This weekly seminar aims to bridge the gap between the senior seminars in London (here and here) and run-of-the-mill number theory by providing a relaxed atmosphere for talks that is mainly post-graduate student driven.
If you would like to suggest a talk (that you could give or want someone else to give), please speak to one of the current organisers: Johannes Girsch (Imperial) or Ashwin Iyengar (King's). We also have a mailing list via which we spread the seminar announcements.

### Summer 2019

Place: Imperial College London, Huxley 130
Time: Tuesdays, 5-6pm Please check individual events for changes of place or time!

2 July 2019 — Matthew Honnor — Huxley 130
Gross-Stark units and Shintani zeta functions
In the 1980s Tate stated the Brumer-Stark conjecture which, for a totally real field $$F$$ with prime ideal $$\mathfrak{p}$$, conjectures the existence of a $$\mathfrak{p}$$-unit, the so called Gross-Stark unit, which has $$\mathfrak{P}$$ order equal to the value of a partial zeta function at $$0$$, for a prime $$\mathfrak{P}$$ above $$\mathfrak{p}$$. In this talk I will introduce this conjecture and a conjecture of Dasgupta which uses Shintani zeta functions to give a conjectural formula for these units in the form of a $$\mathfrak{p}$$-adic integral.

21 May 2019 — Lukas Melninkas (Université de Strasbourg) — Huxley 130
Local root numbers of abelian varieties with real multiplication.
Global root numbers appear in the conjectural functional equations of $$L$$-functions and can be computed as products of local factors. I will define local root numbers in terms of Weil-Deligne representations and will explain how a structure of real multiplication allows us to find formulas relying these numbers with other invariants of abelian varieties.

14 May 2019 — Nicolas Müller (ETH Zürich) — King's College London - Strand 2.39, This week exceptionally at King's from 5:50pm to 6:50pm!
Hyperelliptic curves with many automorphisms
A smooth projective connected complex algebraic curve is said to have "many automorphisms" if it cannot be deformed non-trivially together with its automorphism group. It is known that they correspond to isolated points in the moduli space of curves. Given his life-long interest in special points on moduli spaces, Frans Oort asked whether these points are special, i.e., whether the jacobian has complex multiplication. In joint work with Richard Pink we answer this question in the special case of hyperelliptic curves.

9 May 2019 — Kevin Buzzard — Huxley 340, This week exceptionally Thursday 5pm-6pm!
What is a perfectoid space?: Video
I will explain what a perfectoid space is.

### Spring 2019

Place: University College London, Math Dept. Room 707
Time: Thursdays, 5-6pm Please check individual events for changes of place or time!

28 March 2019 — Alexandre Daoud — UCL Maths 707
Weil-étale Cohomology and Higher Rank Euler Systems
In 2005 Lichtenbaum conjectured the existence of a 'Weil-étale topology' for arithmetic schemes which which should, in many ways, be better suited to the study of special values of derivatives of motivic $$L$$-functions than the étale topology. No such topology has so far been constructed which gives the right cohomology in degrees $$>= 4$$. However, the recent work of Burns, Kurihara and Sano has led to the construction of a perfect complex $$C$$ which computes the cohomology of the constant Weil-étale sheaf $$\mathbf{Z}$$ over the $$S$$-integers. In this talk I will briefly state Lichtenbaum's conjecture and say something about the construction of $$C$$, as well as outline a link to the equivariant Tamagawa Number Conjecture (a.k.a equivariant Bloch-Kato).

In the second half of the talk I will report on joint work with Burns, Sano and Seo concerning the existence of (globally-valued) higher rank Euler systems which follows as a purely algebraic consequence of the existence of $$C$$.

21 March 2019 — Lorenzo La Porta — UCL Maths 707
In the classical theory of (elliptic) modular forms mod p the action of the operator $$\theta = q\cdot d/dq$$ on the weight filtration is a useful tool, with some interesting applications, such as the study of the so called "theta cycles". Because of this, one would like to define similar operators on more general classes of automorphic forms. In order to achieve such a generalisation, a sensible thing to do is to describe the classical $$\theta$$ in a more geometric fashion. In this talk I will discuss one way of doing this, following a construction due to N. M. Katz.

14 March 2019 — Alex Torzewski — UCL Maths 707
Hermite-Minkowski and Geometry
In nice situations, the classical Hermite-Minkowski theorem ensures that there are only finitely many Galois representations satisfying given properties. We explain how this can be used to prove finiteness results in arithmetic geometry. We also discuss some truly geometric analogues of Hermite-Minkowski.

28 February 2019 — Chris Williams — UCL Maths 707
The Bianchi Eigenvariety
In addition to being conceptually beautiful, the theory of $$p$$-adic families of modular forms has proved extremely useful. These families are captured geometrically in objects known as 'eigenvarieties'. The original construction of the eigenvariety uses underlying geometric structure, and thus doesn't generalise to settings with no geometry e.g. the Bianchi setting of $$\mathrm{GL}(2)$$ over imaginary quadratic fields. I will explain a different approach of Ash-Stevens via (overconvergent) cohomology, and how it adapts to the Bianchi setting. I will also give some examples of how the Bianchi eigenvariety is 'badly behaved'. This should tie in neatly with the ongoing study groups on the work of Venkatesh, and in particular with Ana's upcoming talk in the 'Hecke' track (though I will not assume facts from the study group).

21 February 2019 — Chris Birkbeck — UCL Maths 707
Diamonds and some Applications
In Scholze’s 2014 course at Berkeley, he introduced the class of geometric objects known as diamonds. In my talk I will define them and give some of their properties. I will also give some "real world" applications to classifying extensions of (semistable) vector bundles on the Fargues-Fontaine curve.

