Niven Achenjang (credit to Ashvin Swaminathan)
Lecture 1: Introduction to counting orbits of coregular representations: 3-torsion in quadratic class groups
In our first lecture on counting orbits of coregular representations, our goal is to compute the average size of the 3-torsion subgroup of the ideal class group of a quadratic field (or of a quadratic order). Following Davenport–Heilbronn, we translate this into an orbit-counting problem for SL_2(\Z) on the space V_\Z^* of integer-matrix binary cubic forms. The key input is the Bhargava–Varma parameterization, which encodes an orbit on V_\Z^* as an equivalence class of triples (O, I, \delta) for a quadratic order O, an ideal I, and a cube-defect \delta; under this parametrization, 3-torsion ideal classes of a maximal order are literally orbits. Counting the irreducible orbits (corresponding to non-identity elements of the class group) by Bhargava's averaging method, sieving for maximal orders, and applying a class-number formula will yield the theorem for quadratic fields. Afterwards, we will show that to obtain an average for quadratic orders, one needs to also count the reducible orbits, which live in the cuspidal region of the relevant fundamental domain.
Lecture 2: Introduction to counting orbits of coregular representations: counting in the cusp and 2-Selmer groups
Continuing from Lecture 1, we will use the slicing method of Shankar-Siad-Swaminathan-Varma to count reducible orbits in Gauss's fundamental domain. We apply this count to finish our determination of the Bhargava-Varma average size of Cl_3(O) over quadratic orders. Switching gears, we will then describe how this same machinery (parameterization, Bhargava averaging in the main body, slicing in the cuspidal region) can be applied to other representations. Namely, we will take up the action of PGL_2 on the space of binary quartic forms, which parametrizes (modulo local-solubility conditions) the 2-Selmer groups of elliptic curves. We will sketch how the machinery yields the theorem of Bhargava and Shankar that the average size of the 2-Selmer group of an elliptic curve is 3. We then observe that via the accidental isomorphism PGL_2 \cong SO_3, the binary quartic representation is isomorphic to SO_3 acting on symmetric 3×3 matrices of trace zero; the analogous action of SO_ on symmetric (2n+1)x(2n+1) matrices parametrizes 2-Selmer groups of hyperelliptic Jacobians. Remarkably, the counting machinery carries over verbatim, yielding the Bhargava-Gross theorem: the average size of the 2-Selmer group of the Jacobian of a hyperelliptic curve with a rational Weierstrass point is at most 3, independently of the genus.
Louis-Pierre Arguin
The goal of this lecture series is to introduce the theory of extrema of log-correlated processes and explain their deep connections with conjectures on the Riemann zeta function. We aim to present key probabilistic techniques used to study extreme values, illustrate them in tractable models such as the Steinhaus model, and provide insight into the methods used in the study of the zeta function itself.
Lecture 1
In Lecture 1, we will explain the general theory of extreme values in probability and how the FHK conjecture is closely tied to extrema of branching random walks and log-correlated processes.
Lecture 2
In Lecture 2, we will prove a version of the conjecture for the Steinhaus model of the zeta function.
Lecture 3
In Lecture 3, we will sketch the proof of the FHK conjecture for the zeta function. The techniques developed throughout the proof (moments of Steinhaus model & Dirichlet polynomials, ballot theorems, Berry-Essen estimates, etc) turn out to be also useful in the study of random multiplicative functions and multiplicative chaos. They will be explained in the probability working sessions I and II.
Tim Browning
Points on varieties
I will survey a few of the tools from analytic number theory that have been developed to study questions about local and global solubility of Diophantine equations.
The first lecture will survey the scope of the circle method as well as provide a whistle stop tour of the method, as it applies to diagonal cubic forms.
The second lecture will discuss the large sieve and its application to a result of Serre from the 1990s, showing that random conics don't contain rational points. I'll also discuss a conjecture of Loughran and Smeets from 2016 and some recent progress towards it (and its refinements).
