In this talk we determine the average (arithmetic) Mordell--Weil rank of elliptic surfaces over number fields, hereby proving a conjecture of Alex Cowan.

For this we need to study the behaviour of the Néron-Severi group in families of surfaces. Over the complex numbers this can be done using Noether-Lefschetz theory. However, Noether--Lefschetz type results typically give rise to countably many exceptional loci, and therefore are of little use over countable fields, like $\overline{\mathbb{Q}}$.

Results by André and by Maulik-Poonen give a preciser description of the jump loci of the Picard number than Noether-Lefschetz theory. This allows us to study the geometric Mordell-Weil rank in families of elliptic surfaces. We then use a variant of Hilbert's irreducibility theorem to determine the average arithmetic Mordell-Weil rank.

In this talk we probe the space of automorphic forms for PGL3 over elliptic function fields. In particular, we show how to describe the action of Hecke operators on this space and how we can use it to investigate periodical conditions. This is about a joint work (in progress) with O. Lorscheid and V. Pereira Jr.

Finding an explicit isogeny between two given isogenous elliptic curves over a finite field is considered a hard problem, even for quantum computers. In 2011 this led Jao and De Feo to propose a key exchange protocol that became known as SIDH, short for Supersingular Isogeny Diffie-Hellman. The security of SIDH does not rely on a pure isogeny problem, due to certain "auxiliary" elliptic curve points that are exchanged during the protocol (for constructive reasons).

In 2017 SIDH was submitted to the NIST standardization effort for post-quantum cryptography, and since then it has attracted a lot of attention. Early July, it advanced to the fourth round. In this talk I will discuss a break of SIDH that was discovered in collaboration with Thomas Decru about three weeks later. The attack uses isogenies between abelian surfaces and exploits the aforementioned auxiliary points, so it does not break the pure isogeny problem. It allows for a full key recovery at the highest security level in a few hours. As time permits, I will also discuss some more recent improvements and follow-up work due to Maino-Martindale, Wesolowski, and Robert.

• 11:00, room 293: Martin Lüdtke (Groningen): Non-abelian Chabauty for the thrice-punctured line
• 11:30, room 293: Philipp Schläger (Oldenburg): The anticanonical complex of 3-dimensional Fano varieties with (\C^*)^2-action
• 12:00, room 293: Manoel Zanoelo Jarra (Groningen): Linear algebra, matroids and geometry
• 12:30: lunch
• 14:00, room 253: Aline Bartel (Oldenburg): Towards a classification of simple non-isolated CMC2 singularities
• 14:30, room 253: Maicom Varella (Oldenburg): Local Euler obstruction for essentially isolated determinantal singularities
• 15:00, room 253: Sabrina Alexandra Gaube (Oldenburg): Algorithmic strategies for resolution of determinantal singularities
• 15:30: coffee
• 16:00, room 253: Manoy Trip (Groningen): A naive p-adic height on Jacobians of genus 2 curves
• 16:30, room 253: Konstantin Meiwald (Oldenburg): Gluing curves from irreducible components
• 17:00, room 253: Sven Bootsma (Groningen): An Elliptic Surface in Characteristic p>0

We construct a one dimensional analytic space over a p-adic field. A discrete co-compact subgroup of the unitary group in three variables acts discontinuously on this space. The quotients of the space by the discrete co-compact group are algebraic curves. The construction uses the building of the unitary group, which is a tree. The components of the reduction of the space correspond to the vertices of the tree. These components are hermitian curves and projective lines.

Abstract: In this talk I will explain the concept of an (n,n)-split Jacobian of a genus-2 curve. A full description for n=2 is classically known, dating back as far as Legendre and Jacobi. The case n=3 was treated by Kuhn in the 80s, but misses some important nuances that I will clarify; this involves some very interesting families of curves that have not been properly analysed so far. Time permitting, I will talk about gluing curves along the n-torsion and present some results for n=3 and n=4.

Abstract: We consider certain families of sextic twists of the elliptic curve y^2=x^3+1 that are not defined over Q, but over Q[sqrt(-3)]. We compute a formula that relates the central value of their L-functions L(E, 1) to the square of a trace of a modular function evaluated at a CM point. When the value above is non-zero, we recover the order of the Tate-Shafarevich group, and we show that the value is indeed an integer square.

Abstract: In this talk, I give an outline of a new method to study matroid representations with an algebraic tool that we call the "foundation of a matroid". The emphasis is on the ideas en large rather than on specific results.

