Density of rational points on certain K3 surfaces

 Let V be a nonsingular projective K3 surface defined over Q and having at least two elliptic fibrations defined over Q. We also assume that V(Q) is not empty.  We prove, under a weak hypothesis, the Zariski density of V(Q) and study the closure of V(Q) under the real and p-adic topologies.

Counting n-coverings of elliptic curves over the rationals

Let E/Q be an elliptic curve.  Let C be a smooth genus one curve representing an element in the n-Selmer group of E/Q for n≤4.  It is known that we can produce explicit global minimal genus one equations describing C.  I will explain how to count equivalence classes of these equations over the rational integers Z.

Quadratic polynomials represented by norms

Let K/Q be an extension of number fields. The Hasse norm theorem states that when K is cyclic any non-zero element of Q can be represented as a norm from K globally if and only if it can be represented everywhere locally. In this talk I will discuss the harder problem of representing non-constant polynomials by norms, focusing on the case of irreducible quadratics. Here it transpires that ideas stemming from Linnik's dispersion method and the theory of bilinear sums can lead to a proof of the Hasse principle in this case. This is joint work with Roger Heath-Brown.

A de Rham–Witt complex for rigid cohomology

For a smooth variety X over a perfect field k we construct an overconvergent de Rham–Witt complex by imposing a growth condition on the de Rham–Witt complex of Deligne–Illusie. It is a complex of étale sheaves and a differential graded algebra over the ring of overconvergent Witt vectors which is suitable to compute Monsky–Washnitzer cohomology if X is affine, and more generally the rigid cohomology of X.

Mordell–Weil generators for cubic surfaces

Let C be a smooth plane cubic curve over the rationals. The Mordell–Weil Theorem can be restated as follows: there is a finite subset B of rational points such that all rational points of C can be obtained from this subset by successive tangent and secant constructions. It is conjectured that a minimal such B can be arbitrarily large; this is indeed the well-known conjecture that there are elliptic curves with arbitrarily large ranks. This talk is concerned with the corresponding problem for cubic surfaces.

What I did in the holidays

Over the Christmas break I was working with the Magma group in Sydney on implementing algorithms for computing the Brauer–Manin obstruction on algebraic varieties.  Various interesting sub-problems in computational arithmetic geometry, of independent interest, came up along the way.  I will talk about some of these, including Galois descent for varieties and for divisor classes.

Conics on the Fermat quintic threefold

Many interesting features of algebraic varieties are encoded in the spaces of rational curves that they contain.  For instance, a smooth cubic surface in complex projective three-dimensional space contains
exactly 27 lines; exploiting the configuration of these lines it is possible to find a (rational)  parameterization of the points of the cubic by the points in the complex projective plane.

After a general overview, we focus on the Fermat quintic threefold X, namely the hypersurface in four-dimensional projective space with equation x⁵+y⁵+z⁵+u⁵+v⁵=0.  The space of lines on X is
well-known. I will explain how to use a mix of algebraic geometry, number theory and computer-assisted calculations to study the space of conics on X.

Cyclotomic Matrices and Graphs

Lehmer's problem on the Mahler measure of integer polynomials motivates the study of matrices satsifying certain eigenvalue constraints.  For integer symmetric matrices, a complete classification of cyclotomic matrices – those with all eigenvalues in [−2,2] – and minimal noncyclotomic matrices was obtained by McKee and Smyth.  These results confirm Lehmer's conjecture for a broad class of polynomials, but the general problem remains open.  I'll give an overview of the rational integer case, as well as some recent work generalising their approach – based on charged signed graphs – to matrices / graphs over the rings of integers of some imaginary quadratic fields.

Height bounds on coverings of elliptic curves

Explicit n-descent can be used to help find generators for the Mordell–Weil group of an elliptic curve.  Indeed the logarithmic height of a rational point on the covering curve is expected to be smaller by a factor of 2n when compared to its image on the elliptic curve. Standard results from the theory of heights prove this up to the addition of a constant, depending only on the n-covering. I will describe joint work with my recent student Graham Sills, determining these constants in the cases n=2,3,4.

Average values of shifted convolutions of d₃(n)

Work of Conrey and Gonek relates the conjectural asymptotics of the sixth moment of the Riemann zeta function to asymptotics for D(x,h), the sum for n up to x of  d₃(n) d₃(n+h), where d₃(n) is the number of ways of writing n=a.b.c.  In joint work with T. D. Browning, S. Baier and L. Zhao, we investigate the average and mean square of the error term in the conjectural asymptotic for D(x,h), as h varies.  The main tool in this work is the Hardy-Littlewood circle method.

Integrality and rigidity for postcritically finite polynomials

A rational map f: P¹ → P¹ is said to be postcritically finite if every critical point of f is eventually periodic.  Over C, this is a well-studied, but extremely rich class of dynamical systems.  A fundamental rigidity theorem of Thurston asserts that (with well-understood exceptions) such maps are isolated in the moduli space of projective conjugacy classes.  This theorem is an algebraic assertion whose only known proof is transcendental – that is to say, the question makes sense over the field of algebraic numbers, but there is no known Galois invariant proof.  The special case of postcritically finite polynomials is now better understood.  I will discuss Patrick Ingram's arithmetic proof, and related results of my own.  The arguments are elementary, from basic properties of valuations and heights.

Parity conjectures for elliptic curves in positive characteristic

I will be talking about joint work with Fabien Trihan on the parity conjectures in positive characteristic.  These conjectures are a weak version of the Birch and Swinnerton-Dyer conjecture modulo 2.  Namely it asserts that the analytic rank of an elliptic curve has the same parity as the corank of the l-primary Selmer group.  Recently, a lot of work has been done proving them for number fields.  I will show how we can prove this conjecture fully for elliptic curves over global field of characteristic p>3 when l = p and I will also discuss some partial results on the case when l is different from p.

Effective bounds for Frobenius structures on connections

Recent methods to compute the zeta function of a variety over a finite field exploit the Frobenius structure on the relative cohomology of a family of varieties.  When using these methods, one needs an effective bound on the rate of convergence of the Frobenius structure at the bad fibers of the family.  Using the theory of p-adic differential equations, we obtain a bound which is much better than all previous bounds and probably even sharp.  This is joint work with Kiran Kedlaya.

Elementary divisors and Iwasawa theory for modular forms

I will explain some applications of the theory of elementary divisors in p-adic Hodge theory. In particular, I will give a reformulation of Kato's main conjecture for modular forms.

Towards explicit L-packets

The local Langlands correspondence for general linear groups is a bijection but this is no longer the case for other reductive groups, where the representations must be split into L-packets. In the tame case, Bushnell and Henniart have given an explicit description of the correspondence and we will discuss to what extent this can be done for other groups.

The André–Oort conjecture

We present a proof of the André–Oort conjecture on Zariski closure of sets of special points in Shimura varieties assuming the generalised Riemann hypothesis. We also present recent ideas of Jonathan Pila stemming from model theory that might lead to an uncondional proof.

Zeros of the Riemann zeta function

I will discuss the distribution of the zeros of the Riemann zeta function in the critical strip. This is joint work with Brian Conrey and Matt Young.

Realisable Galois module classes for some metabelian extensions

If N/k is a tame Galois extension of number fields with Galois group G, the ring of integers in N is a locally free module over the group ring of G (with coefficients in the ring of integers of k), and we wish to determine the possibilities for the structure of this module as N varies (with k and G fixed). For abelian G, this was done by Leon McCulloh in the 1980's. I will describe joint work with Bouchaib Sodaigui, where we answer this question for a family of metabelian groups G.

Zeros of L-functions of elliptic curves

We discuss a formula for higher derivatives of the p-adic L-function attached to an elliptic curve, with split multiplicative reduction at p>3. We will work over certain non-abelian extensions of the rationals. The proof uses families of Hilbert modular forms, together with a little deformation theory.