14 February 2019 — Pol van Hoften — UCL Maths 707
On Ribet's Proof of the $$\epsilon$$-conjecture
Gerhard Frey suggested in the 1980s that the (then conjectural) modularity of elliptic curves over the rational numbers could be used to prove Fermat's last theorem. Serre showed that his argument worked up to a small conjecture (the epsilon conjecture), which was proven by Ribet in 1986. In this talk, I will discuss the statement of the conjecture and some of the ingredients of the proof.

7 February 2019 — Andy Graham — UCL Maths 707
Perrin-Riou's Logarithm Map
Let $$V$$ be a crystalline $$p$$-adic representation of the absolute Galois group of $$K$$, where $$K$$ is a finite extension of $$\mathbf{Q}_p$$. In her seminal work, Perrin-Riou constructed a "big logarithm map" which interpolates the Bloch-Kato logarithm and dual exponential maps for twists of $$V$$. This map allows one to associate a $$p$$-adic $$L$$-function to compatible systems of Galois cohomology classes for $$V$$. In this talk I will describe this map (and all of the above terminology) and perhaps say a few things about Berger's construction using $$(\varphi, \Gamma)$$-modules.

31 January 2019 — Eran Assaf — UCL Maths 707
Serre Weight Conjectures
Serre's modularity conjecture, as originally formulated in 1973, predicted that every odd irreducible two dimensional continuous Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the level and weight of the modular form. It is now known as the modularity theorem, after being proved by Khare and Winterberger in 2009. In this talk, we will review the original conjecture, relate it to the Langlands programme, and introduce several generalizations. If time allows, we will discuss the difference between generic and non-generic representations and talk about work in progress trying to extend the picture to non-generic representations.

24 January 2019 — Nikoleta Kalaydzhieva — UCL Maths 707
Continued Fractions in Hyperelliptic Function Fields
In 1826 Abel observed the elementary integral
$$\int\frac{dx}{\sqrt{x^2+2bx+c}}=\log\left(x+b+\sqrt{x^2+2bx+c}\right)$$
Later, Chebyshev explained that this is a simple example of a more general phenomenon related to the theory of continued fractions. In this talk I will give you a quick introduction into the world of continued fractions over function fields. We will then discuss periodicity and their connection to hyperelliptic curves and units.

### Autumn 2018

Place: King's College London, Strand 2.28 (subject to change: the exact locations will be posted here and e-mailed out each week).
Time: Tuesdays, 5-6pm. Please check individual events for changes of place or time!

11 December 2018 — Rebecca Bellovin — Strand 2.29
Honda-Tate Theory
A theorem of Honda and Tate gives a remarkably simple classification of isogeny classes of abelian varieties over a finite field. I will sketch the statement and proof of the classification, and if time permits, I will discuss some consequences.

4 December 2018 — Andrea Dotto — Strand 2.28
Many words, some of which are "shtuka".
The cohomology of modular curves realizes instances of the Langlands correspondence over the rationals. Over function fields, you get a similar statement for the cohomology of moduli spaces of elliptic modules and shtukas. Elliptic modules are formal analogues of elliptic curves, they admit uniformization, level structures, etc. But what about shtukas? The aim of this talk is to present a geometric perspective on these objects (following Mumford) that goes a step beyond the usual "let's modify a vector bundle" viewpoint, and connects them to a very concrete problem about polynomial equations.

27 November 2018 — Rob Kurinczuk — Strand 2.29
Local Langlands in Families
The local Langlands correspondence provides a remarkable connection between the representation theory of Galois groups and general linear groups over $$p$$-adic fields. For a prime $$\ell$$ different to $$p$$, this correspondence was interpolated in $$\ell$$-adic families in the recent pioneering work of Emerton-Helm-Moss. I will explain what all of this means, and describe current joint work with Jean-François Dat, David Helm, and Gil Moss on a conjectural generalisation to a quite general class of reductive $$p$$-adic groups. There will be some overlap with David Helm's talk at the 24th London-Paris Number Theory Seminar; we will include more background at the start and give less detailed statements at the end.

20 November 2018 — Sarah Nowell — Strand 2.28
Models of Hyperelliptic Curves Arising from Cluster Pictures
For an elliptic curve Tate's algorithm allows us to find the special fiber of the minimal regular model from a minimal Weierstrass equation. It is natural to ask if there a way of generalising this to higher genus curves. It turns out that the method used in Tate's algorithm does not generalise to higher genus. However for a hyperelliptic curve $$C: y^2=f(x)$$, a huge amount of information can be extracted from the $$p$$-adic distances between the roots of $$f(x)$$. I will introduce a way of visualising these $$p$$-adic distances through cluster pictures and touch briefly on different models of curves. I will conclude by giving a rough idea of how one can use cluster pictures to obtain the minimal regular model of any hyperelliptic curve, this is joint work with O. Faraggi.

13 November 2018 — Ben Heuer — Strand 2.29
$$p$$-adic modular forms via non-archimedean curves
Ask Wikipedia and it will tell you that a modular form is a function on the complex upper half plane satisfying a few random conditions. A more algebraic perspective is that modular forms are functions on moduli spaces of elliptic curves equipped with extra data which tell you something about the differentials of that elliptic curbe. The upper half plane is a complex analytic instance of such a moduli space. In this talk, I will discuss how one can do a similar thing in the $$p$$-adic world, using non-archimedean moduli spaces like rigid analytic Igusa curves to define $$p$$-adic modular forms.

6 November 2018 — Damián Gvirtz — Strand 2.28
Brauer Groups of Surfaces
The Brauer group of an algebraic variety is an important invariant in the obstruction theory of rational points. To understand it, one can find a filtration into several parts, constant, algebraic and transcendental. The third one is the least understood. Relying on homological algebra work by J.-L. Colliot-Thélène and A. Skorobogatov and complex multiplication on K3 surfaces by Rizov-Valloni, I will show how to fully classify the transcendental part of diagonal quartic surfaces over $$\mathbf{Q}$$ and sketch further cases. The case of diagonal quartics is joint work with A. Skorobogatov.

