Starts 5 Mar 2026 10:30
Ends 5 Mar 2026 17:30
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Number Theory Day 2026
ICTP
Leonardo Building - Luigi Stasi Seminar Room
Organizers: Emanuel Carneiro (ICTP), Pietro Corvaja (University of Udine) and Umberto Zannier (SNS - Pisa)
Program:
10:30 - 11:20: Francesco Amoroso (Univ. of Torino, Italy)
Title: Bounded height problems and applications
Abstract: PDF
11:30 - 12:20: Christoph Aistleitner (TU - Graz, Austria)
Title: Conitnued fraction expansion of rationals with fixed denominator
Abstract: In this talk I will report on recent joint work with Bence Borda and Manuel Hauke on the distribution of partial quotients of reduced fractions p/q. Here q is understood to be fixed, and p ranges through the set of coprime integers mod q. I will explain some of the history of the problem, and point out the relation to Zaremba's conjecture and low-discrepancy sampling with the "good lattice points" method. Then I will explain the method of proof, which is relatively elementary, and uses Legendre's characterization of convergents and the behavior of continued fractions under an reversion of their ordering.
14:00 - 14:50: Aleksander Simonic (Univ. of Primorska, Slovenia)
Title: An explicit form of Ingham's zero density estimate
Abstract: Ingham (1940) proved that $N(\sigma,T)\ll T^{3(1-\sigma)/(2-\sigma)}\log^{5}{T}$, where $N(\sigma,T)$ counts the number of the non-trivial zeros $\rho$ of the Riemann zeta-function with $\Re\{\rho\}\geq\sigma\geq 1/2$ and $0<\Im\{\rho\}\leq T$. Such estimates are often valuable in the distribution theory of prime numbers. In this talk I will present an explicit version of this result with the exponent $(7-5\sigma)/(2-\sigma)$ of the logarithmic factor. The crucial ingredient in the proof is an explicit estimate with asymptotically correct main term for the fourth power moment of the Riemann zeta-function on the critical line, a result which is of independent interest. This is joint work with Shashi Chourasiya (UNSW Canberra).
15:00 - 15:50: Julian Demeio (Univ. of Hannover, Germany)
Title: Hilbert Modular Surfaces, Hilbert Property, and the Inverse Galois Problem
Abstract: I will talk on a recent work in collaboration with Damián Gvirtz-Chen, where we show that the fourteen modular Hilbert surfaces of K3 type possess abundant rational curves. In particular, we show that they have enough rational curves to obtain the Hilbert property. We deduce a positive answer to the regular Inverse Galois Problem for $\operatorname{PSL}_2(\mathbb{F}_{p^2})$ for a set of primes p of density $1-2^{-12}$.
16:30 - 17:20: Martin Widmer (TU - Graz, Austria)
Title: Universal quadratic forms over infinite extensions
Abstract: Every positive integer is the sum of four squares. An integral positive definite quadratic form that represents every positive integer is called universal (over the rationals). This notion generalizes to arbitrary totally real fields. It is well-known that that every totally real number field admits a universal quadratic form. For infinite extensions the situation is fundamentally different. Daans, Kala and Man showed that in this case the Northcott property is an obstruction to the existence of such a form. However, Northcott fields are very rare (in a suitable topological sense). We present a necessary condition for the existence of a universal quadratic form in a given number of variables which is new, even in the case of number fields. As an application we show that most totally real fields do not admit a universal quadratic form. This is joint work with Nicolas Daans, Siu Hang Man, Vitezslav Kala, and Pavlo Yatsyna.
Program:
10:30 - 11:20: Francesco Amoroso (Univ. of Torino, Italy)
Title: Bounded height problems and applications
Abstract: PDF
11:30 - 12:20: Christoph Aistleitner (TU - Graz, Austria)
Title: Conitnued fraction expansion of rationals with fixed denominator
Abstract: In this talk I will report on recent joint work with Bence Borda and Manuel Hauke on the distribution of partial quotients of reduced fractions p/q. Here q is understood to be fixed, and p ranges through the set of coprime integers mod q. I will explain some of the history of the problem, and point out the relation to Zaremba's conjecture and low-discrepancy sampling with the "good lattice points" method. Then I will explain the method of proof, which is relatively elementary, and uses Legendre's characterization of convergents and the behavior of continued fractions under an reversion of their ordering.
14:00 - 14:50: Aleksander Simonic (Univ. of Primorska, Slovenia)
Title: An explicit form of Ingham's zero density estimate
Abstract: Ingham (1940) proved that $N(\sigma,T)\ll T^{3(1-\sigma)/(2-\sigma)}\log^{5}{T}$, where $N(\sigma,T)$ counts the number of the non-trivial zeros $\rho$ of the Riemann zeta-function with $\Re\{\rho\}\geq\sigma\geq 1/2$ and $0<\Im\{\rho\}\leq T$. Such estimates are often valuable in the distribution theory of prime numbers. In this talk I will present an explicit version of this result with the exponent $(7-5\sigma)/(2-\sigma)$ of the logarithmic factor. The crucial ingredient in the proof is an explicit estimate with asymptotically correct main term for the fourth power moment of the Riemann zeta-function on the critical line, a result which is of independent interest. This is joint work with Shashi Chourasiya (UNSW Canberra).
15:00 - 15:50: Julian Demeio (Univ. of Hannover, Germany)
Title: Hilbert Modular Surfaces, Hilbert Property, and the Inverse Galois Problem
Abstract: I will talk on a recent work in collaboration with Damián Gvirtz-Chen, where we show that the fourteen modular Hilbert surfaces of K3 type possess abundant rational curves. In particular, we show that they have enough rational curves to obtain the Hilbert property. We deduce a positive answer to the regular Inverse Galois Problem for $\operatorname{PSL}_2(\mathbb{F}_{p^2})$ for a set of primes p of density $1-2^{-12}$.
16:30 - 17:20: Martin Widmer (TU - Graz, Austria)
Title: Universal quadratic forms over infinite extensions
Abstract: Every positive integer is the sum of four squares. An integral positive definite quadratic form that represents every positive integer is called universal (over the rationals). This notion generalizes to arbitrary totally real fields. It is well-known that that every totally real number field admits a universal quadratic form. For infinite extensions the situation is fundamentally different. Daans, Kala and Man showed that in this case the Northcott property is an obstruction to the existence of such a form. However, Northcott fields are very rare (in a suitable topological sense). We present a necessary condition for the existence of a universal quadratic form in a given number of variables which is new, even in the case of number fields. As an application we show that most totally real fields do not admit a universal quadratic form. This is joint work with Nicolas Daans, Siu Hang Man, Vitezslav Kala, and Pavlo Yatsyna.