We begin with a gentle introduction to matroids without and with coefficients (the latter theory goes by the name of Baker-Bowler theory). The story continues with moduli spaces of matroids and their residue fields. The foundation of a matroid appears as a variation on residue fields, which knows about the representation theory of the matroid and which is accessible to explicit computations.

In the final part of the talk, we discuss how structural insights into foundations can be turned into results about matroid representations. All of this is joint work with Matthew Baker.

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In the informal introduction before the research talk, I give a classical introduction to matroids to complement the rather algebraically minded viewpoint of Baker-Bowler theory.

Abstract: Matroids encode the combinatorics of independency in linear subspaces. In analogy, flag matroids encode the combinatorics of flags of subspaces. There is a way back: when representing a (flag) matroid over a field, we get a (flag of) linear subspace(s). In recent years, Baker and Bowler generalized this picture in the case of matroids to a more general type of algebraic object that includes fields, hyperfields and partial fields as particular cases. In this talk we explain how to extend this theory to flag matroids, including a geometric interpretation in terms of their moduli spaces. This is a joint work with Oliver Lorscheid.

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In the introductory talk we discuss the Plücker relations of Grassmannians and flag varieties.

Abstract: A smooth hyperbolic curve X/Q is known to have only finitely many S-integral points, for any finite set of primes S, but finding these is in general very difficult. The non-abelian Chabauty method is conjectured to achieve this by producing functions on the p-adic points X(Z_p) which cut out precisely the set of S-integral points. This is known as Kim's conjecture. I present a proof that Kim's conjecture holds for the thrice-punctured line if S is the empty set and p is arbitrary. For the proof it is necessary to go beyond linear and quadratic Chabauty. I shall explain how calculations in higher depth become accessible by working with Corwin--Dan-Cohen's motivic Selmer scheme.

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In the introductory talk I will discuss torsors and explain how G-equivariant torsors are parametrized by non-abelian H^1.

Abstract:Quantum topology studies low-dimensional topology via invariants produced with the auxiliary data of an algebraic gadget. In this talk I will focus on two such invariants for knots: one is the "universal tangle invariant" Z_D, which uses a particular topological (ribbon) Hopf algebra D depending on some parameter epsilon; and the other is the "Kontsevich invariant", which uses some formal power series as input. It is a conjecture by Bar-Natan and van der Veen that the second term in the expansion in epsilon of Z_D of a knot is tantamount to the second term in the so-called "loop expansion" of the Kontsevich invariant of the knot. I will outline recent work showing that the conjecture is true for the class of knots that bound a compact, connected, orientable surface of genus one.

In 1967 Shafarevich and Tate published a three and a half page short note, mainly based on Tate's 1966 proof of what is now called "the Tate conjecture for abelian varieties over finite fields". The note proves that in any characteristic p>2 elliptic curves over the rational function field of the algebraic closure of a finite field exist, with arbitrary large Mordell-Weil rank. Elkies (and others) extended this to characteristic two; a key ingredient in both cases is quadratic twists over the function field of a hyperelliptic curve. In the talk it is discussed how analogous results involving so-called quartic twists and sextic twists are obtained.

In the introductory part, I will introduce multiple zeta values, and discuss some of the motivation, main results and conjectures in this area. I will also briefly explain how the period polynomials of cusp forms give rise to (surprising) relations on double zeta values, as shown by Gangl, Kaneko and Zagier.

In the research part, I will explain the block filtration on motivic multiple zeta values (defined by Keilthy, following work of Brown, and some ideas I introduced earlier). I will then discuss a recent project with Keilthy where we were able to understand the structure in block degree 2 by evaluating ζ(2, ..., 2, 4, 2, ..., 2) in terms of double zeta values, and where we showed how the famous period polynomial relations for double zeta values arise in an explicit way from the so-called block relations introduced in Keilthy’s thesis.

Abstract: Given a smooth projective surface, we look for an upper bound for the regulator, i.e. the determinant of the intersection pairing on the Néron-Severi group, in terms of the geometric genus of the surface. While this seems out of reach over a general field, we show that, when the field of definition of the surface is finite, such a bound can be attained via the Weil zeta function of the surface, conditional to a well known conjecture of Tate. Precisely, if the surface is defined over a field of cardinality $q$, we obtain, under some numerical constraints on the invariants of the surface, that the logarithm of the regulator is (essentially) bounded by a constant times the geometric genus, where the constant can be any number $>\log q$. This bound comes together with a bound for the cardinality of the Brauer group of the surface. This bound is essentially optimal, as shown by examples of Griffon (2018). When considering higher dimensional varieties, we also investigate the determinant of the intersection pairing on the groups of algebraic cycles modulo numerical equivalence; in this general case, results are even more conditional.

