Monday, 28 October 2024

waring's problem and Goldbach's conjecture

In these talks, I will explain how the smooth delta-function form of the circle method can be used to count solutions to integral quadratic forms.

We review the theory of exponential sums due to Weyl and van der Corput and consider several applications. If time permits, we also look at the theory of p-adic exponent pairs, as developed by Milićević.

TBA

In this series of lectures, I will give an introduction to the theory of moments of L-functions. I will focus on important examples, such as the moments of the Riemann zeta function and Dirichlet L-functions, as well as some GL_2 families. I will also present some of the important tools for understanding moments, as well as applications of moments.

Tuesday, 29 October 2024

waring's problem and Goldbach's conjecture

In these talks, I will explain how the smooth delta-function form of the circle method can be used to count solutions to integral quadratic forms.

TBA

The notion of congruence (modulo an integer q) was formalised by C. F. Gauss in his Disquisitiones arithmeticae. This is a basic yet fundamental concept in all aspects of number theory. Indeed congruences allow to evaluate and compare integers in way considerably richer than the archimedean order alone permits.

In analytic number theory, several outstanding question -starting with Dirichlet’s theorem on primes in arithmetic progressions- reduce to the of measuring whether some classical arithmetic function (say the characteristic function of prime numbers) correlate with suitable q periodic functions for instance Gauss sums, Jacobi sums or Kloosterman sums. It turns out that these functions, when the modulus q is a prime (to which one can reduce via the Chinese Reminder Theorem) can be recognised as « trace functions». The study of trace functions was initiated by A. Weil in the 1940’s and was pursued by A. Grothendieck in the second half of the century with his refoundation of algebraic geometry and the invention of étale cohomology; it culminated with P. Deligne’s proof of the Riemann Hypothesis for algebraic varieties over finite fields.

In these lectures, we will explain, through various examples, how the theory of trace functions and its subsequent developments in l-adic cohomology, notably the works of Katz and Laumon, have made their way into modern analytic number theory. Several (but not all) of the examples discussed during these lectures originate from joint works with E. Fouvry, E. Kowalski and W. Sawin.

In this series of lectures, I will give an introduction to the theory of moments of L-functions. I will focus on important examples, such as the moments of the Riemann zeta function and Dirichlet L-functions, as well as some GL_2 families. I will also present some of the important tools for understanding moments, as well as applications of moments.

Wednesday, 30 October 2024

waring's problem and Goldbach's conjecture

TBA

We review the theory of exponential sums due to Weyl and van der Corput and consider several applications. If time permits, we also look at the theory of p-adic exponent pairs, as developed by Milićević.

The notion of congruence (modulo an integer q) was formalised by C. F. Gauss in his Disquisitiones arithmeticae. This is a basic yet fundamental concept in all aspects of number theory. Indeed congruences allow to evaluate and compare integers in way considerably richer than the archimedean order alone permits.

In analytic number theory, several outstanding question -starting with Dirichlet’s theorem on primes in arithmetic progressions- reduce to the of measuring whether some classical arithmetic function (say the characteristic function of prime numbers) correlate with suitable q periodic functions for instance Gauss sums, Jacobi sums or Kloosterman sums. It turns out that these functions, when the modulus q is a prime (to which one can reduce via the Chinese Reminder Theorem) can be recognised as « trace functions». The study of trace functions was initiated by A. Weil in the 1940’s and was pursued by A. Grothendieck in the second half of the century with his refoundation of algebraic geometry and the invention of étale cohomology; it culminated with P. Deligne’s proof of the Riemann Hypothesis for algebraic varieties over finite fields.

In these lectures, we will explain, through various examples, how the theory of trace functions and its subsequent developments in l-adic cohomology, notably the works of Katz and Laumon, have made their way into modern analytic number theory. Several (but not all) of the examples discussed during these lectures originate from joint works with E. Fouvry, E. Kowalski and W. Sawin.

