Rational Points on Schiermonnikoog

Artin conjectured around 1930 that a rational form of degree d in more than d² variables has nontrivial zeros in all p-adic fields. This turned out to be not quite true, but has been confirmed in important special cases, for example for diagonal forms. We propose a diagonalisation approach and then describe joint work with Robert that establishes Artin’s conjecture for forms that can be obtained by restricting a diagonal form to a hyperplane.

The first step in checking computationally whether a variety over the rationals has a rational zero is to check whether it has a p-adic point for all primes p. In recent work with Daniel Loughran we studied the typical behavior of the primes for which there is no p-adic point by studying their cardinality. In upcoming work we describe the typical behaviour of such primes by studying their location. The new input comes from Brownian motion.

The circle method over F_q(t) allowed Pankaj Vishe and I to calculate the dimension and irreducibility of the moduli space of rational curves on any smooth projective hypersurface of low enough degree. In this talk I will report on joint work with Will Sawin, in which we go one step further and exploit a relationship with a particular system of two equations that allows us to bound the dimension of its singular locus.

In 2014, Skorobogatov and Ieronymou asked if odd-torsion classes in the Brauer group of a K3 surface over a number field could obstruct the existence of rational points. Corn and Nakahara answered this question affirmatively with a 3-torsion algebraic class on a K3 surface of degree 2. We show that transcendental 3-torsion classes arising from a cubic fourfold containing a sextic elliptic surface can also obstruct the existence of rational points. Our approach does not require explicit cyclic Azumaya representatives of the class; it is instead a purely geometric argument. The 3-torsion classes we construct via a geometric correspondence fit into framework that describes level structures of small level on low-degree K3 surfaces. This is joint work with Jennifer Berg.

The following is joint work with Rachel Newton. Let k be a number field. Building on results of Yongqi Liang and, more recently, of Evis Ieronymou and of our own, we relate, via the Brauer–Manin obstruction, the arithmetic of rational points over finite extensions of k to that of zero-cycles over k for any variety X= Πⁿ_i=1 X_i that is a finite product of Kummer varieties, K3 surfaces, and smooth projective geometrically rationally connected varieties over k. In particular, we show that if the Brauer–Manin obstruction is the only obstruction to the existence of rational points on X_i over all finite extensions of k for each i=1,…, n, then the Brauer–Manin obstruction is the only obstruction to the existence of a zero-cycle of degree 1 on X.

Let X be a projective variety over Q. Manin used the Brauer group of X to develop the Brauer–Manin obstruction, which can explain the absence of rational points. Now consider a del Pezzo surface X of degree 5, which always has a rational point. This implies that an obstruction will not exist, which also follows from the fact that the Brauer group of such a surface is always trivial. The situation is completely different in the following case: we start with a del Pezzo surface X of degree 5 with its embedding X → P⁵(Q). Let D ⊆ X be a hyperplane section and consider the complement U=X\D which is an example of a log K3 surface. If one has explicit equations or equivalently a model over Z for the affine surface U we can study the existence of integral points on U using the Brauer–Manin obstruction as before. The Brauer group of such surfaces can contain elements of order 5 and I will show how to construct such an affine surface over Z for which these elements obstruct the existence of integral points.

Suppose one wants to count points (x,y) on a line ax+by=c, in which x=u/w and y=v/w are rational numbers. One might require that u,v,w are integers with no common factor, and count solutions with |u|,|v|,|w| ≤ B. How does this number grow as B tends to infinity? This question is fairly easy and “classical”. It turns out that the counting function is roughly C_a,b,cB, for some positive constant C_a,b,c. However to see this behaviour, B must be large enough compared to the coefficients a,b,c; and it turns out that the point at which the correct asymptotic behaviour begins will depend on the size of the smallest solution to the original equation ax+by=c. All this will be described in the talk, which will go on to discuss the analogous situation for conics.

In this talk I describe some progress towards Serre’s problem on the number of varieties in a family with a rational point using tools from additive combinatorics. This is joint work with Lilian Matthiesen.