30 October 2018 — Giada Grossi — Strand 2.28
Iwasawa main conjectures for elliptic curves and arithmetic consequences
Iwasawa theory was introduced around 1960 in the context of class groups of $$\mathbf{Z}_p$$-extensions of number fields. In the 1970's the ideas of Iwasawa theory were extended to elliptic curves: the Main Conjectures relate certain Galois cohomology groups (the algebraic side) with $$p$$-adic $$L$$-functions (the analytic side). Like the original Main Conjecture of Iwasawa, these Main Conjectures have consequences, via the so-called control theorems, for the related special value formulas. In this talk, I’ll give an overview of these ideas and explain how control theorems combined with Iwasawa main conjectures in the case of elliptic curves imply the $$p$$-part of the Birch-Swinnerton-Dyer formula when the analytic rank is at most one.

23 October 2018 — Petru Constantinescu — Bush House 4.04
The Selberg Trace Formula
My goal is to explain how to develop Selberg's trace formula for compact hyperbolic surfaces and present some of its important applications. A generalisation of Poisson summation formula, it relates the spectrum of the Laplace- Beltrami operator ("analytic side") and the lengths of geodesics on the Riemannian surface ("geometric side"). Time permitting, I will introduce the Selberg Zeta function and give a few hints about the non-compact case.

16 October 2018 — Misja Steinmetz — Strand 2.28
An invitation to p-adic Hodge theory via étale $$\varphi$$-modules
Fontaine's p-adic Hodge theory is quite scary with its huge period rings and mysterious functors. In this talk we will take some baby steps in this field. I will steer clear of the scariness and focus on the more explicit case of étale $$\varphi$$-modules, where the period rings are easier and it is more feasible to do explicit calculations. Nonetheless it gives a nice illustration of techniques used throughout the rest of the theory. Time permitting we will prove an interesting result relating certain étale $$\varphi$$-modules to Artin-Schreier extensions by an explicit calculation. I will try to keep prerequisites to a minimum, although I may assume some familiarity with Galois cohomology towards the end.

9 October 2018 — Raffael Singer — Strand 2.28
Understanding the automorphic side of Local Langlands
Given a $$p$$-adic field $$F$$ and a connected reductive group $$G/F$$, Langlands conjectures a certain natural "correspondence" between the set of Langlands parameters and the set of smooth irreducible representations of $$G(F)$$. We will define smooth representations and a subclass of irreducible representations called cuspidal. We will then discuss how to obtain all irreducible representations from cuspidal ones via parabolic induction.

### Summer 2018

Place: Imperial College, Huxley 139.
Time: Mondays, 5-6pm. Please check individual events for changes of place or time!

25th June 2018 - Hanneke Wiersema
A modular approach to some Diophantine equations
The modular approach to Diophantine equations involves elliptic curves, modular forms and their Galois representations. It was famously used to prove Fermat’s Last Theorem and it is the approach we will use to study a certain class of Diophantine equations. We will associate a Frey curve to a hypothetical solution of a chosen Diophantine equation. Using the modularity theorem and level-lowering, we show the Frey curve is modular of a level dividing the conductor of the Frey curve. Contradicting this modularity for supposed solutions, we are able to prove an explicit result on the existence of solutions for some Diophantine equations.

18th June 2018 - Gregorio Baldi
Independence of points on elliptic curves coming from modular curves
Modular curves naturally parametrise elliptic curves, in particular it makes sense to consider isogeny classes inside such curves. Given a correspondence between a modular curve $$S$$ and an elliptic curve $$E$$, we prove that the intersection of any finite rank subgroup of $$E$$ with the set of points on $$E$$ coming from an isogney class on $$S$$ is finite. The proof relies on Serre’s open image theorem and various equidistribution results.

11th June 2018 - Omri Faraggi
Minimal Regular Models of Hyperelliptic Curves
The Galois representation associated to a hyperelliptic curve is intimately related to the so called minimal regular model of the curve; therefore, if we are interested in Galois representations (as many modern number theorists are), it might make sense to look at models as well. In this talk, I will explore the relation between minimal regular models and so called cluster pictures, which are completely combinatorial objects associated to hyperelliptic curves, carrying information about the $$p$$-adic distances between the roots of the equation defining the curve. It has been shown that the minimal regular model of a hyperelliptic curve with semistable reduction is determined entirely by its cluster picture, and the hope is that this will be true for all hyperelliptic curves as well.

4th June 2018 - Sarah Nowell
Torsion of elliptic curves
One very well known theorem about torsion points on elliptic curves is the Nagell-Lutz Theorem. It is natural to ask if this generalises to elliptic curves defined over number fields. The answer is yes, so I will begin by sketching an explicit proof of a generalisation of the Nagell-Lutz Theorem. Another useful tool, when working with torsion points of elliptic curves, are modular curves. So I will introduce these and show how you may calculate them explicitly. As an example of their application I will give a proof, via some $$2$$-descent, that no elliptic curve defined over the rationals has a torsion point of order $$11$$.

28th May 2018
No seminar due to bank holiday.

21st May 2018 - Andrew Graham
Introduction to Euler Systems
In the late 1980s Kolyvagin introduced the concept of an Euler system - a tool which relates the arithmetic of Galois representations to values of its associated L-function. His construction was inspired by Thaine's work on bounding ideal class groups and his own work on elliptic curves. In this talk I will describe these motivating examples before giving the modern definition of an Euler system due to Rubin. If time permits I will also describe the current method of constructing Euler systems from Siegel units.