In the final lecture I'll go over the proof of the Ekedahl (aka geometric) sieve and I'll show how it can be used to address the conjecture of Loughran-Smeets for families of varieties with no reducible fibres in codimension 1.
This lecture course is introductory and will be suitable for graduate students at any stage.
Alexandra Florea
Statistics of L-functions over finite fields
These lectures will survey statistical properties of L-functions over function fields (with a focus on F_q[t]). Results in this setting often mirror phenomena in the integer setting and can provide valuable insight into classical problems in analytic number theory. In recent years, there has been progress on such questions over function fields, and we will highlight some of these developments.
A tentative outline of the lectures (subject to change) is as follows:
Lecture 1: Introduction to L-functions over function fields and their basic properties; overview of one-level density results for zeros in families of L-functions over both number fields and function fields.
Lecture 2: Discussion of the Katz-Sarnak density conjectures and their function field analogues, including cases where these conjectures are known to hold.
Lecture 3: Applications of one-level density results to nonvanishing of L-functions over function fields; further topics might include the distribution of points on curves over finite fields and fluctuations in the zeros of zeta functions of such curves.
Adam Harper
Introduction to random multiplicative functions
Random multiplicative functions provide a model for certain functions of number theoretic interest, like Dirichlet characters. They can also be important tools for proving results about those functions, as well as an interesting probabilistic object in their own right.
In these lectures, I will try to introduce and motivate random multiplicative functions; explain some ways of thinking about them, which have proved to be useful in various recent works; and present details of some proofs.
A rough plan for the three lectures (but this will vary depending on how things go) is:
Lecture 1: definition of RMFs, sample results (probabilistic and number theoretic), some simple calculations, thinking about RMFs via conditioning.
Lecture 2: the main steps, and some details, of the "better than squareroot cancellation" upper bound for low moments.
Dimitris Koukoulopoulos
Poissonian multiplicative chaos and divisor problems
The complex divisor function τ(n;ξ)=∑_d^ has many analogous properties to the Riemann zeta function. It is related to Poissonian multiplicative chaos, an object closely related to the Gaussian multiplicative chaos that rules the distribution of zeta. I will discuss the theory of the complex divisor function and study the distribution of its maximum value when ξ ranges over the interval [1,2]. This is based on ongoing joint work with Louis-Pierre Arguin and Paul Bourgade.
Melanie Matchett-Wood
Random profinite groups and arithmetic statistics
Lecture 1: Random Finite Abelian Groups
We describe the distribution on finite abelian groups that Cohen and Lenstra conjecture the (odd part of the) class groups of real quadratic fields are distributed as. We will discuss the factor involving the automorphism group in the denominator. We will discuss how this distribution arises from taking cokernels of random matrices with uniform/Haar coefficients, but also from cokernels of much more general random matrices in a universality result. We discuss the notion of moments of random groups.
Lecture 2: Random Profinite Groups
Even considering the distribution of class groups of imaginary quadratic fields will already lead us to considering distributions on profinite groups. More generally, we will consider random profinite groups, both in natural models from generators and relations, and as arise in conjectures on distributions of Galois groups of maximal unramified extensions of number fields.
Lecture 3: Decorated Groups and Reconstruction of Random Algebraic Objects from Moments
We discuss how the conjectures in number theory often require one to consider the distribution of objects with all their structure, including pairings, and actions of other groups.
We also discuss what the moments of such random algebraic objects with additional structure are, and how the distribution of a random object can be explicitly constructed from its moments.
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Carlo Pagano
Hilbert 10 for ring of integers of number fields
We will overview joint work with Peter Koymans settling Hilbert 10th problem for finitely generated rings. We will focus largely on our main results about diophantine stability of elliptic curves under quadratic extensions, and our subsequent work on elliptic curves of rank exactly equal to 1 over general number fields. We will explain in detail the Markov chain controlling 2-descent under quadratic twisting, overview how it has been used by a number of authors to establish spectacular results about the statistical behavior of 2-Selmer in quadratic twist families, and finally we will articulate how these tools, once combined with results from additive combinatorics, can be used to prove the theorems above.