Graphs and Diophantine Equations

The gonality of a curve (minimal degree of a morphism to the projective line) is a geometric invariant that can be used to control the finiteness of its set of rational points of bounded degree. For a compact Riemann surface, Li and Yau have given a (differential-geometric) lower bound on the gonality, which Abramovich has applied to modular curves. In this talk, we present a graph theoretical analogue of this inequality, its relation to the algebraic geometry of curves over non-archimedean fields, and applications to certain diophantine problems on modular and elliptic curves over function fields. (Joint work with Fumiharu Kato and Janne Kool.)

Effective Chabauty for $\mathrm{Sym}^2$

Abstract: While we know by Faltings' theorem that curves of genus at least $2$ have finitely many rational points, his theorem is not effective. In 1985, R. Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is small, can be used to give a good effective bound on the number of rational points of curves of genus $g > 1$. In this talk, we draw ideas from tropical geometry to show that we can also give an effective bound on the number of rational points of $\mathrm{Sym}^2(X)$ that are not parametrized by a projective line or an elliptic curve, where $X$ is a (hyperelliptic) curve of genus $g > 2$, when the Mordell-Weil rank of the Jacobian of the curve is less than $g-2$.

Determination of modular forms by fundamental Fourier coefficients

It is an interesting question when a natural subset of the Fourier coefficients is sufficient to uniquely determine a modular form. I will talk about a special case of this question for two types of forms: modular forms of half-integral weight, and Siegel modular forms of degree 2 and integral weight. These two apparently very different scenarios turn out to be closely related, and the results obtained have interesting consequences for the automorphic representation and central L-value attached to an eigencuspform. Part of this is joint work with Ralf Schmidt.

One point suffices: generating rational points on cubic surfaces over finite fields

Let $F_q$ be a finite field with q at least 25. Let S be a smooth cubic surface defined over $F_q$ containing at least one rational line. I will explain how I use a pigeonhole principle to prove that all of the rational points on S can be generated via secant and tangent operations from a single point.

Unlikely intersections in abelian varieties and Shimura varieties

The Manin-Mumford conjecture, which is a theorem of Raynaud, states that a curve of genus at least 2 in an abelian variety contains only finitely many torsion points. Analogues of this, such as the André-Oort and Zilber-Pink conjectures, have been stated for Shimura varieties in place of abelian varieties. In their most general form these imply many Diophantine results such as the Mordell-Lang conjecture.

In this talk I will outline these conjectures and discuss one method of attacking them, due to Pila and Zannier and using results from model theory. In particular I will apply this method to a problem about curves in the moduli space of principally polarised abelian varieties.

Selmer groups of elliptic curves in degree p extensions

I will attempt to motivate the study of Selmer groups of elliptic curves, as well as talk about some of the recent remarkable progress that has been made in their study. After that I will discuss some results on the growth of Selmer ranks in certain cyclic extensions of number fields. In particular, under suitable conditions one can determine the dimension of the Galois invariants of the p-Selmer group as a sum of local contributions.

On higher regulators of Siegel threefolds: the vanishing on the boundary

I will explain the construction of some 1-extensions of mixed Hodge structures in the image of Beilinson's regulator. These are conjecturally related to special values of the degree 4 L-function of some automorphic representations of GSp(4). The key ingredient is the computations of higher direct images of variations of Hodge structures on the boundary of the Baily-Borel compactification. I announced this kind of result some time ago but my proof contained an error.

Overconvergent cohomology of Hilbert modular varieties and p-adic L-functions

For a Hilbert modular form that satisfies a condition of non critical slope we construct a p-adic distribution on the Galois group of the maximal abelian extension of the associated totally real field, unramified outside p and infinity. We prove that the distribution is admissible and interpoles the critical values of the L-function of the form. This construction is based on the study of the overconvergent cohomology of the Hilbert modular variety and certain cycles on this variety.

P-adic heights and integral points on hyperelliptic curves

We give a Chabauty-like method for finding p-adic approximations to integral points on hyperelliptic curves when the Mordell-Weil rank of the Jacobian equals the genus. The method uses an interpretation of the component at p of the p-adic height pairing in terms of iterated Coleman integrals. This is joint work with Amnon Besser and Steffen Müller.

Universal deformation rings and the inverse deformation problem

A good way of understanding groups is to look at its representations into the group of invertible n by n matrices. One can organise such representations into families by first fixing a representation into $GL_n$ of a finite field and then lifting it to `bigger' rings. By a theorem of Mazur, under certain hypothesis the liftings fit into a nice universal family. I will discuss the inverse problem of realizing rings as the coefficient rings of such a universal family and results in this direction (joint work with Tim Eardley).

Parity of 2-Selmer ranks for hyperelliptic curves

I will outline what we know about the parity of ranks of higher dimensional abelian varietes and the existing techniques for controlling it. As for elliptic curves, all such results are conjectural: the unconditional statements concern either root numbers ("parity of the analytic rank") or dimensions of Selmer groups. At the end I will sketch the proof of a recent result with Tim Dokchitser, that a positive proportion of hyperelliptic curves have Jacobians with odd (respectively, even) 2-Selmer rank.

Residual modular Galois representations and their images

In this talk I will outline an algorithm for computing the image of a residual modular 2-dimensional semi-simple Galois representation. In almost all cases, the computations use only Hecke operators up to the Sturm bound at the given level.

Parity of two Selmer ranks of hyperelliptic curves over quadratic extensions

For an elliptic curve E over a number field K, a consequence of the Birch and Swinnerton-Dyer conjecture is the 2-parity conjecture; the global root number agrees with the parity of the 2-infinity Selmer rank. Over a quadratic extension of K, both quantities may be expressed as a product of local contributions. T. and V. Dokchitser, building on work of Kramer and Tunnell, showed that these local contributions are related by a specific correction term which vanishes globally, thus giving parity results. I will discuss extensions of this to hyperelliptic curves of higher genus, where the possibility that the Tate-Shafarevich group of the Jacobian (if finite) has order twice a square adds an extra component to the local analysis.

Multiple polylogarithms in weight 4

It is well-known that all multiple polylogarithms of weight 2 and 3 are expressible in terms of Li_2 and Li_3, respectively, while the interplay between the weight 4 functions is considerably more complicated--in particular, they are not all expressed in terms of Li_4.

We give some of the first functional equations among weight 4 polylogarithms, involving in particular Li_{2,2}, Li_{3,1} and Li_{1,1,1,1}. We are also led to a functional equation in 4 variables for Li_4 which should play an important role in an explicit definition of the corresponding higher Bloch group.

A Deuring criterion for abelian varieties

Let A be an abelian variety defined over a number field with complex multiplication by a CM field F. If A is an elliptic curve, a famous criterion of Deuring provides a direct link between the splitting of a prime number p in F and the reduction type of A at any prime of good reduction above p. With a bit of thought, it is easy to see that there can be no such simple relationship as soon as the dimension of A is greater than 1. Nonetheless, in this talk we will prove several generalisations of the Deuring reduction criterion to abelian varieties of arbitrary dimension.

Darmon points for fields of mixed signature

Let E be an elliptic curve defined over a number field F, and let K be a quadratic extension of F. We present a conjectural construction of algebraic points on abelian extensions of K, which generalizes work of Darmon, Logan, Gartner, Greenberg and Trifkovic. We will sketch the construction, explain how one can perform computations, and provide numerical evidence.

Computing transcendental Brauer groups of products of CM elliptic curves

In 1971, Manin showed that the Brauer group Br(X) of a variety X over a number field K can obstruct the Hasse principle on X. In other words, the existence of points everywhere locally on X despite the lack of a global point is sometimes explained by non-trivial elements in Br(X). Since Manin's observation, the Brauer group has been the subject of a great deal of research.

The 'algebraic' part of the Brauer group is the part which becomes trivial upon base change to an algebraic closure of K. It is generally easier to handle than the remaining 'transcendental' part and a substantial portion of the literature is devoted to its study. In contrast, until recently the transcendental part of the Brauer group had not been computed for a single variety. The transcendental part of the Brauer group is known to have arithmetic importance – it can give non-trivial obstructions to the Hasse principle and weak approximation.