14th May 2018 - Joaquin Rodrigues Jacinto
Introduction to the $$p$$-adic Langlands correspondence for $$\mathrm{GL}_2(\mathbf{Q}_p)$$
The $$p$$-adic Langlands correspondence for $$\mathrm{GL}_2(\mathbf{Q}_p)$$ establishes a bijection between certain class of continuous $$p$$-adic Galois representations of dimension $$2$$ and certain class of Banach representations of the group $$\mathrm{GL}_2(\mathbf{Q}_p)$$. We will state this correspondence precisely and give an outline of the main techniques involved in its proof.

7th May 2018
No seminar due to bank holiday.

30th April 2018 - Chris Birkbeck
$$p$$-adic Langlands functoriality
I will explain what $$p$$-adic Langlands functoriality is and give some examples. In particular, we will look at how one can interpolate the classical Jacquet—Langlands correspondence and obtain closed immersions between certain eigenvarieties. Lastly, I will give some applications of this to the study of overconvergent modular forms.

### Spring 2018

Place: UCL Maths Department, room 706.
Time: Mondays, 5-6pm.

19th March 2018 - Enrica Mazzon
Introduction to Berkovich Spaces
Non-archimedean geometry is a theory of analytic geometry over fields equipped with a non-archimedean absolute value, such as the field of $$p$$-adic numbers $$\mathbf{Q}_p$$ or the field of complex Laurent series $$\mathbf{C}((t))$$. Naïve attempts to mimic the definition of a holomorphic function from the complex case do not lead to satisfactory results, due to the fact that the metric topology on a non-archimedean field is totally disconnected, which destroys the global nature of complex analytic geometry. Several approaches have been developed to overcome this difficulty; on of them is Berkovich’s theory of analytic spaces.
In this talk I will give an introduction to this theory. I will define the Berkovich's analytification functor and the basic topological properties of Berkovich spaces, showing some explicit example in low dimension.

12th March 2018 - Jack Lamplugh
$$\ell$$-indivisibily of class numbers in $$\mathbf{Z}_p$$-extensions
I will talk about how one can prove a non-vanishing result for certain $$L$$-values modulo a prime $$\ell \neq p$$. This proves that in certain $$\mathbf{Z}_p$$-extensions the $$\ell$$-part of the class group is bounded. There are also similar results for Selmer groups of elliptic curves. I will also talk about how one can show that for these extensions there is an explicit set of primes of density $$1$$ which do not divide any class numbers in the $$\mathbf{Z}_p$$-extension.

5th March 2018 - Andre Macedo
The Hasse Norm Principle
Given an extension $$L/K$$ of number fields, we say that the Hasse norm principle holds for $$L/K$$ if every element of $$K^\times$$ which is a local norm everywhere is in fact a global norm from $$L^\times$$. In this talk, I’ll present Hasse’s classic norm theorem for cyclic extensions and I'll outline Tate’s description of the knot group (an object measuring the failure of the Hasse norm principle) for Galois extensions. I’ll characterize the validity of this principle in biquadratic extensions and mention some frequency results. If time permits, I'll also talk about the weak approximation property on the associated norm one tori and how it relates to the Hasse norm principle.

26th February 2018 - Carl Wang-Erickson
Ribet's Converse to Herbrand's Theorem
Ribet's proof of the converse to Herbrand's theorem, published in 1976, initiated an approach to the relationship modular forms to arithmetic that has proved very fruitful. I will overview Ribet's result and discuss subsequent developments.

19th February 2018 - Ashwin Iyengar
Basic Arakelov Intersection Theory
This talk will give a gentle introduction to Arakelov theory by explaining Arakelov's original formulation of arithmetic intersection theory in the case of surface. I'll give a brief summary of intersection theory for surfaces over algebraically closed fields, before moving onto Arakelov's paper. I'll explain how to define analogous notions of divisors and intersection pairings on arithmetic varieties. I'll mention the arithmetic Hodge index theorem as analogous to the classical Hodge index theorem, and if time permits I'll talk about how to generalize the picture to higher dimensions to construct arithmetic Chow groups.

12th February 2018 - Raffael Singer
Explicit Local Class Field Theory
Tate gave a construction of the local Artin reciprocity map from Galois cohomological methods. This construction is useful for describing the maximal abelian quotient of the Galois group, but it is too abstract to reconstruct from it the maximal abelian extension or the action of $$K^\times$$ on it. Lubin and Tate showed how to obtain explicit totally ramified abelian extensions by adjoining torsion points of Lubin-Tate formal groups.
In my talk, I will briefly recall the cohomological construction and then give a summary of the theory of Lubin-Tate formal groups. Finally, if there is time, I will try to give a modern point of view via the Lubin-Tate tower and talk about generalizations of local CFT to local Langlands for $$\mathrm{GL}_n$$.

5th February 2018 - Kwok-Wing 'Ghaleo' Tsoi
On the Equivariant Tamagawa Number Conjecture (Mostly Commutative)
In this talk, I will give an overview on the formulation of the Equivariant Tamagawa Number Conjecture (eTNC) with commutative coefficients. In the case of (untwisted) Tate motive, the statement of eTNC can be made very explicit and has been used extensively for making very refined predictions on the structure of arithmetic objects. In this talk, I will survey these ideas. If time permits, I will also describe how this conjectural framework has inspired the current development of (non-commutative) Iwasawa Theory.

29th January 2018 - Ardavan Afshar
The Function Field Sathé-Selberg Formula
In the 1950s Selberg invented an ingenious analytic technique, which would later be developed into the so-called Selberg-Delange method, for enumerating certain arithmetic quantities. In particular, he used it to derive an asymptotic formula, originally due to Sathé, for the number of numbers less than $$x$$ with exactly $$k$$ prime factors, uniformly in $$k$$ (a generalisation of the prime number theorem). We have adapted this technique to the setting of the rational function field over a finite field, where one can eliminate some technicalities and get stronger results, the latter on account of Weil's Riemann Hypothesis for curves over finite fields. We count an analogous quantity to that of Sathé and Selberg, and then refine it to the case of fixed arithmetic progressions or short intervals. This is joint work with Sam Porritt.

