Alfes-Neumann, J. Funke and M. Mertens.
Thomas Decru KU Leuven, Breaking the Supersingular Isogeny Diffie-Hellman protocol
Due to the advent of quantum computers, new mathematical hard problems need to be considered for public-key encryption to replace the well-established integer factorisation and discrete logarithm problems. One promising approach is based on finding isogenies between elliptic curves, which is conjectured to be hard, even for a quantum computer. In 2011, Jao and De Feo created a practical Diffie–Hellman style key-exchange based on a weaker variant of this assumption, where additional torsion point information is revealed. The security of the scheme was mostly left unscathed for over a decade, but last summer Wouter Castryck and I managed to break it by using isogenies between abelian surfaces.
Martin Lüdtke Groningen, Chabauty–Kim and the locally geometric section conjecture
Let X be a smooth projective curve of genus at least 2 over the rational numbers. A natural variant of Grothendieck’s Section Conjecture postulates that every section of the fundamental exact sequence for X which everywhere locally comes from a point of X in fact globally comes from a point of X. In this talk, I will explain a new strategy for verifying this conjecture by relating it to Chabauty–Kim calculations. Namely, if X satisfies Kim’s conjecture for almost all choices of auxiliary prime p, then it satisfies the locally geometric section conjecture. After giving the appropriate extension to affine hyperbolic curves, I will explain how we carry out this strategy for the thrice-punctured line over ℤ[1/2]. This is joint work with Alex Betts and Theresa Kumpitsch.

Intercity Number Theory Seminar

30 June, Groningen. Bernoulliborg, room 253
Oliver Lorscheid Groningen, Descartes, Newton and hyperfields

Descartes’ rule of signs bounds the number of positive roots of a polynomial with real coefficients by the number of sign changes in the coefficients, Newton found a bound for the number of roots of a polynomial over a nonarchimedean field in terms of what is nowadays called its Newton polygon.

In this talk, we review these two classical theorems and explain how they follow by the same principle from assertions about the multiplicities of roots for polynomials over the sign and the tropical hyperfield, respectively. This is joint work with Matthew Baker.

Milena Wrobel Oldenburg, The anticanonical complex for non-degenerate toric complete intersections
The anticanonical complex has been introduced as a combinatorial tool, extending the correspondence between toric Fano varieties and Fano polytopes and has so far been developed for certain classes of Fano varieties. In this talk, we introduce a general construction for the anticanonical complex, or more general the anticanonical region, which also leads to variants for non-Fano varieties. We explicitly work out the structure of the anticanonical complex in the case of non-degenerated toric complete intersections and present classification results.
Nils Bruin Simon Fraser, Arithmetic properties of some low-level quartic modular threefolds
Two Siegel modular threefolds of low level allow for quartic birational models in projective space: A2(2) is birational to the Igusa (also called Castelnuovo–Richmond) quartic and A2(3) is birational to the Burkhardt quartic.

These models have some beautiful geometric properties that have been studied extensively. Their arithmetic is equally interesting. For instance, for the Burkhardt quartic one can ask which of its twists are rational or unirational, and also what moduli interpretation they admit. For Igusa quartics there are interesting questions concerning their moduli interpretations as well. We will give an overview of the various results one can obtain.

This talk is based on joint works with Brett Nasserden, Eugene Filatov, and Avinash Kulkarni.

Emre Sertöz Hannover/Leiden, Limit periods, arithmetic, and combinatorics
With Spencer Bloch and Robin de Jong, we recently proved that in a nodal degeneration of smooth curves, the periods of the resulting limit mixed Hodge structure (LMHS) contain arithmetic information. More precisely, if the nodal fiber is identified with a smooth curve C glued at two points p and q then the LMHS relates to the Neron–Tate height of p−q in the Jacobian of C. In making this relation precise, we observed that a “tropical correction term” is required that is based on the degenerate fiber. In this talk, I will explain this circle of ideas with the goal of arriving at the tropical correction term.