I will use class field theory together with results of Ieronymou, Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the product E x E for an elliptic curve E with complex multiplication. The results for the odd-order torsion descend to the Kummer surface Kum(E x E).

Galois theory of radical groups

We present a new approach to the notion of "entangled radicals", i.e., unexpected additive relations between roots, such as 1+ \sqrt{-1} = 4th root of -4. As an application, we give a general method to obtain the rational correction factor in Artin's primitives root conjecture. The same method applies to computing Galois representations on division points of tori of rank 1, which also gives rise to a corresponding version of Artin's conjecture. This is joint work with Willem Jan Palenstijn, who will defend his PhD in Leiden this spring.

Arithmetic of large dimensional complete intersections

The circle method, when it works, answers most sensible questions about the arithmetic of smooth varieties of large dimension compared to the degree. In this talk, which is joint work with Heath-Brown, I'll discuss how Birch's famous 1961 paper can be generalised to handle smooth complete intersections cut out by forms of differing degree.

Elliptic Curves over Real Quadratic Fields are Modular

We combine recent breakthroughs in modularity lifting with a 3-5-7 modularity switching argument to deduce modularity of elliptic curves over real quadratic fields. We discuss the implications for the Fermat equation. In particular we show that if d is congruent to 3 modulo 8, or congruent to 6 or 10 modulo 16, and K=Q(\sqrt{d}) then there is an effectively computable constant B depending on K, such that if p>B is prime, and a^p+b^p+c^p=0 with a,b,c in K, then abc=0. This is based on joint work with Nuno Freitas (Bayreuth) and Bao Le Hung (Harvard).

Diophantine Geometry and Non-Abelian Reciprocity

We use non-abelian fundamental groups to define a sequence of higher reciprocity maps on the adelic points of a variety over a number field satisfying certain conditions in Galois cohomology. The non-abelian reciprocity law then states that the global points are contained in the kernel of all the reciprocity maps.

The Brauer-Manin obstruction and ranks of quadratic twists

Let B be an abelian variety over a number field k and let X be the Kummer variety Km(B) of B, which is a smooth projective model of the quotient B/<-1>. Roughly speaking, rational points on X correspond to rational points on quadratic twists of B. We use this to show that if the Brauer–Manin obstruction controls the failure of weak approximation on all Kummer varieties, then for every positive-dimensional abelian variety A over a number field, the ranks of quadratic twists of A are unbounded. This is a joint work with David Holmes (Leiden University).

Arithmetic of Drinfeld modules

Drinfeld modules are function field objects that are analogs of elliptic curves (or of the multiplicative group) over number fields. They were introduced by Drinfeld in the 1970s, although parts of the theory had been discovered already in the 1930s by Carlitz. In this talk I will survey the more arithmetic aspects of Drinfeld modules, focussing on some recent finiteness theorems analogous to the Mordell-Weil theorem (or Dirichlet's unit theorem) and the finiteness of Sha (or of the class group), and on special values of L-functions. I will not assume any prior knowledge of Drinfeld modules.

On Some Arithmetic Properties of Twists of the Klein Quartic

We start by introducing the Klein quartic: $x^3y+y^3z+z^3x=0$, and describe some of its properties, stressing the fact that it is isomorphic over $\mathbb{Q}$ to the modular curve $X(7)$. We study its twists via two different approaches, via the algorithm based on joint work with Lario, and via a modular interpretations of the twists. Some of the arithmetic properties that we will study are: the Generalized Sato-Tate Conjecture (joint work with Fite and Sutherland), and the set of rational points and of $\mathbb{Q}_v-$points, for all the valuations $v$ of $\mathbb{Q}$, of the twists of the Klein quartic. In this way, we will find some counterexamples to the Hasse Principle.

Level optimisation for p-adic Hilbert modular forms

In the 1980s, Mazur and Ribet, motivated by conjectures of Serre, proved level optimisation results relating the conductors of mod p Galois representations and modular forms. Suppose p, q are distinct primes and N is a positive integer coprime to pq. Their level optimisation results have the following form: suppose f is a weight 2 Hecke eigenform of level Nq, such that the Galois representation associated to f becomes unramified at q when it is reduced modulo p. Then there exists a weight 2 Hecke eigenform of level N which is congruent to f modulo p. I will explain an analogue of these results in the setting of (overconvergent) p-adic Hilbert modular forms, together with an application to local-global compatibility for Hilbert modular forms.

Solubility and insolubility of certain diagonal quartic surfaces

We consider the quartic surface
a₀X₀⁴+a₁X₁⁴+a₂X₂⁴+a₃X₃⁴=0
subject to the additional condition a₀a₁a₂a₃ is a square. The effect of this condition is that the surface is fibred by curves of genus 1. Subject to two major hypotheses and a minor constraint on the surface, I shall show that the surface contains rational points if and only if it has a fibre which is everywhere locally soluble.

The mock modularity of q-hypergeometric series

An intriguing and almost completely unsolved problem is to understand the overlap between classes of q-hypergeometric series and modular forms. This challenge was the subject of George Andrews' plenary address at the UIUC Millennial Conference on Number Theory and has its origin in Ramanujan's last letter to G.H. Hardy on January 12, 1920 whereby 17 mock theta functions were introduced. We discuss recent work concerning the explicit construction of new individual examples and infinite families of mock theta functions (in the modern sense of Zagier). Additionally, we discuss various ways to produce q-hypergeometric series which are mixed mock modular forms and a recent connection between q-series and knots. This is joint work with Jeremy Lovejoy (Paris 7).

Computation aspects of modular Galois representations

Let f be a Hecke eigenform of weight k for the group Gamma_1(n) with coefficients in a finite field F. To f one can attach a continuous semi-simple two-dimensional F-linear representation rho of the absolute Galois group of Q. I will explain algorithms of Couveignes, Edixhoven, de Jong, Merkl and Bosman (n=1) and myself (n>1) to compute rho in polynomial time in n, k and #F. These involve "approximate" computations in Jacobians of modular curves, either over C or over finite fields. I will also touch upon the progress made so far by several people in doing actual computations.

Subgroups of the modular group and symmetries

I will report on the result of recent extensive computations of subgroups (the complete list up to index 17 contains 39831 groups) of the modular group and their associated discrete spectra. In particular I will present numerical support for a conjecture stating that even for subgroups of the modular group there is no "new" discrete spectrum (i.e. no new Maass waveforms) unless there exist certain explicit symmetries.

The finiteness of p-parts of Tate--Shafarevich groups

The Tate-Shafarevich groups of elliptic curves are among the most mysterious objects in number theory. It is conjectured that they are always finite. I will describe a recent joint result with Guido Kings and Sarah Zerbes, which gives finiteness for the p-part of the Tate-Shafarevich group of an elliptic curve twisted by a dihedral Artin representation (under some mild hypotheses on the prime p) when the associated L-function is non-vanishing at s = 1.

Torsion points on cm elliptic curves over prime degree fields

In this talk, which should be accessible to graduate students, we answer the following question of Schuett. Let p be a prime, F a degree p number field, and E an elliptic curve over F with cm. As p grows, how does the torsion in E(F) grow? We introduce some basics on cm elliptic curves and reduce this question in part to one on cyclotomic fields. This is joint work with A. Bourdon and P. Clark.

Supercuspidal types and the Jacquet-Langlands correspondence
In this talk, we revisit the Jacquet-Langlands correspondence for $\mathrm{GL}_2$. We restate this correspondence using the notion of supercuspidal types. Our reformulation was motivated by algorithmic questions. So we will give several examples which illustrate its usefulness

Computing Klein modular forms
Arithmetic Kleinian groups are arithmetic lattices in PSL_2(C). I will present an algorithm that uses the cohomology of such groups to compute systems of Hecke eigenvalues of certain automorphic forms for GL_2 over number fields.