In this series of lectures, I will give an introduction to the theory of moments of L-functions. I will focus on important examples, such as the moments of the Riemann zeta function and Dirichlet L-functions, as well as some GL_2 families. I will also present some of the important tools for understanding moments, as well as applications of moments.

Thursday, 31 October 2024

waring's problem and Goldbach's conjecture

We review the theory of exponential sums due to Weyl and van der Corput and consider several applications. If time permits, we also look at the theory of p-adic exponent pairs, as developed by Milićević.

In these talks, I will explain how the smooth delta-function form of the circle method can be used to count solutions to integral quadratic forms.

TBA

Friday, 01 November 2024

TBA

The notion of congruence (modulo an integer q) was formalised by C. F. Gauss in his Disquisitiones arithmeticae. This is a basic yet fundamental concept in all aspects of number theory. Indeed congruences allow to evaluate and compare integers in way considerably richer than the archimedean order alone permits.

In analytic number theory, several outstanding question -starting with Dirichlet’s theorem on primes in arithmetic progressions- reduce to the of measuring whether some classical arithmetic function (say the characteristic function of prime numbers) correlate with suitable q periodic functions for instance Gauss sums, Jacobi sums or Kloosterman sums. It turns out that these functions, when the modulus q is a prime (to which one can reduce via the Chinese Reminder Theorem) can be recognised as « trace functions». The study of trace functions was initiated by A. Weil in the 1940’s and was pursued by A. Grothendieck in the second half of the century with his refoundation of algebraic geometry and the invention of étale cohomology; it culminated with P. Deligne’s proof of the Riemann Hypothesis for algebraic varieties over finite fields.

In these lectures, we will explain, through various examples, how the theory of trace functions and its subsequent developments in l-adic cohomology, notably the works of Katz and Laumon, have made their way into modern analytic number theory. Several (but not all) of the examples discussed during these lectures originate from joint works with E. Fouvry, E. Kowalski and W. Sawin.

Monday, 04 November 2024

I will present a new zero-free region for all $\mathrm{GL}(1)$-twists of $\mathrm{GL}(m)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions. The proof is inspired by Siegel's celebrated lower bound for Dirichlet $L$-functions at $s=1$. I will also discuss two applications briefly. Joint work with Jesse Thorner.

For a $SL(2,\mathbb{Z})$ form $f$, we obtain a sub-Weyl bound: \begin{equation*}L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-\delta+\varepsilon}\end{equation*} for some explicit $\delta>0$, crossing the Weyl barrier for the first time beyond $GL(1)$. The proof uses a refinement of the `trivial' delta method.

We use the Petersson trace formula over totally real number fields as a delta symbol to prove a t-aspect subconvexity bound for L-functions of Hilbert modular forms. This seems to be the first instance of using a delta symbol approach over number fields for proving a subconvexity result. This is an ongoing work, joint with Naomi Tanabe.

Tuesday, 05 November 2024

In practice, L-functions appear as generating functions encapsulating information about various objects, such as Galois representations, elliptic curves, arithmetic functions, modular forms, Maass forms, etc. Studying L-functions is therefore of utmost importance in number theory at large. Two of their attached data carry critical information: their zeros, which govern the distributional behavior of underlying objects; and their central values, which are related to invariants such as the class number of a field extension. We discuss a connection between low-lying zeros and central values of L-functions, in particular showing that results about the distribution of low-lying zeros (towards the density conjecture of Katz-Sarnak) implies results about the distribution of the central values (towards the normal distribution conjecture of Keating-Snaith). Even though we discuss this principle in general, we instanciate it in the case of modular forms in the level aspect to give a statement and explain the arguments of the proof.

n this talk, I will discuss the problem of obtaining lower bounds for the number of rational points of bounded height on cubic hypersufaces. Our main tools will be the circle method, the Ekedahl sieve and the geometry of numbers.