Intercity Number Theory Seminar

22 September, Utrecht. Minnaert building, room 2.02
Ekin Özman Boğaziçi Üniversitesi, Local-global trace relations and their implications
Let E be an elliptic curve defined over the rational numbers and K be a quadratic number field. In this talk we will explore the necessary and sufficient conditions for local-global trace obstructions of the trace map from E(K) to E(ℚ). Then we will mention some statistical results and heuristics implied by these observations. This is joint work with Mirela Çiperiani.
Ziyang Gao Leibniz Universität Hannover, Degeneracy loci in families of abelian varieties and their applications
Given an abelian scheme in characteristic zero and an irreducible subvariety X, one can define the t-th degeneracy locus of X for each integer t. This geometric concept of degeneracy loci has recently seen many applications in Diophantine Geometry, notably when t is 0 or 1, in the recent developments on the uniformity of the number of rational points on curves, on the solutions of the Uniform Mordell–Lang Conjecture and of the Relative Manin–Mumford Conjecture. In this talk, I will define the degeneracy loci in the universal abelian variety, and explain how they are used in the applications mentioned above.
Sandro Bettin Università di Genova, Approximations by signed harmonic sums
We consider the problem of approximating a fixed real number a by sums of the shape ∑n ≤ Nc(n)/n with c(n) ∈ {±1}, as N goes to infinity. A possible approach is that of choosing the c(n) greedily as in Riemann’s rearrangement theorem. We discuss the rate of convergence to a using this strategy and highlight a relationship with the Thue–Morse sequence. We also discuss an alternative approach using probabilistic methods. This is joint work with Giuseppe Molteni and Carlo Sanna.
Sam Chow Warwick University, Why dispersion beats discrepancy
I will discuss some problems related to Littlewood’s conjecture in Diophantine approximation, and the role hitherto played by discrepancy theory. I’ll explain why our new dispersion-theoretic approach should, and does, deliver stronger results. Our dispersion estimate is proved using Poisson summation and diophantine inequalities. This is joint work with Niclas Technau.

Getaltheorie in het Vlakkeland – Arithmétique en Plat Pays

9 October, Gent. See web site for details.

Intercity Number Theory Seminar

17 November, Leiden. Snellius building, room 401 (talks 1 and 4) and 405 (talks 2 and 3)
Alex Bartel Glasgow, Isospectral manifolds from orders in division algebras
Kac’s catchy question “can you hear the shape of a drum” has surprisingly many points of contact with number theory. Like number fields, Riemannian manifolds have zeta functions, and two manifolds “sound the same” if and only if they have equal zeta functions, exhibiting a parallel with arithmetically equivalent number fields. A completely different connection with number theory is an arithmetic construction of pairs of manifolds that “sound the same”. The construction is originally due to Vigneras and dates back to 1980, but there are still many open questions surrounding it, chiefly among them: when does it actually do what it promises? I will report on progress, jointly with Aurel Page, on several of those open questions. I will not assume that you already care about, or, for that matter, have encountered isospectral manifolds.
David Lilienfeldt Leiden, The Gross–Zagier formula for generalized Heegner cycles
In the 1980s, Gross and Zagier famously proved a formula equating on the one hand the central value of the first derivative of the Rankin–Selberg convolution L-function of a weight 2 eigenform with the theta series of a class group character of an imaginary quadratic field, and on the other hand the height of a Heegner point on the corresponding modular curve. This equality was a key ingredient in the proof of the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rationals in analytic rank 0 and 1. Two important generalizations present themselves: to allow eigenforms of higher weight, and to further allow Hecke characters of infinite order. The former generalization is due to Shou-Wu Zhang. The latter one is the subject of this talk and requires the calculation of the Beilinson–Bloch heights of generalized Heegner cycles. This is joint work with Ari Shnidman.
Sarah Arpin Leiden, The scheme of monogenic generators
Given an extension of algebras B/A, when is B generated by a single element θ ∈ B over A? We show there is a scheme MB/A parameterizing the choice of a generator θ ∈ B, a “moduli space” of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. Several notions of local monogenicity emerge from this perspective, which we consider in particular in the case when B/A is étale. We give explicit equations and ample examples. This is joint work with Sebastian Bozlee, Leo Herr, and Hanson Smith.
Adam Morgan Glasgow, Hasse principle for Kummer varieties in the case of generic 2-torsion
Conditional on finiteness of relevant Shafarevich–Tate groups, Harpaz and Skorobogatov established the Hasse principle for Kummer varieties associated to a 2-covering of a principally polarised abelian variety A, under certain large image assumptions on the Galois action on A[2]. However, their method stops short of treating the case where the image is the full symplectic group, due to the possible failure of the Shafarevich–Tate group to have square order in this setting. I will explain recent work which overcomes this obstruction by combining second descent ideas of Harpaz with new results on the 2-parity conjecture.