Galois groups of local fields, Lie algebras and ramification
Let K be a complete discrete valuation field with a finite residue field of characteristic p>0.
If G_K(p) is the Galois group of the maximal p-extension of K then its structure is completely known: it is either free or Demushkin's group. However, this result is not completely satisfactory because the appropriate functor from the fields K to the pro-p-groups G_K(p) is not fully faithful. In other words, in this setting the Galois group does not reflect essential invariants of the original field K. The situation becomes completely different if we take into account an additional structure on G_K(p) given by its decreasing filtration by ramification subgroups G_K(p)^{(v)}, v\ge 0. The importance of explicit description of this filtration was pointed out in 1960-1970's by A.Weil, I.Shafarevich, P.Deligne etc. In particular, if K has characteristic p then we should invent a way to specify a special choice of free generators of G_K(p) which reflects arithmetic properties of K. Suppose C_s, where s\ge 1, is the closure of the subgroup of commutators of order at least s in G_K(p). Then the above problem of "arithmetic description" of G_K(p) can be considered modulo subgroups C_s. If s=1 it is trivial and if s=2 the answer is given via class field theory. In the case s>2 we obtain a long-standing problem of constructing a "nilpotent class field theory". In the talk we discuss the case s=p, in particular, the author new results related to the mixed characteristic case (i.e. when K is a finite extension of Q_p). The quotient G_K(p)/C_p (together with the induced ramification filtration) is complicated enough to reflect the invariants of K. At the same time this quotient comes from a profinite Lie algebra via Campbell-Hausdorff composition law. The description of the appropriate ramification filtration essentially uses this structure of Lie algebra.

Galois scaffolds and (generalised) Galois module structure
In a wildly ramified Galois extension of local fields, it is usually difficult to determine the structure of the valuation ring (or one of its ideals) in the larger field as a module over its associated order in the group algebra. A Galois scaffold consists of a system of special generators of the group algebra whose action on the field has a particularly simple form. When a Galois scaffold exists, one can determine which ideals are free over their associated orders. Moreover, the same machinery can be applied more widely, e.g. to inseparable extensions with an action by a Hopf algebra. This is joint work with Griff Elder and Lindsay Childs.

Computing modular Galois representations
We will see how to efficiently compute explicitly the Galois representation attached to a classical newform f modulo a small prime l. This originates from work of Jean-Marc Couveignes's and Bas Edixhoven's, and we will show how to make their insight practical. This can be used to deduce the Fourier coefficients a(p) of f modulo l for huge primes p.

Dwork's congruences for periods of families of Calabi-Yau hypersurfaces
For a Laurent polynomial whose Newton polytope contains the origin as the unique interior integral point, we prove that the constant terms of its powers satisfy certain congruences modulo prime powers. The generating series of these numbers is a solution of the Picard-Fuchs equation for the respective family of hypersurfaces. Our congruences allow to construct an explicit p-adic analytic continuation of this solution, following a method pioneered by B. Dwork for a class of hypergeometric series. The talk is based on a joint work with A. Mellit. This result was also proved recently by A. Adolphson and S. Sperber using a different method. We discuss possible generalizations and relation to the deformation method by A. Lauder.

E_6 and integral points on genus 3 curves
We will discuss the (average) arithmetic of a family of non-hyperelliptic curves of genus 3. These curves appear as the nearby fibres of the universal deformation of a simple curve singularity of type E_6, and can be studied using Vinberg's representations arising from the simple Lie group of the same name.

The weight part of Serre's conjecture
I'll discuss joint work in progress with Matthew Emerton, Florian Herzig and David Savitt, which attempts to formulate analogues of the weight part of Serre's conjecture in considerable generality.

Counting rational points on smooth cubic surfaces
A popular subject in Diophantine geometry is the existence of rational points in varieties, a subject which includes topics such as the Hasse principle and the Brauer-Manin obstruction. On the opposite end of the spectrum there are varieties whose rational points are Zariski dense and where one is interested in more refined properties of their distribution. Manin's conjecture attempts to describe the distribution of points in specific subvarieties via the use of heights. It has never been established for a single smooth cubic surface although it is known for particular del Pezzo surfaces of any degree d>3. In this talk we will prove the one sided estimate predicted by Manin's conjecture for a family of smooth cubic surfaces containing a rational line.

Galois representations attached to cohomological automorphic forms
Conjectures of Langlands and Clozel predict the existence of Galois representations attached to cuspidal automorphic forms of GL_n(A_Q) which are cohomological, in the sense that they contribute to the cohomology of certain arithmetic locally symmetric spaces. In the case n = 2, these spaces are modular curves and the proof of these conjectures, due to Deligne, goes by the understanding their etale cohomology. I will discuss the solution to this problem in the general case. This is joint work with Harris, Lan, and Taylor.

A quadratic Chabauty's method for rational points on higher genus curves
In this talk we will review the nonabelian generalisation of Chabauty's method for finding rational points on curves, some new approaches to making it explicit for rational points on higher genus curves, and joint work with Jennifer Balakrishnan implementing the method to find rational points on some genus 2 curves.

Applications of hybrid p-adic group rings and hybrid Iwasawa algebras to arithmetic conjectures
We introduce the closely related notions of hybrid p-adic group ring and hybrid Iwasawa algebra. When certain group-theoretic criteria are met, hybrid p-adic group rings allow one to reduce the proof of the equivariant Tamagawa number conjecture for a particular Galois extension of number fields to the proofs of two easier conjectures; this leads to unconditional proofs in certain cases. A similar story holds for hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture for totally real fields, though there are some important differences. This is joint work with Andreas Nickel (Bielefeld).

Some density results in number theory
I will describe joint work with Manjul Bhargava (Princeton) and Tom Fisher (Cambridge) in which we determine the probability that a random equation has a solution either locally (over the reals or the p-adics) or globally. Three kinds of equation will be considered: quadratics in any number of variables, ternary cubics and hyperelliptic quartics. Some of the results, such as the probability that a random real symmetric matrix is positive definite, for which we obtain a simple closed formula for any dimension, may be of wider interest.

Regularized theta lifts and currents on orthogonal Shimura varieties
We will start by reviewing a conjecture of Beilinson relating the regulator of a higher Chow group of a variety defined over with the special value of an L-function attached to . Then I will consider the case where is a Shimura variety attached to an orthogonal group . I will briefly review the notion of a theta lift and will explain how to construct certain currents on related to regulator maps as regularised theta lifts for the dual pair . We will show that in some cases these currents can be evaluated and that the result involves the special value of a certain automorphic L-function times certain a certain period on .

The ternary Goldbach conjecture
The ternary Goldbach conjecture (1742) asserts that every odd number greater than 5 can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant C satisfies the conjecture. In the years since then, there has been a succession of results reducing C, but only to levels much too high for a verification by computer up to C to be possible (C> 10^1300). (Work by Ramare and Tao solved the corresponding problems for six and five prime numbers instead of three.) My recent work proves the conjecture. We will go over the main ideas in the proof.

Units and Class Modules in the Arithmetic of Global Function Fields
We present an overview of some recent developments in the arithmetic of function fields over finites fields. Our main purpose in this talk will be the connection between the special values of some non-archimedean $L$-series and some unit and class modules introduced by L.Taelman in 2010.

Sup norms of modular forms in the level aspect
Recently, there has been much activity on the problem of establishing bounds on the maximum size of a newform as the conductor (level) of the form varies. This talk will especially focus on the powerful level aspect; this is when high powers of primes divide the increasing level. After some survey remarks, I will describe two recent results of mine. One of them is a lower bound for the ratio that is ``large" whenever the central character is sufficiently ramified with respect to the level. This result extends some previous work of N. Templier, and being purely local, applies to general number fields. The other is an upper bound for the L^\infty norm in the classical case (where the base field is Q) that improves upon the trivial bound. This extends previous work of Blomer-Holowinsky, Harcos-Templier, and others, who dealt with the case of increasing squarefree levels.