We develop a two dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over Q defined over at least 10 variables. This is a joint work with Simon L. Rydin Myerson (warwick) and Junxian Li (UC Davis).

We will present some similarity relations between various delta methods depending on the rank and arithmetic of the object to which they are applied. These will be given in the context of subconvexity results for Rankin-Selberg convolutions.

Consider a classical ensemble of complete exponential sums exhibiting square-root cancellation, such as a natural family of Kloosterman sums, Gauss sums, or character sums. Polygonal paths traced by their normalized incomplete sums give a fascinating insight into their chaotic formation. In this talk, we will present our joint results describing convergence in law in two families of Kloosterman and Gauss paths, in which we find that each family in fact splits into multiple distinct ensembles, each converging in law to an explicit random complex-valued Fourier series. The key arithmetic inputs are estimates on sums of products, which are also of broader interest in families of L-functions. There will be many pretty pictures!

Wednesday, 06 November 2024

I will discuss the problem of obtaining an asymptotic formula for counting integral solutions to an equation of the form f(x, y, z, w)=N in an expanding box, where N is a non-zero integer and f is an indefinite quadratic form over the integers. For forms of the shape axy-bzw, I will then explain how we can obtain a significantly strong error term by applying deep methods from the spectral theory of automorphic forms. This is a joint work with Rachita Guria.

TBA

We explore approaches to systems of forms with differing degrees which use the ‘repulsion’ technique. This would allow for example asymptotic formulas for the density of solutions to nonsingular systems of Diophantine inequalities in sufficiently many variables.

I'll discuss recent joint work with Florian Wilsch in which we use the circle method to develop a new heuristic for the number of integral points of bounded height on affine cubic surfaces, together with some of the evidence for it that we accrued.

TBA

Thursday, 07 November 2024

It is trivial to show that exponential sums are large near the origin. Since the resolution of the main conjecture associated with Vinogradov’s mean value theorem, we now also have sharp bounds on the average size of exponential sums, taken over the entire unit torus. However, intermediate situations in which the average is taken only over suitable subsets of the unit torus are much less well understood. In the talk, I will present some results and conjectures regarding averages of exponential sums in the vicinity of the origin, and briefly look into potential applications.

We will discuss ongoing work with Alexandra Florea, Matilde Lalín, and Amita Malik on the shifted convolution problem for divisor functions in function fields. This involves studying the average value of $d(f) d(f+h)$ where $h$ is a fixed polynomial (having possibly large degree $m$) in $\mathbb{F}_q[T]$ and $f$ runs over all monic polynomials in $\mathbb{F}_q[T]$ of degree $n$, where $n$ goes to infinity. Our techniques mirror the classical approach of Estermann in the integer setting. The main new ingredient is a functional equation for the Estermann function (equivalently, a Voronoi summation formula for the divisor function) that was not previously available in function fields. If time permits, we will discuss a related result involving the shifted convolution of the norm-counting functions of quadratic extensions. The talk should be accessible to those unfamiliar with function fields.

TBA

We explore approaches to systems of forms with differing degrees which use the ‘repulsion’ technique. This would allow for example asymptotic formulas for the density of solutions to nonsingular systems of Diophantine inequalities in sufficiently many variables.

The second and fourth moments of the Riemann zeta function have been known for about a century, but the sixth moment remains elusive. The sixth moment of zeta can be thought of as the second moment of a GL_3 Eisenstein series, and it is natural to consider variants of the problem where the Eisenstein series is replaced by a cusp form. I will discuss recent work with Agniva Dasgupta and Wing Hong Leung where we obtain a nontrivial bound on this second moment. I will also discuss some applications, including an improvement on the Rankin-Selberg problem.

Friday, 08 November 2024

TBA

TBA

In an upcoming work with Mallesham, Munshi, Singh, we will try to improve the bound for the error term of the general Rankin-Selberg problem of degree 4

TBA

TBA