Compositio Prize Symposium

24 November, UvA Amsterdam. Speakers: Wushi Goldring, Maarten Solleveld, Benjamin Schraen, Pol van Hoften, Guy Henniart. More information on the website.

Intercity Number Theory Seminar

15 December, UvA and VU Amsterdam. Location: Lab42, room L1.02, Science Park Amsterdam. In collaboration with Computer Algebra Nederland. The Schoonschip Prize will be awarded before the final talk and there will be drinks afterwards.
Assia Mahboubi VU/Inria, Nantes, Proof transfer for free, with or without univalence

Libraries of formalized mathematics use a possibly broad range of different representations for a same mathematical concept. Yet light to major manual input from users remains most often required for obtaining the corresponding variants of theorems, when such obvious replacements are typically left implicit on paper. This input represents a significant part of the bureaucratic work plaguing the activity of formalizing mathematics. In this talk, we propose a novel framework for proof transfer for proof assistants based on dependent type theory, such as Lean, Coq or Agda. This approach relies on original results in type theory (*), as well as on the design and implementation of an automation device, which puts them at work on concrete problems.

This talk will strive to assume little familiarity from the audience with dependent type theory, explaining in particular all the words in the title, and trying to illustrate how properties of the foundations backing a proof assistant may impact the practice of formalizing mathematics.

This is a joint work with Enzo Crance and Cyril Cohen.

(*) for a knowledgeable audience: we propose a novel formulation of type equivalence, used to generalize Sozeau–Tanter–Tabareau’s univalent parametricity translation.

Johan Commelin Utrecht, Condensed Type Theory

Condensed sets form a topos, and hence admit an internal type theory. In this talk I will describe a list of axioms satisfied by this particular type theory. In particular, we will see two predicates on types, that single out a class CHaus of “compact Hausdorff” types and a class ODisc of “overt and discrete” types, respectively. A handful of axioms describe how theses classes interact. The resulting type theory is spiritually related Taylor’s “Abstract Stone Duality”.

As an application I will explain that ODisc is naturally a category, and furthermore, every function ODisc → ODisc is automatically functorial. I will explain what this result means externally, in the language of condensed sets. If time permits, I will comment on some of the techniques that go into the proof.

Joint work with Reid Barton.

Tim Dokchitser Bristol, A classification for reduction types of curves
The primary invariant for a family of curves is the combinatorial description of 'bad' fibers. When the curves are elliptic, the classification of possible geometric configurations ('reduction types') is due to Kodaira and Neron, in genus 2 to Namikawa-Ueno, and in genus 3 to Ashikaga-Ishizaka. In this talk, I would like to describe a possible classification for curves of arbitrary genus.
Joey van Langen FIQAS, Applying the modular method in a single button press, a step towards the dream
Ever since the connection between modular forms and elliptic curves was proven to solve Fermat's Last Theorem, the same technique has repeatedly been used to solve other exponential Diophantine equations. This technique, now known as the modular method, can be a powerful tool to prove the non-existence of solutions or at least bound the set of exponents for which solutions could exist. Until thus far, various steps required to apply the modular method had to be performed manually. In this talk I will go over the steps of this procedure I have managed to automate using the computer algebra package Sage. These automations include computing the conductor of a Frey curve, and calculating invariants of Q-curves.