In a 1990 paper, Grant derives equations for the Jacobian $J$ of a genus 2 curve $C$ and its group law. If the curve is defined over $\mathbb{Z}_p$, the kernel of reduction of the Jacobian modulo $p$ is the group attached to a formal group law of dimension 2, for which Grant describes explicit parameters.

We implement Grant’s formal group law and discuss applications to explicit $p$-adic integration and $p$-adic heights on $C$ and $J$ (no assumptions on the reduction). This essentially gives an explicit version (in the genus 2 setting) of some of Colmez’s theory of $p$-adic integration and heights.

In the introductory talk, I will discuss formal group laws (and if time permits, other preliminaries).

We show that the 68 independent sections on the elliptic surface over $\mathbb{C}$ with affine equation $ y^2 = x^3 + t^{360} + 1$ all exist over $\mathbb{F}_p$ after a reduction modulo $p = 44460001$. Moreover, we show that the corresponding Mordell-Weil group has rank 68 over $\mathbb{F}_p$. Using the Tate conjecture for abelian varieties over finite fields we prove that the family of elliptic surfaces over $\mathbb{F}_p$ with affine equation $y^2 = x^3 + t^{p+1} + 1$ has Mordell-Weil rank $p - 1$.

The Jacobian $J_0(N)$ of the modular curve $X_0(N)$ has received much attention within arithmetic geometry for its relation with cusp forms and elliptic curves. In particular, the group of $\mathbb{Q}$-rational points on $X_0(N)$ controls the cyclic $N$-isogenies of elliptic curves. A conjecture of Ogg predicted that, for $N$ prime, the torsion of this group comes all from the cusps. The statement was proved by Mazur and later generalised to arbitrary level $N$ into what we call generalised Ogg’s conjecture.

Consider now the generalised Jacobian $J_0(N)\mathfrak{m}$ with respect to a modulus $\mathfrak{m}$. This algebraic group also seems to be related to the arithmetic of $X_0(N)$ through the theory of modular forms. In the talk we will present new results that compute the $\mathbb{Q}$-rational torsion of $J_0(N)$ for $N$ an odd integer with respect to a cuspidal modulus $\mathfrak{m}$. These generalise previous results of Yamazaki, Yang and Wei. In doing so, we will also discuss how our results relate to generalised Ogg’s conjecture.

For a separable binary form of degree n over a complete non-archimedean field, there is a canonical metric tree on n leaves associated to it. Furthermore, for every degree n, there are only finitely many tree types, which gives rise to a partition of the space of all non-archimedean binary forms of degree n. In this talk, we will focus on the particular case where n = 5. Then, there are exactly 3 unmarked and 5 marked tree types. We will give a set of tropical invariants, valuations of certain elements coming from invariant theory for binary quintics, and show that these invariants allow us to distinguish the tree types algorithmically. As an application, we will express the reduction types of Picard curves in terms of tropical invariants of the associated binary quintics. This is joint work with Yassine El Maazouz and Paul Alexander Helminck.

The Mahler measure of polynomials is ubiquitous in mathematics: it appears naturally and frequently in algebraic number theory, analysis, dynamical systems, and even statistical physics and electrical engineering. In the first part of the talk I will discuss the definition and basic properties of the Mahler measure of polynomials in one and several variables. In the second part of the talk I will present a couple of problems from my work which I hope are of some interest to algebraic / algebraic geometry specialists.

Given an order in a number field, or, more generally, in an étale algebra over $\mathbb{Q}$, we will study its (Cohen-Macaulay) type. We give two applications about abelian varieties over finite fields, with commutative endomorphism algebra, which are ordinary or over a prime field. The first is a sufficient condition, which involves the type of $\mathrm{End}(A)$, for $A$ not being isomorphic to its dual. In particular, such an $A$ cannot be principally polarised or a Jacobian. As a second application, we show that the group of points of $A$ is uniquely determined by $\mathrm{End}(A)$ if the type of $\mathrm{End}(A)$ is $< 3$.

Sequence A011784 in the Online Encyclopedia of Integer Sequences OEIS

got attention not only from Lionel Levine who introduced it, but also from, e.g., Richard Guy, Colin L. Mallows, Bjorn Poonen, Eric Rains, and others. "Numberphile" in this video presents the definition of the sequence.

The talk discusses the (unique) invertible, differentiable function $h\colon [0,1] \to [0,1]$ with the following property: If one rotates the graph of h clockwise by 90 degrees, one obtains the graph of a positive constant times the derivative of h. Particularly, why such h exists, how to construct it, and how it relates to A011784 will be explained.