22nd January 2018 - Robin Bartlett
Filtered Isocrystals and Galois representations
In this this talk I'm going to explain why filtered isocrystals can be interesting things to think about. I will discuss how they arise from geometry and how the condition of weak admissibility singles out those which have something to do with Galois representations. I will conclude by sketching a short proof of Fontaine--Rapoport which describes a necessary and sufficient condition for the existence of a weakly admissible filtration on an isocrystal, of a given Hodge type.

15th January 2018 - Pol van Hoften
Barsotti-Tate Groups
This talk be be an introduction to Barsotti-Tate Groups (or $$p$$-divisible groups) and we will focus on their relation to abelian varieties. I will first give a crash course in finite flat group schemes before discussing the basic properties of $$p$$-divisible groups. In the end I will indicate how this relates to good Tate modules and good reduction of abelian varieties over $$p$$-adic fields.

### Autumn 2017

Place: King's College, Norfolk Building G.01.
Directions: From main Strand reception, exit the building and turn right. Turn right onto Surrey Street (the street that leads down to Temple station) and walk down to a red brick building; it is the Norfolk Building. Signs are outside the door. Room G.01 is on the left when you walk into the entrance area. Or use this map.
Time: Tuesdays, 5:15-6:15pm.

5th December 2017 - James Newton
Potential modularity of elliptic curves
Elliptic curves defined over the rational numbers have famously been proven to be modular (by Wiles, Breuil, Conrad, Diamond and Taylor). This means that the $$L$$-function coincides with the $$L$$-function of a modular form. One important consequence of this result is that the $$L$$-function of the elliptic curve has analytic continuation to the entire complex plane and a functional equation.
It is not yet known whether elliptic curves over all totally real number fields are modular (in this case "modular" means that the $$L$$-function coincides with the $$L$$-function of a Hilbert modular form). However we do know that they are "potentially modular", which means that they become modular after making some extension of the field of definition. I will say something about how this result is proved, and discuss some applications, including showing that the $$L$$-function of the elliptic curve has meromorphic continuation to the entire complex plane and a functional equation.

28th November 2017
JNT goes on hiatus for the London-Paris Number Theory Seminar.

21st November 2017 - Adam Morgan
Birch and Swinnerton-Dyer conjecture over function fields
Over global function fields one can say much about the Birch and Swinnerton-Dyer conjecture that is known over number fields, largely thanks to the Weil conjectures. I will survey what is known and explain the relevance of the Weil conjectures, introducing the basic theory of elliptic surfaces along the way.

14th November 2017 - Matt Bisatt
How to win a million dollars: Part VII
One of the Millennium Problems is the famed conjecture of Birch and Swinnerton-Dyer which connects the rank of an elliptic curve to the order of vanishing of its $$L$$-function. Combining this with a conjecture of Deligne, we explore twisted $$L$$-functions and how the rank changes under field extension; we manipulate this new relation to force large jumps in the analytic rank in certain cases. The overarching question, and our road to the million dollars, is whether these new predictions match up with what we know; time permitting, we shall discuss a particular case when this is true.
This talk is rated PG for PostGrad (or PleaseGo!): there will be no mention of any schemes, categories or perfectoid spaces at all, only friendly elliptic curves!

7th November 2017 - David Solomon
Lifting the Jacobi Reciprocity Law with Dedekind Sums
Originally introduced by Dedekind in their most basic form to express the transformation law of his eta-function, so-called Dedekind Sums are objects of 'classical mathematics' whose properties and applications deserve to be much better known to contemporary number theorists. A large number of generalisations of the basic sum have been defined and studied by Rademacher, Sczech, Zagier and many other authors up to the present. They find number theoretic applications in areas as diverse as modular forms, the partition function, lattice-point counting and the $$L$$-functions and arithmetic of totally real number fields, to mention only a few. They also crop up in statistics and elsewhere in mathematics. A central feature of each variant is the appropriate variant of a 'Reciprocity Law' that it satisfies. I shall briefly review basic Dedekind Sums, define a new generalisation and discuss its reciprocity law and transformation under matrices in $$\mathrm{GL}_2(\mathbf{Z})$$. I shall also recall the quadratic Jacobi Symbol from elementary number theory and its reciprocity law which may be viewed as a as a congruence in $$\mathbb{Z}/2\mathbf{Z}$$, via a generalisation of the Gauss Lemma. I shall then show how Dedekind sums allow us to lift this form of the Jacobi Reciprocity law to a rather neat equation in the integers related to a certain finite continued fraction expression. If time allows I shall consider what happens when we replace quadratic symbols with cubic and higher powers and explain a possible connection with work of Robert Sczech. All the mathematics in this talk will be of an 'elementary' nature but with connections and applications to deep and interesting number theory.

31st October 2017 - Giada Grossi
Selmer groups and Bloch-Kato conjecture: an introduction
From a $$p$$-adic Galois representation $$V$$, one can construct, under some assumptions, the Bloch—Kato Selmer group, which is an algebraic object, and an $$L$$-function, which is a more analytic one. The Bloch—Kato conjecture relates these different incarnations of $$V$$. In this talk we will recall the standard definition of the Selmer group attached to an abelian variety and try to understand how the Bloch—Kato Selmer group provides its right generalisation. We will then state the (weak form of the) Bloch—Kato conjecture for $$V$$ and see how one recovers the Dirichlet’s unit theorem in the case where $$V$$ is the trivial representation and the Birch—Swinnerton-Dyer conjecture when $$V$$ is the Tate module of an abelian variety.