The average elliptic curve has few integral points
It is a theorem of Siegel that the Weierstrass model $y^2 = x^3 + Ax + B$ of an elliptic curve has finitely many integral points. A "random" such curve should have no points at all. I will show that the average number of integral points on such curves (ordered by height) is bounded --- in fact, by 67. The methods combine a Mumford-type gap principle, LP bounds in sphere packing, and results in Diophantine approximation. The same result also holds (though I have not computed an explicit constant) for the families $y^2 = x^3 + Ax$, $y^2 = x^3 + B$, and $y^2=x^3-n^2x$.

On abelian varieties over F_p
The general theme of this joint work with Jakob Stix is studying the category of abelian varieties over a finite field by means of "simple" linear algebra. The first result obtained concerns only the prime field case and the full subcategory C of abelian varieties whose Frobenius eigenvalues are different from $\pm\sqrt p$. It asserts that C is equivalent to the category of pairs $(T,F)$ where $T$ is a finite free abelian group and $F:T\to T$ is a linear map which is semi-simple and whose eigenvalues are non-real Weil $p$-numbers.

This theorem is inspired by a result of Waterhouse and generalizes the $q=p$ case of a theorem of Deligne, who showed that a description in the above style is possible for the category of ordinary abelian varieties. While Deligne's strategy is based on Serre-Tate's canonical lifting, our method relies on the Gorenstein property of the order generated by a Weil number and its
conjugate.

Semi-stable reduction of correspondences
Correspondences between curves induce linear maps on cohomology, whose spectral properties are often of interest. Using Hecke operators on modular forms and their p-adic deformations as our motivating example, we prove a potential semi-stable reduction theorem for correspondences analogous to the result for curves by Deligne-Mumford. This generalises and strengthens work of Coleman and Liu. Time permitting, we explicitly work out the example of Hecke operators by making a careful study of the adic geometry of modular curves.

Expanding pretension in analytic number theory
For the last 5 years, Soundararajan and I have been developing an alternative approach (sometimes called the "pretentious approach") to classical analytic number theory. Now that this approach covers most of the classical questions, we have begun to explore more modern problems, for example higher weight L-functions, and L-functions over finite fields. In this talk we will begin by explaining the "pretentious Riemann Hypothesis", and see how this is as useful as the classical approach, and then discuss recent work with Harper, looking at analytic number theory in F_q[t].

Quadratic twists of elliptic curves
I will discuss joint work with Y. Li, Y. Tian, and S. Zhai on the generalization of the method, going back to Heegner and Birch, of proving that certain elliptic curves defined over Q have many quadratic twists where the order of the zero of their complex L-series at s=1 is at most 1.

Constructing Heegner points when the "Heegner hypothesis" is not satisfied
Let $E/Q$ be an elliptic curve of conductor $N$, and $K/Q$ an imaginary quadratic field such that the functional equation of $E/K$ has sign $-1$. Under this general hypothesis one expects to construct systems of (non-torsion) Heegner points attached to the pair $(E,K)$. Suppose that the following conditions are satisfied:

1) If $p \mid N$ and it is unramified in $K$ then it is split.
2) If $p \mid N$ and it ramifies in $K$ then $p^2\nmid N$.

Then one can construct Heegner points in the curve $X_0(N)$ and map them to $E$ under the canonical map. Such a construction is explicit, easy and fast to implement. In this talk we will explain other constructions that hold when the previous hypotheses are not satisfied.

On K_2 of curves
Let C be a curve over a field k and F=k(C) its function field. We discuss some subgroups of K_2(F) that contain the kernel of the tame symbol, and some of their properties. In joint work with Bogdan Banu, we use this method to compute non-torsion elements in the kernel of the tame symbol in some cases where no theoretical construction of such elements is known. Finally, time permitting, we also discuss some constructions (made jointly with Hang Liu) of examples of curves over the rationals with as many 'integral' elements as predicted by Beilinson, and describe some limit results for the resulting regulator.

Local arithmetic of hyperelliptic curves
I will report on a joint work with Vladimir Dokchitser, Celine Maistret and Adam Morgan. For a hyperelliptic curve C over a p-adic field K (with p odd), the project is to understand its Galois representation and the associated invariants of C/K – Tamagawa numbers, root numbers, existence of p-adic points, the shape of the semistable model and so on. I will talk about two results, a necessary and sufficient criterion for the curve to be semistable, and a description of its Galois representation in general.

Prime Number Races with very many competitors
The prime number race is the competition between different coprime residue classes mod $q$ to contain the most primes, up to a point $x$. Rubinstein and Sarnak showed, assuming two $L$-function conjectures, that as $x$ varies the problem is equivalent to a problem about orderings of certain random variables, having weak correlations coming from number theory. In particular, as $q \rightarrow \infty$ the number of primes in any fixed set of $r$ coprime classes will achieve any given ordering for $\sim 1/r!$ values of $x$. In this talk I will try to explain what happens when $r$ is allowed to grow as a function of $q$. It turns out that one still sees uniformity of orderings in many situations, but not always. The proofs involve various probabilistic ideas, and also some harmonic analysis related to the circle method. This is joint work with Youness Lamzouri.

Sums of two squares in polynomial rings over finite fields
The integers that are a sum of two squares are characterized by the classical theorem of Fermat as those with a prime factorization in which every prime congruent 3 mod 4 appears with even multiplicity. A theorem of Landau determines the asymptotic behavior of the number of such integers up to x, but their behavior in short intervals is not fully understood. I will report on joint work with Efrat Bank and Lior Bary-Soroker in which we address a function field analogue of the latter problem in the large finite field limit

Anti-cyclotomic p-adic L-functions and the exceptional zero phenomenon
Let E be a modular elliptic curve over a totally real field F and let K/F be a totally imaginary quadratic extension. In event of exceptional zero phenomenon, we prove a formula for the gradient of the multivariable anti-cyclotomic p-adic L-function attached to (E,K) in terms of the Hasse-Weil L-function and certain p-adic periods attached to the respective automorphic forms. Our methods are based on a new construction of the anti-cyclotomic p-adic L-function by means of the corresponding automorphic representation.

Local L-functions, local factors, and their reductions modulo-l
I will describe some joint work with Nadir Matringe, in which we associate Rankin-Selberg local L-functions and local epsilon factors to pairs of generic l-modular representations of general linear groups over a locally compact non-archimedean local field of residual characteristic different to l. I will explain some of their interesting properties, including the relationship with l-adic Rankin-Selberg local L-functions via reduction modulo l.

Serre weights and wild ramification in two-dimensional Galois representations
For a mod p Galois representation arising from a Hilbert modular form, the weight part of Serre's Conjecture (proved by Gee at al) determines the weights of forms from which it arises. However the dependence of the weights on wild ramification is not explicit. I'll discuss joint work with Dembele and Roberts towards making it so, or equivalently, describing wild ramification in reductions of two-dimensional crystalline Galois representations.

Polynomials representing primes
It is a famous conjecture that any one variable polynomial satisfying some simple conditions should take infinitely many prime values. Unfortunately, this isn't known in any case except for linear polynomials - the sparsity of values of higher degree polynomials causes substantial difficulties. If we look at polynomials in multiple variables, then there are a few polynomials known to represent infinitely many primes whilst still taking on "few" values; Friedlander-Iwaniec showed X^2+Y^4 is prime infinitely often, and Heath-Brown showed the same for X^3 + 2Y^3. We will describe recent work which gives a family of multivariate polynomials all of which take infinitely many prime values.

Pre-adic and adic spaces
I will give an introduction to the theory of adic spaces, a subject which was created in the 1990s by Huber but which only came to the forefront of number theory after recent work of Scholze. The original foundations of the theory were set up with Noetherian hypotheses, but unfortunately for a lot of modern applications (e.g. the theory of perfectoid spaces) these hypotheses are usually not satisfied. I will explain how this causes problems and how, in some cases, they can be resolved.

