ICTS

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.

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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.

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.

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.

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.

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.

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 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.

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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.

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.

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.

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