24th October 2017 - Alex Betts
Introduction to the Etale Fundamental Group
Some of the most useful tools in arithmetic geometry are various analogues of algebro-topological tools, most famously the theory of etale cohomology as used in Grothendieck's and Deligne's proofs of the Weil conjectures. In this talk we will define and survey some of the basic theory of etale cohomology's lesser-known cousin, the etale fundamental group. This group is conjectured to be tied very closely to the Diophantine geometry of hyperbolic curves, and provides finer information on such curves than simply etale cohomology alone.

17th October 2017 - Galen Voysey
Geometric and $$p$$-adic modular forms
I was introduced to modular forms as functions from the upper half plane to the complex numbers satisfying certain conditions. This definition fails to encompass/emphasize the arithmetic properties and link to modular curves. In this talk, we will examine modular forms as sections of sheaves on modular schemes. We'll talk about Katz's geometric definition of modular forms, and see how this perspective lets us use powerful tools in (non-archimedean) algebraic geometry to study modular forms.

10th October 2017 - Andrea Sartori
Vinogradov's conjecture on the least primitive root
In 1930 Vinogradov conjectured that the smallest element which generates $$(\mathbf{Z}/p \mathbf{Z})^\times$$ (the smallest primitive root) is smaller then any power of $$p$$ if $$p$$ is sufficiently large. In general we only have partial results, due to Vinogradov himself and Burgess, towards this conjecture. However, if we allow small exceptional sets much more is known and one could say that Vinogradov conjecture is true for almost every prime. This naturally rises the question weather we can find better bounds on the smallest primitive root given some extra informations about $$p$$. I will discuss what can be said if the factorization of $$p-1$$ is brought into play.

3rd October 2017
JNT goes on hiatus during LSGNT induction week.

26th September 2017 - Dougal Davis
What is a principally polarised abelian variety (and what can they do for me)?
Principally polarised abelian varieties are higher dimensional analogues of elliptic curves that turn up all over the place in both number theory and geometry. In this talk, I will describe how they can be thought of in terms of algebraic geometry, and in terms of analysis, explain some of their basic properties, and give a taste of some of the cool things they can do for you.

### Summer 2017

Place: Imperial College, usually Huxley 140 (except for 2/5, 13/6, 20/6 when we meet in Huxley 213).
Time: Tuesdays, 5-6pm.

27th June 2017 - Raffael Singer
Tate's Thesis
In his doctoral thesis Tate gave a new proof of the meromorphic continuation and functional equation for the modified Riemann Zeta function by interpreting it as an idelic integral. The advantage of his method is that it vastly generalises and gives meromorphic continuation and functional equation for a large class of $$L$$-functions. I will give the main ingredients of Tate's proof and - time permitting - discuss applications such as the analytic class number formula and the Riemann Roch Theorem for curves over finite fields.

NOTE: This week the seminar meets in Huxley 213.
20th June 2017 - Damián Gvirtz
Alterations
Resolution of singularities in mixed and positive characteristic by birational modifications was considered by Grothendieck to be one of the pressing issues of algebraic geometry and in spite of active research, the general result still seems out of reach at present. However, in 1995 Aise Johan de Jong showed to an amazed audience that if one relaxes the birationality condition of isomorphic function fields to allow finite extensions, any variety can be made regular. In this talk I will try to answer two questions: 1. What can de Jong's result do for me? In many arithmetic cases, resolution by alterations may be enough to draw conclusions. But even in characteristic 0, there are some new results 2. What can de Jong's proof do for me? One source of amazement in 1995 must have been that the tools used had been around for quite a while, but it took the creativity of de Jong to put them together. It is certainly worthwhile knowing them.

NOTE: This week the seminar meets in Huxley 213.
13th June 2017 - Domenico Valloni
Absolute Hodge Classes
In this talk we are going to talk about absolute Hodge classes. Starting from the definition, we are going to explain how to build a category of motives using these classes as correspondences, and why it behaves very well for practical computations (i.e. we are going to show that some of the standard conjectures are automatically true in this category and so on). Later we are going to talk about the Variational Hodge Conjecture and Deligne's Principle B, and as an example we will show the the Kuga-Satake correspondence is Absolute Hodge. Finally we are going to state Deligne's Theorem on Absolute Hodge Classes for Abelian varieties and show how it implies an inclusion of the Mumford Tate Conjecture.

NOTE: Due to the London-Paris Number Theory seminar, this week we meet in Room 500 of the UCL Maths Department.
6th June 2017 - Marco D'Addezio
Survey on Deligne Conjectures for lisse sheaves
In Weil 2 Deligne has formulated some deep conjectures about lisse sheaves on normal varieties over a finite field. The Langlands Correspondence for $$\mathrm{GL}_n$$ over a function fields gives as a consequence a positive answer to the conjectures for smooth curves. We will explain briefly how to extend this result to varieties of higher dimension. Because of the lack of a Langlands Correspondence this is performed via a clever reduction to the case of curves. At the end of the talk I will present the problems appearing when the varieties are singular.

30th May 2017 - Michele Giacomini
O-minimality and the Manin-Mumford conjecture
In 2008 Pila and Zannier gave a new proof of the Manin-Mumford conjecture about torsion points in abelian varieties. The interest in this new proof is due to the fact that it is quite elementary and it can be applied also to similar questions. The proof is based on counting rational points in sets definable in an o-minimal structure. In this talk, I will give a quick introduction to o-minimal structures and Pila and Wilkie’s result counting rational points in definable sets. Then I will prove the Manin-Mumford conjecture.