In this talk I will first discuss a general asymptotic results on linear correlations of multiplicative functions that satisfy certain natural conditions. A simple example of the type of correlations that we consider would be Σ_n,d<x b(n) b(n+d) … b(n+rd), where b denotes the characteristic function of sums of two squares. In the second part, we then discuss how this result is used within the joint project with Dan Loughran on the number of varieties in a family with a rational point.

We consider the number of rational points on a variety with appropriate height function in a large box – specifically emphasising the generic points. We obtain upper bounds in the case of systems of diagonal equations via recent work on relatives of Vinogradov’s mean value theorem (via efficient congruencing and decoupling). There are connections with certain biprojective varieties.

Fouvry and Iwaniec showed that there are infinitely many primes that are a sum of a square and a prime square. In this talk a proof of Vinogradov’s three prime theorem restricted to these primes is outlined. The main techniques used are a transference principle version of the circle method, the linear sieve with sieve switching, and the combinatorical dissection of sums over primes given in a work of Duke, Friedlander, and Iwaniec.

In recent years there has been a lively interest in Serre’s problem on the density of fibres that have a rational point among a family of varieties. The literature contains a wide range of results varying from upper and lower bounds to asymptotics, and from simpler to more complicated bases over which the families are defined.

In joint work with Efthymios Sofos, we analyse some conic bundles over general hypersurfaces of low degree. Using circle method results of Birch in combination with sieving methods, we manage to find asymptotics for the number of fibres of bounded height containing a rational point. Moreover, we are able to factor the leading constant in p-adic parts, obtaining similar behaviour as observed by Loughran for other families of varieties.

The problem of computing endomorphism rings of supersingular elliptic curves is an important one in post-quantum cryptography. In this talk, we expand on the work of Kohel and study the cycles in supersingular l-isogeny graphs that could generate the endomorphism rings (or a full-rank suborder) of supersingular elliptic curves. This work is joint with Efrat Bank, Catalina Camacho-Navarro, Kirsten Eisentraeger, and Travis Morrison.

A folklore conjecture is that the number N_d(K,X) of degree-d extensions of K with discriminant at most d is on order c_d X. In the case K = Q, this is easy for d=2, a theorem of Davenport and Heilbronn for d=3, a much harder theorem of Bhargava for d=4 and 5, and completely out of reach for d>5. More generally, one can ask about extensions with a specified Galois group G; in this case, a conjecture of Malle predicts that the asymptotic growth is on order X^a(log X)^b for specified constants a,b.

The form of Malle’s conjecture is reminiscent of the Batyrev–Manin conjecture, which says that the number of rational points of height at most X on a Batyrev–Manin variety also grows like X^a(log X)^b for specified constants a,b. What’s more, an extension of Q with Galois group G is a rational point on a Deligne–Mumford stack called BG, the classifying stack of G. A natural reaction is to say “the two conjectures is the same; to count number fields is just to count points on the stack BG with bounded height?” The problem: there is no definition of the height of a rational point on a stack. I’ll explain what we think the right definition is, and explain how it suggests a heuristic which has both the Malle conjecture and the Batyrev–Manin conjecture as special cases.

A folklore conjecture is that the number N_d(K,X) of degree-d extensions of K with discriminant at most d is on order c_d X. In the case K = Q, this is easy for d=2, a theorem of Davenport and Heilbronn for d=3, a much harder theorem of Bhargava for d=4 and 5, and completely out of reach for d>5. More generally, one can ask about extensions with a specified Galois group G; in this case, a conjecture of Malle predicts that the asymptotic growth is on order X^a(log X)^b for specified constants a,b.

The form of Malle’s conjecture is reminiscent of the Batyrev–Manin conjecture, which says that the number of rational points of height at most X on a Batyrev–Manin variety also grows like X^a(log X)^b for specified constants a,b. What’s more, an extension of Q with Galois group G is a rational point on a Deligne–Mumford stack called BG, the classifying stack of G. A natural reaction is to say “the two conjectures is the same; to count number fields is just to count points on the stack BG with bounded height?” The problem: there is no definition of the height of a rational point on a stack. I’ll explain what we think the right definition is, and explain how it suggests a heuristic which has both the Malle conjecture and the Batyrev–Manin conjecture as special cases.