Eigenvarieties for non-cuspidal Siegel modular forms.
In a recent work Andreata, Iovita, and Pilloni constructed the eigenvariety for cuspidal Siegel modular forms. This eigenvariety has the expected dimension (the genus of the Siegel forms) but it parametrizes only cuspidal forms. We explain how to generalize the construction to the non-cuspidal case. To be precise, we introduce the notion of "degree of cuspidality" and we construct an eigenvariety that parametrizes forms of a given degree of cuspidability. The dimension of these eigenvarieties depends on the degree of cuspidality we want to consider: the more non-cuspidal the forms, the smaller the dimension. This is a joint work with Riccardo Brasca.

Darmon cycles and the Kohnen-Shintani lifting
Let f(q) be a Coleman family of cusp forms of tame level N. Let k0 be the p-new classical weight of the Coleman family f(q). By the Kohnen- Shintani correspondence, we associate to every even classical weight k a half- integral weight form gk having fourier coefficients c(D,k).
We first prove that the Fourier coefficients c(D,k) for even k can be interpo- lated by a p-adic analytic function, in a neighbourhood of k0 in the p-adic weight space.
Based on the eigenvalue of the Atkin-Lehner operator at p, we partition the discriminants D appearing in the Fourier expansion, into two types (Type I and Type II). For any Type II discriminant D, we show that the derivative along the weight at k0, is related to certain algebraic cycles associated to the motive of cusp forms of weight k0 and level Np. These algebraic cycles appear in the theory of Darmon cycles.

Variations on quadratic Chabauty
Let C be a curve over the rationals of genus g at least 2. By Faltings' theorem, we know that C has finitely many rational points. When the Mordell-Weil rank of the Jacobian of C is less than g, the Chabauty-Coleman method can often be used to find these rational points through the construction of certain p-adic integrals.
When the rank is equal to g, we can use the theory of p-adic height pairings to produce p-adic double integrals that allow us to find integral points on curves. In particular, I will discuss how to carry out this ``quadratic Chabauty'' method on hyperelliptic curves over number fields (joint work with Amnon Besser and Steffen Mueller) and present related ideas to find rational points on bielliptic genus 2 curves (joint work with Netan Dogra).

Parity of Selmer ranks in quadratic extensions
For an abelian variety A over a number field K, the Birch and Swinnerton-Dyer conjecture predicts that the root number of A/K is 1 or -1 according to whether the rank of A/K is even or odd respectively. Replacing the rank with the p-infinity Selmer rank for a prime p, one obtains the p-parity conjecture. Whilst there has been a lot of progress towards this conjecture for elliptic curves, much less is known for higher dimensional abelian varieties. I will survey existing results and describe new work allowing one to prove the 2-parity conjecture over quadratic extensions of the base field for certain principally polarised abelian varieties. In particular, the methods work for Jacobians of hyperelliptic curves, under some mild assumptions on their reduction.

On the reciprocity law for p-adic Green functions
The reciprocity law for Green functions (also known as "integrals of differentials of the third kind") on p-adic curves was established by Coleman (1989) and Colmez (1998) by means of p-adic integration. We will discuss a new approach to this reciprocity law by means of p-adic functional analysis.

Explicit refinements of Boecherer's conjecture for Siegel modular forms of square-free level
In the 80s, S. Boecherer conjectured that there should be a connection between certain sums of Fourier coefficients of Siegel modular forms and central values of the $L$-function attached to that modular form. I will describe recent work giving a precise formulation of this conjecture. I will also outline how this formulation follows from a special case of a recent conjecture of Y. Liu which refines the Gan--Gross--Prasad conjectures on periods of automorphic forms for orthogonal groups. This is joint work with A. Pitale, A. Saha, and R. Schmidt.

Oddness of residually reducible Galois representations
I will show how to apply Ribet's construction of extensions of Galois representations to obtain results on the parity of residually reducible polarizable Galois representations, providing evidence for Fontaine-Mazur style conjectures.

The Hasse norm principle for abelian extensions
Let L/K be an extension of number fields and let J_L and J_K be the associated groups of ideles. Using the diagonal embedding, we view L* and K* as subgroups of J_L and J_K respectively. The norm map N: J_L--> J_K restricts to the usual field norm N: L*--> K* on L*. Thus, if an element of K* is a norm from L*, then it is a norm from J_L. We say that the Hasse norm principle holds for L/K if the converse holds, i.e. if every element of K* which is a norm from J_L is in fact a norm from L*.

The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of K fail the Hasse norm principle? More generally, for an abelian group G, what proportion of extensions of K with Galois group G fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis.

This is joint work with Christopher Frei and Daniel Loughran.

Congruences
The theory of congruences between modular forms is a central topic in contemporary number theory, lying at the basis of the proof of Mazur's theorem on torsion in elliptic curves, Fermat's Last Theorem, and Sato-Tate, amongst others. Congruences between modular forms, or more in general between systems of eigenvalues, play a crucial role in understanding links between geometry and arithmetic.
In this talk I will recall classical congruences between modular forms and then present some ideas I am currently developing in various collaborations about congruences between half integral weight modular forms, between Hilbert modular forms, between Bianchi modular forms and between Siegel forms. I will also show how to construct graphs to encode such congruences.

Companion points and locally analytic socle for GL2(L)
Let p>2, L be a finite extension of Qp, we prove Breuil's locally analytic socle conjecture for GL2(L), showing the existence of all the companion points on the definite (patched) eigenvariety. This work relies on some infinitesimal "R=T" results for the patched eigenvariety, and the comparison between (partially) de Rham families and (partially) Hodge-Tate families. This method allows in particular to find companion points of non-classical points.

Cohomology of arithmetic groups via Noncommutative Geometry
Cohomology of arithmetic groups plays a prominent role in modern number theory. The standard way to study the cohomology of an arithmetic group G goes through studying its action on the associated global symmetric space X. In this talk, we instead consider the action of G on the "boundary" of X. As this action is topologically not good, we employ the approach of Noncommutative Geometry in the style of Connes. In joint work with Bram Mesland (Hannover), we show that the cohomology of G, as a Hecke module, can be captured in the K-groups of a certain noncommutative C*-algebra which encodes the action of G on the boundary of X.

Fields of definition of elliptic k-curves with CM and Sato-Tate groups of abelian surfaces
Let A be an abelian surface over a number field k that is isogenous over the algebraic closure to the square of an elliptic curve E. If E does not have CM, by results of Ribet and Elkies concerning fields of definition of elliptic k-curves, E is isogenous to a curve defined over a polyquadratic extension of k. We show that one can adapt Ribet's methods to study the field of definition of E up to isogeny also in the CM case, as long as k contains the field of CM. As an application of this analysis, we provide a number field over which abelian surfaces can be found realizing each of the 52 possible Sato-Tate groups of abelian surfaces. This is joint work with Francesc Fité.

Arthur's multiplicity formula for automorphic representations of certain inner forms of special orthogonal and symplectic groups
I will explain the formulation and proof of Arthur’s multiplicity formula for automorphic representations of certain special orthogonal groups and certain inner forms of symplectic groups G over a number field F. I work under an assumption that substantially simplifies the use of the stabilisation of the trace formula, namely that there exists a non-empty set S of real places of F such that G has discrete series at places in S and is quasi-split at places outside S, and by restricting to automorphic representations of G(A_F) which have algebraic regular infinitesimal character at all places in S. In particular, I prove the general multiplicity formula for groups G such that F is totally real, G is compact at all real places of F and quasi-split at all finite places of F. Crucially, the formulation of Arthur’s multiplicity formula is made possible by Kaletha’s recent work on local and global Galois gerbes and their application to the normalisation of Kottwitz-Langlands-Shelstad transfer factors. I will begin by explaining why this particular case is useful for applications.

Vanishing Order of Automorphic L-Functions
One way to think of an L-function is as a generating function of arithmetic information. As with any generating function, we are interested in its analytic properties so as to hopefully infer information about the underlying structures. In this talk, we will be interested in the orders of zeros of automorphic L-functions. In particular, I will explain how to use Andrew Booker's notion of "L-Data" in this context. The main theorem presented here is a classification of low degree L-Data (up to degree 2).