23rd May 2017 - Joe Kramer-Miller
Crystalline Riemann-Hurwitz
Given a covering of curves in char $$p$$ $$C\to X$$ we know how the genus changes in terms of the degree of the map and ramification data. This is basically telling us the dimension of $$H^1_{\mathrm{cris}}(C)$$. The question is then: can we say anything about the Newton polygon of $$H^1_{\mathrm{cris}}(C)$$ in terms of the Newton polygon of $$H^1_{\mathrm{cris}}(X)$$ and some ramification data? We describe Crew's result for the unit root subcrystals and we'll share some of our hopes/dreams about how to approach the more general situation. If time permits we will explain the relationship to arithmetic progressions in slopes of "abstract eigencurves".

16th May 2017 - Andrea Dotto
Langlands parameters for unramified representations
The local Langlands conjectures formulate a parametrization of irreducible representations of $$p$$-adic groups in terms of arithmetic invariants, closely related to representations of the absolute Galois group of the base field. I'll show how the Satake isomorphism motivates their definition, and sketch how it yields a construction of this correspondence for a certain class of representations.

NOTE: This week the seminar meets as usual in Huxley 140.
9th May 2017 - Tibor Backhausz
Local-global compatibility in completed cohomology for $$\mathrm{GL}_2$$
Completed cohomology is a certain limit of the cohomology of symmetric spaces obtained from reductive groups, used as a replacement for a space of $$p$$-adic automorphic forms. In the $$\mathrm{GL}_2$$ case, we have commuting Galois and adelic $$\mathrm{GL}_2$$ actions on completed cohomology. Emerton has shown, with a few technical hypotheses, that the homomorphism from a Galois representation $$V$$ (over a $$p$$-adic coefficient field) into the completed $$H^1$$ of modular curves decomposes (compatibly with the adelic $$\mathrm{GL}_2$$ action) into a restricted tensor product with each term of the product depending only on the restriction of $$V$$ to the decomposition group of some prime. Moreover, this dependence is given by the $$p$$-adic and (modified) classical local Langlands correspondences; this is an instance of local-global compatibility. There is a modulo $$p$$ version of this theorem, and also an integral statement at the heart of Emerton's paper, describing the structure of the unit ball of the completed $$H^1$$ in terms of the "local Langlands correspondence in families", which I will discuss if time permits.

NOTE: This week the seminar meets in Huxley 213.
2nd May 2017 - Wansu Kim
Introduction to affine Deligne-Lusztig varieties
Affine Deligne-Lusztig varieties are certain group-theoretically defined “spaces”, motivated by the study of the mod $$p$$ reduction of the moduli spaces of abelian varieties with extra structure (such as modular curves, Siegel modular varieties, etc). The definition involves a nice group-theoretic interpretation of the theory of Dieudonné modules. We motivate the definition from the simplest case of modular curves, and work out a few simple examples of more general affine Deligne-Lusztig varieties.

### Spring 2017

Place: UCL Maths Department, room 706.
Time: Mondays, 5-6pm.

There will be no other seminar this term due to the AWS.

6th March 2017 - Martin Orr
Unlikely intersections and point counting
"Unlikely intersections" refers to results of the following form: if an algebraic variety contains lots of "special" points, then this must be explained by the geometrical structure of the variety. The meaning of the word "special" here must be given a precise definition relevant to a particular situation (for example, it could refer to torsion points in an algebraic group). In this talk, I will give an overview of several theorems and conjectures of this form, such as the Manin-Mumford conjecture. Then I will sketch a method of proving such results due to Pila and Zannier, based upon counting rational points in definable sets in an o-minimal structure (an idea which comes from mathematical logic).

27th Ferburary 2017 - Ben Heuer
Eigencurves
An eigencurve is a geometric space that parametrises and $$p$$-adically interpolates Hecke eigenforms. But what does that even mean, "parametrise and $$p$$-adically interpolate", and if I have an eigencurve, then what? In this talk we want to give an introduction to "eigenworld". We will start with different ways to think about $$p$$-adic families of eigenforms, and briefly talk about why people are interested in these sort of things (Galois representations, congruences between modular forms, $$L$$-functions...). We will then talk about the main ideas in the construction of the Coleman-Mazur eigencurve. If time permits, we will also talk about recent generalisations and some open questions about the geometry of eigenvarieties, such as the "spectral halo conjecture".

20th February 2017 - Chris Williams
Modular symbols and why we should care about them
In a recent 'senior' seminar, Christian Johansson gave an introduction to modular symbols and explained why they were useful in the context of constructing eigenvarieties, that is, constructing $$p$$-adic families of modular forms. In this talk I'll give a slightly different (but equivalent) version of the theory and discuss the role of modular symbols in a construction of $$p$$-adic $$L$$-functions, an application that Christian alluded to in his seminar, following work of Pollack and Stevens. If time permits, I'll discuss why modular symbol methods really are easier than the more classical geometric methods for considering generalisations of this theory, particularly in the context of $$\mathrm{GL}(2)$$ over number fields.

13th February 2017 - Emiliano Ambrosi
Specialization of representations of the étale fundamental group and applications
Let $$X \to S$$ be a one dimensional family of smooth projective varieties over a finitely generated field $$k$$. For every rational point $$s$$ of $$S$$ we have an $$\ell$$-adic representation of the absolute Galois group of $$k$$ on the $$\ell$$-adic étale cohomology of the fiber $$X_s$$ of the morphism. We will discuss how the image of these representations vary when $$s$$ is varying in $$S(k)$$. Finally we will show how these results can be used to study problems related to the uniform boundedness of the $$\ell$$-primary torsion of abelian varieties and of the $$\ell$$-primary torsion in the Brauer group of families of K3 surfaces.