Manin’s conjecture predicts the asymptotic behavior of the number of rational points of bounded height on Fano varieties over number fields. We prove this conjecture for a family of nonsplit singular quartic del Pezzo surfaces over arbitrary number fields. For the proof, we parameterize the rational points on such a del Pezzo surface by integral points on a nonuniversal torsor (which is determined explicitly using a Cox ring of a certain type), and we count them using a result of Barroero–Widmer on lattice points in o-minimal structures. This is joint work in progress with Marta Pieropan.

When one wants to study the arithmetic of a given field K, it is often useful to understand the cohomology of the Galois module of roots of unity Q/Z(1) or, more generally, the cohomology of its twists Q/Z(r). In this talk, we will be interested in the situation when K is a finite extension of the Laurent series field in m variables k((x₁,…,x_m)) with coefficients in a finite field, a p-adic field or a number field. We will discuss some vanishing theorems as well as some exact sequences that play the role of the Brauer–Hasse–Noether exact sequence for the field K.

In the 1950s Lang studied the properties of C₁ fields, that is, fields over which every hypersurface of degree at most n in a projective space of dimension n has a rational point. Later he conjectured that every smooth proper rationally connected variety over a C₁ field has a rational point. I will explain how to find rational points on some Fano threefolds over C₁ fields of characteristic 0.

Manin’s Conjecture predicts an asymptotic formula for the counting function of rational points on a Fano variety after removing an exceptional set. We propose the construction of the exceptional set and prove that it is contained in a thin subset of rational points using the boundedness of singular Fano varieties and Hilbert Irreducibility Theorem. This is joint work with Brian Lehmann and Akash Sengupta.

Let d and g be positive integers with 1<d<g. If d is odd, we prove there exists B(d)>0 such that a positive proportion of odd genus g hyperelliptic curves over Q have at most B(d) points of degree d. If d is even, we similarly bound the degree d points not pulled back from degree d/2 points of the projective line. Our proof proceeds by refining Park’s recent application of tropical geometry to symmetric power Chabauty, and then applying results of Bhargava and Gross on average ranks of Jacobians of hyperelliptic curves. This is joint work with Joseph Gunther.

When studying the possible torsion structures of elliptic curves over number fields of a fixed degree, one is naturally led to the question of existence of points of low degree on modular curves. This question is related to the gonality of such curves, defined as the lowest degree of a rational map from the curve to the projective line, by a theorem of Frey: if a smooth projective curve over Q has infinitely many points of degree d, then there exists a non-constant map from this curve to the projective line of degree at most 2d. Hence, lower bounds on the gonality of a curve imply results on the finiteness of points of low degree on that curve. However, in general it is a hard problem to compute the gonality of a given curve. In this talk I will outline a strategy to obtain lower bounds on the gonality of a modular curve, by using reduction modulo primes dividing the level of the curve. This reduction is not smooth, but often admits a detailed geometric description obtained by studing p-torsion on elliptic curves in characteristic p. This allows us, amongst others, to prove that there are only finitely many elliptic curves defined over number fields of degree 7 whose torsion structure is isomorphic to Z/2Z × Z/2Z, partially answering a question by Derickx and Sutherland. This is joint work with Maarten Derickx and Bas Edixhoven.

Elements in the Brauer group of a variety over a number field which are annihilated by base change to an algebraic closure are said to be algebraic. In general it is possible for the Brauer group to provide an obstruction to the Hasse Principle or Weak Approximation which cannot be explained by algebraic classes. I will describe a recent result that such transcendental obstructions do not occur for torsors under abelian varieties.

Algebraic Brauer groups of diagonal quartic surfaces have been classified by Martin Bright. To determine their transcendental Brauer groups over Q one first needs to understand the Galois representation on the transcendental part of the second cohomology group. This can be done more generally for K3 surfaces with complex multiplication. A more subtle problem is to determine the image of the Brauer group in the geometric Brauer group. This can be done using some homological algebra from an earlier joint work of Jean-Louis Colliot-Thélène and the speaker. The case of odd order torsion was settled by E. Ieronymou and the speaker. It turns out that the 2-primary subgroup of the Brauer group of any diagonal quartic surface over Q is algebraic, with the exception of only two cases that can be explicitly described. This is a joint work with Damián Gvirtz.