Badly approximable numbers in Twisted Diophantine Approximation
I will talk about the approximation of an n-dimensional real vector y by the integer multiples qx of another fixed arbitrary real vector x. This problem can be viewed in terms of toral rotations if we identify the torus with the cube [0,1)^n and we think of qx as the position of the origin after q rotations by x. I will define the concept of badly approximable vector in this context and will discuss the "size" of the set of badly approximable vectors. The case n=1 and a particular case in higher dimension have been recently established with the works of Kim, Tseng and Bugeaud-Harrap-Kristensen-Velani. In a recent work with N. Moshchevitin, we establish the case n>1 in all its generality. With N. Stepanova we study this same problem on manifolds.

A 3-variable main conjecture for the Rankin--Selberg convolution of two CM modular forms
For a pair of modular forms Lei, Loeffler and Zerbes have constructed an Euler system attached to their Rankin--Selberg convolution. I will discuss how to use this Euler system to bound the associated Selmer groups in the case where both modular forms have CM.

Local converse problem for $p$-adic $GL_n (F)$
Let $F$ be a non-Archimedean local field​,​ and ​$\psi$ be a​ non-trivial additive character of $F$.​ ​Let $\pi$ be an irreducible smooth generic representations of $GL_n (F)$. For an irreducible smooth generic representation $\sigma$ of $GL_m (F)$, a complex function $\gamma (s, \pi \times \sigma, \psi)$ was defined by Jacquet, Piatetski-Shapiro and Shalika in 1980s. This complex function $\gamma (s, \pi \times \sigma, \psi)$, called local $\gamma$ factor, inherits important arithmetic information of $\pi$ and $\sigma$, for example it plays a critical role in the formulation of local Langlands correspondence for $GL_n (F)$. A folklore conjecture, due to ​Jacquet, states that the family of $\gamma(s, \pi\times \sigma, \psi)$, for all irreducible smooth generic representations $\sigma$ of $GL_m$, with $m=1, 2, ..., [n/2]$, determines the representation $\pi$ up to isomorphism. In this talk, based on joint work with Moshe Adrian, Baiying Liu, and Shaun Stevens, I will discuss some recent progress ​on​ Jacquet's conjecture. All are welcome!

Real and rational systems of forms
Let $ f = (f_1, \dots, f_R) $ be a system of forms of degree $ d $ in $ n $ variables. A classic result of Birch estimates the density of integral zeroes of $ f $ when $ n \gg_{d,R} 1$ is large and the variety $ f = 0 $ is smooth. We give an improvement when $ R \gg_d 1$ is large, and an extension to systems of Diophantine inequalities $ | f | < 1 $ with real coefficients. Our strategy reduces the problem to an upper bound for the number of solutions to a multilinear auxiliary inequality.

Numerical modular symbols for elliptic curves
Given an elliptic curve E over the rationals, the modular symbol attached to E is a map that sends a rational r to the integral from infinity to r of the differential corresponding to E on the upper half plane divided by the Neron period of E. These symbols [r] are known to be rational numbers. All current implementations start by computing the space of all modular symbols of a given level and then use Hecke operators to cut out the symbol associated to E. Instead one can also compute a numerical approximation of the integral to a proven error bound. The question of the denominators of [r] will show up, too.

Overconvergent modular forms and perfectoid Shimura curves
We show a new approach to overconvergent modular forms and overconvergent Eichler-Shimura map which uses crucially the recent work of Scholze on perfectoid Shimura varieties. This gives a non-archimedean analogue of "cz+d" approach to classical modular forms. This is a joint work with David Hansen (Columbia) and Christian Johansson (IAS).

Some Heegner point constructions
In this talk we will show how to construct certain special points on rational elliptic curves in situations where the so-called "Heegner hypothesis" does not hold. More concretely, given a rational elliptic curve with conductor divisible by the square of a prime p, we show how to construct points associated to an imaginary quadratic field K regardless of the factorization of p in K.

On the arithmetic of a family of quartic K3 surfaces
This talk starts with a family of quartic K3 surface, whose general member contains 320 conics. A theme is arithmetic geometry is the existence of rational points, motivating the study of the Picard group of surfaces. We explain how we found the Picard group of this family. Along the way we take a detour to look at K3 surfaces which contains conics defined over the rationals.

Overconvergent modular symbols over number fields
The theory of overconvergent modular symbols, developed by Rob Pollack and Glenn Stevens, gives a beautiful and effective construction of the p-adic L-function of a modular form. They define a 'specialisation map' from the space of overconvergent modular symbols to the space of classical symbols, and the crux of their theory is a 'control theorem' that says that this map is an isomorphism on the small slope subspace. This gives an analogue of Coleman's small slope theorem in the modular symbol setting. In this talk, I will describe their results, and then discuss an analogue of the theory for the case of modular forms over other number fields, for which similar results exist.

Computations done by Buzzard and Kilford (among others) of slopes of overconvergent modular forms gave us great insights into the geometry of the associated eigenvarieties and are the basis of many conjectures. This is an active area of research and in many cases these conjectures are now proven, yet not much is known in the case of Hilbert modular forms. In my talk I will discuss some recent computations of slopes of overconvergent Hilbert modular forms and what they suggest about the geometry of the associated eigenvarieties.

Period integrals and special values of L-functions

In many ways L-functions have been seen to contain interesting arithmetic information; evaluating them at special points can make this connection very explicit. In this talk we shall ask what information is contained in the central values of certain automorphic L-functions (in the spirit of the Gan-Gross-Prasad conjectures) and report on recent progress. We also describe some surprising applications in analytic number theory regarding the ``size'' of a modular form.

The Cohen-Lenstra-Martinet heuristics make predictions about the behaviour of class groups of number fields in families. The idea is that a "random" algebraic object should be isomorphic to a given object A with probability that is proportional to 1/#Aut(A), and this appears to apply to class groups of imaginary quadratic fields. For more general families, the probability distributions proposed by Cohen-Martinet look rather more complicated, and do not seem to fit this general rule, but they have, so far, appeared to agree very well with numerical data. As I will explain in this talk, the heuristics of Cohen-Martinet are in fact wrong in at least two ways. After disproving the heuristics, I will propose ways to correct them. Finally, I will explain that class groups of number fields do actually always obey the general rule stated above, provided one passes to Arakelov class groups. Some work is needed to make sense of this rule for Arakelov class groups, since they typically have infinitely many automorphisms. This is joint work with Hendrik Lenstra.

Rational Points on Cubic Surfaces

A conjecture due to Batyrev and Manin predicts the asymptotic behaviour of the number of rational points of bounded height on cubic surfaces. In this talk I will report on some modest progress towards this conjecture, in joint work with Christopher Frei and Efthymios Sofos.

Green and Mackey Functors in Number Theory

Given a Galois extension with Galois group G, we may associate important number theoretic modules to each subfield and hence each subgroup of G. In this way we often find ourselves considering structures where we have a module or algebra associated to every subgroup of G with maps between them corresponding to induction and restriction. Some examples include class groups of number fields, the Tate module of an elliptic curve as the field varies, or the group cohomology of a G-module.

Mackey and Green functors offer a general framework to study these questions. In my talk I will define these Mackey and Green functors, giving some examples along the way. I will show how Brauer relations can give relations for these functors motivating work by Bartel and Dokchitser and Bartel and myself.

Serre's Uniformity Conjecture for Elliptic Curves with a Rational Cyclic Isogeny

Serre’s uniformity conjecture asks if, for any elliptic curve E over the rationals without complex multiplication, its residual mod l representation is surjective for all primes l > 37. In this talk, I will show how an argument of Darmon and Merel — which is, itself, based on Mazur’s formal immersion technique — can be adapted to prove the conjecture when E admits a non-trivial cyclic isogeny.

Parity of ranks of abelian surfaces

Let K be a number field and A/K an abelian surface (dimension 2 analogue of an elliptic curve). By the Mordell-Weil theorem, the group of K-rational points on A is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the L-series determines the parity of the rank of A/K. Under suitable local constraints and finiteness of the Shafarevich-Tate group, we prove the parity conjecture for principally polarized abelian surfaces. We also prove analogous unconditional results for Selmer groups.