Maksym Radziwill
Lecture 1 — Mollified Moments of the Riemann Zeta Function and Applications
After recalling basic properties of zeta and standard approximations inside the critical strip, I will discuss moments estimates and introduce the method of mollified moments. I will then discuss the analogy between the Selberg sieve and the problem of finding the optimal mollifier M(s) minimizing
\int_^ |1 – \zeta(1/2+it)M(1/2+it)|^2 dt.
I will then show how these ideas can be applied to zero density estimates and to counting zeros on the critical line.
Lecture 2 — Brun–Hooley Mollifiers and Fractional Moments
I will introduce a different kind of mollifier, modelled on the Brun-Hooley sieve rather than the Selberg sieve, as in my joint work with Soundararajan. After explaining the relevant features of the Brun-Hooley sieve and how to construct this mollifier, I will give an application to upper and lower bounds for fractional moments of zeta. These bounds admit a natural interpretation as distributional results for \log|\zeta(1/2+it)|.
Lecture 3 — Selberg's Central Limit Theorem via Mollifiers
Using the Brun-Hooley mollifiers from the previous lecture, I will give a proof of Selberg's central limit theorem for \log|\zeta(1/2+it))|, following my paper with Soundararajan. There is an instructive analogy with fractional moments
\int_^ |\zeta(1/2+it)|^ dt
when k tends to 0: in this regime, the contribution of large primes ceases to matter, and I will explain why.
Mehtaab Sawhney (Columbia U & Open AI)
Multiplication Table Problem and Permutations fixing a k-Set
We discuss the recent resolution of the multiplication table problem and in particular focus on the model problem of determining the density of permutations fixing a k-set. Based on joint work with Ben Green.
Alex Smith
Part I: The tools
We quickly introduce the basic definitions of Galois cohomology, including H^0 and H^1, the long exact sequence, Shapiro's lemma and the restriction map, and cup product. For finite modules over local fields, the Galois cohomology groups are finite, relatively easy to compute, and satisfy a nice duality principle.
Given a module over a number field F, we introduce the notion of « decorating » a module with local conditions; for each place v of F, this decoration is a choice of a « local conditions' » subgroup of the first cohomology for M over the local field F_v. Such an object has an associated Selmer group, which is the group of elements in the first cohomology for M over F that land in every local conditions subgroup. We specifically look at the Selmer groups associated to elliptic curves and the Selmer groups that calculate portions of class groups. We also introduce a duality on our category of decorated modules.
An Euler characteristic-esque calculation due to Tate gives the Greenberg-Wiles formula, which relates the size of the Selmer group of a decorated module with the size of the Selmer group of the dual module. We mention the implications of this result for reflection theorems for class groups.
Part II: Applications to arithmetic statistics
It has been known for about twenty years that, in families of elliptic curves over the rationals with a rational degree p isogeny, the
p-Selmer groups can have unbounded average size. In particular, this is true for the 2-Selmer groups in the family of elliptic curves with partial 2 torsion.
In all known cases, this phenomenon can be explained using the Greenberg-Wiles formula. We focus on the case of the 2-Selmer groups of quadratic twist families of elliptic curves. In these cases, the distribution of 2-Selmer ranks is known, but can be very complicated due to the influence of the Greenberg-Wiles formula.
We also give more examples for how Galois cohomology can be used to study class groups and their distributions.
Christian Webb
Introduction to multiplicative chaos
The aim of the course is to learn a bit about the use of a probabilistic concept known as multiplicative chaos to describe the statistical behavior of the Riemann zeta function on the critical line.
The course will begin with motivating a probabilistic model for the statistical behavior of the zeta function, known as the Steinhaus model.
We will then discuss some of the general results from the theory of multiplicative chaos, and connect the Steinhaus model to the theory of multiplicative chaos.
Time permitting, we will discuss why the statistical behavior of the zeta function is described by the Steinhaus model (and thus multiplicative chaos) in the setting that is currently understood.