6th February 2017 - Netan Dogra
The Picard-Lefschetz formula for curves and a nonabelian generalisation
In this talk I will discuss the Picard-Lefschetz theorem from a relatively naive point of view and explain how it generalises to a proof of Oda's theorem, which states that a curve over a local field has good reduction if and only if the Galois action on a certain quotient of its fundamental group is unramified. First we will have a brief review of models of curves, and then see how to compute the monodromy of a semistable family of curves over a punctured disc (mostly by drawing pictures). A brief detour through deformation theory and Abhyankar's Lemma will tell us that we've actually computed the action of inertia on the p-adic cohomology of any curve over a local field of mixed characteristic $$(0,l)$$ which has semistable reduction. We'll then notice that the same approach tells us how to compute the action of inertia on quite a lot of the etale fundamental group, and to complete the proof of Oda's theorem.

30th January 2017 - Antonio Cauchi
An introduction to the theory of $$p$$-adic families of ordinary forms
After discussing the example of the $$p$$-adic family of Eisenstein series, I will define the space of $$\lambda$$-adic modular forms, state their properties and I will talk about their associated Galois representation. If time permits, I will also describe Ohta's analogous results on the inverse limit of étale cohomology groups of modular curves and discuss an application on the theory of $$p$$-adic interpolations of Siegel units and Eisenstein classes.

23rd January 2017 - Stephane Bijakowski
What are Hasse invariants used for?
If $$E$$ is an elliptic curve over a field of characteristic $$p$$, then $$E$$ is either ordinary or supersingular. This condition can be seen on the structure of the $$p$$-torsion of $$E$$, or using the Hasse invariant. After presenting these notions and the possible applications, I will show how they can be adapted for general $$p$$-divisible groups.

### Autumn 2016

Place: King's College London, room S4.23 (Strand Building).
Time: Tuesdays, 5:00pm.

NOTE: In light of the LSGNT Christmas party, the seminar has been moved to room 706 in the UCL Maths Dept at 5pm.
13th December 2016 - Gregorio Baldi
All you ever wanted to know about motives, but were afraid to ask
About motives Grothendieck wrote: "Here we enter in a mathematical dream, trying to image what 'could be', being insensately optimistic using the partial knowledge we have about the arithmetic properties of cohomology of algebraic varieties." Starting from classical problems in Arithmetic Geometry such as the Weil and Tate conjectures we will see how to interpret them in a cohomological framework. This will lead us to the definition of Weil cohomology and the insensate optimism will lead to the Standard conjectures. Actually such machinery is the shadow of what is happening in a category "universal among all the cohomological theories", namely the category of (pure) Motives. We will present the construction of the Chow motives and, to get a concrete grasp of it, we will discuss the motives attached to curves and abelian varieties.

NOTE: The seminar will start at 5:30pm this week.
6th December 2016 - Misja Steinmetz
Cooking with Serre - the weight recipe
Upon popular request (I clearly have an issue with peer pressure) I am giving a talk about Serre's modularity conjecture (which is a theorem now). The conjecture predicts that any odd irreducible mod $$p$$ Galois representation arises as the reduction of the $$p$$-adic representation attached to a Hecke eigenform. In Serre's original statement of the conjecture, he gives a precise 'recipe' for deducing the level, character and weight of the eigenform from the mod p Galois representation. The aim of this talk is to give a careful motivated statement of the conjecture following Serre's original paper from 1987; in particular, I want to try to give some intuition for where Serre's weight recipe comes from (which seemed like magic to me the first time I read his paper).

29th November 2016 - Otto Overkamp
Néron Models
Néron models will be introduced and their most basic properties discussed. Then we will apply the theory of Néron models to prove the criterion of Néron-Ogg-Shafarevich relating good reduction of Abelian varieties to properties of their associated $$\ell$$-adic Galois representation.

22nd November 2016 - Matthew Bisatt
Getting to the root of elliptic curves
Given an elliptic curve over the rationals, the sign that occurs in the functional equation of its $$L$$-function is known as the root number. I will discuss how the root number can be computed from a Weierstrass model using epsilon factors and how these can predict whether the curve has infinitely many rational points.
Time permitting, I will generalise this to Jacobians of hyperelliptic curves using the newly developed theory of clusters.

15th November 2016
JNT goes on hiatus for the London-Paris Number Theory Seminar.

8th November 2016 - Ben Heuer and Damián Gvirtz
Two guys talking about Perfectoid Spaces
Since learning (back in kindergarten) about structural similarities between non-archimedean local fields in equal and mixed characteristic, you may have wondered whether these could be put into a rigorous framework to transfer results between the two worlds. Perfectoid Spaces as invented by Peter Scholze are such a framework and have already resulted in great applications. We will start with the classical inspiration for Scholze's definitions of perfectoid fields and algebras, then introduce the transfer tools of tilts and untilts, the tilting equivalence and (briefly) how to glue these things to global objects.

1st November 2016 - Ardavan Afshar
Multiplicative Number Theory in Function Fields
I'll begin with separate introductions to Multiplicative Number Theory and Function Fields (so you don't need to know anything beforehand) and then explain why it can be nice to ask questions from Multiplicative Number Theory in the setting of Function Fields. In particular, I'll try to explore both the combinatorial perspective, which allows us to think about things like the Prime Number Theorem and Dirichlet's Divisor Problem, and the more algebraic aspect, which provides tools like Pellet's formula and helps us to understand sums of multiplicative functions.

25th October 2016 - Robin Bartlett
In this first talk I plan to give a gentle introduction to the study of $$\ell$$-adic representations of a $$p$$-adic Galois group ($$\ell \neq p$$). I'll give examples, talk about things like inertia, tame and wild ramification, etc...., and then finish with some kind of discussion of the $$\ell$$-adic monodromy theorem. This last theorem is some result which restricts the shape these \(\ell)-adic representations can take, to the extent that you can, at least in theory, write them* down on finitely many pieces of paper.