Rational points of varieties with ample cotangent bundle over function fields (with H. Gillet).

Let $K$ be the function field of a smooth curve over an algebraically closed field $k$. Let $X$ be a scheme, which is smooth and projective over $K$. Suppose that the cotangent bundle $\Omega_{X/K}$ is ample. Let $R:=\mathrm{Zar}(X(K)\cap X)$ be the Zariski closure of the set of all $K$-rational points of $X$, endowed with its reduced induced structure. We shall describe a proof of the fact that for each irreducible component ${\mathfrak R}$ of $R$, there is a projective variety ${\mathfrak R}'_0$ over $k$ and a finite and surjective $K^{\mathrm{sep}}$-morphism ${\mathfrak R}'_{0,K^{\mathrm{sep}}}\to {\mathfrak R}_{K^\mathrm{sep}}$, which is birational when the characteristic of $K$ is $0$.

This improves on results of Noguchi and Martin-Deschamps in characteristic $0$. In positive characteristic, our result can be used to give the first examples of varieties, which are not embeddable in abelian varieties and satisfy an analog of the Bombieri-Lang conjecture.

The symplectic argument and the Generalized Fermat Equation

Wiles' proof of Fermat's Last Theorem gave birth to the 'modular method' to study Diophantine equations. Since then many other equations were solved using generalizations of this method. However, the success of the generalizations relies on a final "contradiction step" which is invisible in the original proof.

In this talk, we will discuss why developing methods to distinguish Galois representations is relevant to this contradiction step. In particular, we will explain show how the "symplectic argument" can be used to succeed in this last step. We will illustrate the method with example of applications to special cases of the Generalized Fermat equation x^r + y^q = z^p.

Orders of reductions of non torsion points for Abelian varieties over number fields

Genus stability in $p$-adic towers of curves

The topic of this talk is genus growth in $\mathbb{Z}_p$-towers of curves in characteristic $p$. For example, by work of Katz and Mazur we know that the genus of the $n$-th Igusa curve is given by a quadratic in $p^n$. This
quadratic genus growth property is known as genus stability. We show that any tower arising from the monodromy of a family of varieties is genus stable. This confirms a conjecture of Daqing Wan for ordinary towers, and is the first step towards the geometric Iwasawa theory program devised by Wan. Time permitting,
we will explain how our results allow us to deduce a new proof of a recent theorem of Drinfeld-Kedlaya bounding consecutive slopes of $F$-isocrystals.

Non-abelian Bloch--Kato Selmer sets and heights on abelian varieties

For a smooth variety over a number field, one defines various different homology groups (Betti, de Rham, etale, log-crystalline), which carry various kinds of enriching structure and are thought of as a system of realisations for a putative underlying (mixed) motivic homology group. Following Deligne, one can study fundamental groups in the same way, and the study of specific realisations of the motivic fundamental group have already found Diophantine applications, for instance in the anabelian proof of Siegel's
theorem by Kim.

In this talk, we will discuss recent work examining further the link between the motivic fundamental group and Diophantine geometry, specifically the theory of canonical heights on abelian varieties, presenting a triad of theorems relating local components of heights to specific realisations of certain motivic
fundamental groups. In order to make precise the p-adic instance of this theorem, we need to define certain non-abelian analogues of Bloch--Kato Selmer groups, and the main body of the talk will be devoted to unpacking the basic theory of such objects.

Distribution of ray class groups: 4-ranks and general model

In 1983 Cohen and Lenstra provided a probabilistic model to guess correctly statistical properties of the class group of quadratic number fields, viewed as an abelian group. In 2016 Ila Varma computed the average 3-torsion of ray class groups (of fixed integral conductor) of quadratic number fields. She asked whether it was possible to explain her results by a generalization of the Cohen-Lenstra model. In this talk I will explain how to construct a model to guess correctly statistical properties of ray class groups (of fixed integral conductor) of imaginary quadratic number fields, viewed as short exact sequences of Galois modules. This model agrees with Varma's results for imaginary quadratics.
Then I will explain the main steps of a proof of this new conjecture for an exact sequence related to the 4-rank of the ray class group and the class group, when the discriminants are coprime to the conductor. As a cruder corollary one obtains the joint distribution of the 4-ranks of the two groups. The methods and
the results are a natural extension of the ones of Fouvry and Kluners. This is joint work with Efthymios Sofos.

Generalization of Dasgupta's factorization formula of p-adic Rankin L-series

Recently Dasgupta has proved that $p$-adic $L$-function associated to the tensor square of a p-ordinary eigenform factors as the product of the symmetric square p-adic L-function of the form with a Kubota-Leopoldt p-adic L-function. In this talk, I will try to explain how to generalize this in different contexts. Firstly, I will indicate how to remove the ordinarity assumption. If time permits, I will also indicate how to generalize it into three variable case of $p$-adic $L$-function of $f \otimes g \otimes g$ where $f,g$ are $p$-ordinary eigenforms. This is joint work with Carlos de Vera Piquero.

Local epsilon-isomorphisms in families

Given a representation of $Gal_{Q_p}$ with coefficients in a p-adically complete local ring $R$, Fukaya and Kato have conjectured the existence of a canonical trivialization of the determinant of a certain cohomology complex. When $R=\mathbb{Z}_p$ and the representation is a lattice in a de Rham representation, this trivialization should be related to the $\varepsilon$-factor of the corresponding Weil--Deligne representation. Such a trivialization has been constructed for certain crystalline Galois representations, by the work of a number of authors. I will explain how to extend these trivializations to certain families of crystalline Galois representations. This is joint work with Otmar Venjakob.

L^q norms of Dirichlet polynomials for small q

The L^q norms of Dirichlet polynomials are known to share certain similarities with moments of the Riemann zeta function. For q>1 it is known that these norms have a completely analogous asymptotic formula to the (conjectural) formula for the moments of the zeta function. However, for small q there appears to be some discrepency. In this talk we will discuss these issues and describe some recent results on bounds for the L^q norms when q is small.

Post-quantum cryptography based on supersingular isogeny problems

We review existing cryptographic schemes based on the hardness of computing isogenies between supersingular elliptic curves, and present some attacks against them. In particular, we present new techniques to accelerate the resolution of isogeny problems when the action of the isogeny on a large torsion subgroup is known, and we discuss the impact of these techniques on the supersingular key exchange protocol of Jao-de Feo.

Perfect Powers that are Sums of Consecutive like Powers

In this talk, we present some of the techniques used to tackle subfamilies of the Diophantine equation (x+1)^k + (x+2)^k + ... + (x+d)^k = y^n. We compare two very different approaches which naturally arise when considering the parity of k. We present all integer solutions, (x,y,n) to the equation in the case k=3, 1<d<51 (joint work with Mike Bennett - UBC and Samir Siksek - Warwick), and a (natural) density result when k is a positive even integer, showing that for almost all d at least 2, the equation has no integer solutions, (x,y,n) with n at least 2 (joint work with Samir Siksek - Warwick).

The p-adic Stark conjecture at s=1 and applications

Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at s=1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F. When E=F this is equivalent to Leopoldt’s conjecture for E at p and the ‘p-adic class number formula’ of Colmez. In this talk we discuss the p-adic Stark conjecture at s=1 and applications to certain cases of the equivariant Tamagawa number conjecture (ETNC). This is work in progress joint with Andreas Nickel.

My converse theorem obsession

In January of this year, Min Lee and I hosted a Heilbronn Focused Research Workshop on the "Sarnak Rigidity Conjecture". I will introduce this conjecture and explain why it's hopelessly difficult to make progress on, before turning to some related problems concerned with "converse theorems" in the theory of modular forms that turned out to be more tractable. There will be heavy helpings of philosophy and also some theorems. This is joint work with S. Bettin, J. W. Bober, B. Conrey, M. Lee, G. Molteni, T. Oliver, D. J. Platt, and R. S. Steiner.