For the academic year 2025-26, the seminar is organised by Víťa Kala and myself, and runs on Wednesdays 14:00 in room K2.
The seminar is intended for students and faculty members, so the level of the talks should be accessible to these people.
We have a mailing list for announcing upcoming talks. If you wish to subscribe to the mailing list, please contact any of the organisers.
The Batyrev-Manin conjecture predicts the number of rational points of bounded height on certain varieties. It asserts the existence of constants a and b such that the number of rational points of height less than B is asymptotically CBᵃlog(B)ᵇ for some C>0. Recently, a significant attention has been devoted to the statistical properties of level structures on elliptic curves. For instance, it has been shown that for the 15 values of N for which the modular curve X₀(N) has genus 0, the number of elliptic curves defined over Q admitting rational cyclic N-isogeny also satisfies an asymptotic formula CBᵃlog(B)ᵇ. Despite the reminiscent formulas, these results do not fit in the framework of the standard Batyrev-Manin conjecture because of the presence of non-trivial isomorphisms. We explain how they can be interpreted as instances of a generalization of Batyrev-Manin conjecture to Deligne-Mumford stacks. The talk is based on a joint work with Changho Han and Mohammad Sadek.
The Kuznetsov trace formula is an indispensable tool in studying Maaß forms on GL(2). While abstract trace formulae exist very generally for many groups, it takes a significant amount of work to make them explicit enough for the applications in analytic number theory. In this talk, I will discuss some ideas behind the making of an explicit version of the Kuznetsov formula for higher rank groups, which recently led to an explicit Kuznetsov formula for Maaß forms on GSp(4). The talk is based on joint work with Félicien Comtat and Didier Lesesvre.
We establish effective Erdős–Wintner theorems for digit-additive functions arising from greedy numeration systems.
Our main result concerns linear recurrent bases, namely greedy numeration systems associated with a linear recurrence. For real-valued G-additive functions, additive with respect to the greedy G-expansion, we show that the existence of a limiting distribution along initial segments is equivalent to the convergence of two canonical series: a first-moment series and a quadratic series. This yields an explicit infinite-product factorization of the limiting characteristic function, with local factors depending only on the underlying digit structure.
As a complementary effective example, we discuss Cantor numeration systems: a trailing-window decomposition yields quantitative convergence rates toward a limiting distribution and an explicit infinite-product factorization of the limiting characteristic function.
In 1965, Chowla made the following conjecture.
Chowla's conjecture: If λ is the Liouville function and k≥2 is a fixed integer, then for any fixed distinct non-negative integers h₁, ..., hₖ, we have that
∑n≤x λ(n+h₁) ... λ(n+hₖ) = o(x) as x →∞.
An unconditional answer to this conjecture is yet to be found, and in this talk, we are taking a conditional approach towards it. In particular, we will discuss Chowla's conjecture under the existence of Landau-Siegel zeroes. We will start by introducing the notion of a Landau-Siegel zero and then we will heuristically explain why the existence of these zeroes is a useful assumption when "attacking" Chowla's conjecture. We will continue by describing how one can establish a bound for the k-point correlations ∑n≤x λ(n+h₁) ... λ(n+hₖ) using the presence of Landau-Siegel zeroes. The estimate that we will present constitutes an improvement over the previous respective works of Germán and Kátai, Chinis, and Tao and Teräväinen. This talk is based on joint work with Mikko Jaskari.
Let K be a totally real number field with ring of integers OK. Let UK(n) be the minimal rank of an n-universal OK-lattice; i.e. the smallest positive integer k such that there exists a rank k positive definite OK-lattice which represents all rank n positive definite OK-lattices. With the exception of a finite number of real quadratic fields, we prove an explicit asymptotic formula for log UK(n) as n tends to infinity. We also show that, for any constant C > 0 and n > 2, there are only finitely many totally real fields K such that UK(n) < C, with all such fields being effectively computable. Similarly, for any n > 2, we show that there are only finitely many totally real fields K admitting an n-universal criterion set SK(n) of size less than C, with all such fields likewise being effectively computable. This talk is based on joint work with Dayoon Park, Pavlo Yatsyna, and Jongheun Yoon.
The class groups of number fields is one of the most intriguing objects to mathematicians and it attracts a lot of attention.
This talk is based on the class groups of imaginary biquadratic fields. We exhibit two distinct families of imaginary biquadratic fields with each member having large class groups. So far, very little work is done regarding the structure of class groups of these fields.
Construction of the first family involve elliptic curve and their quadratic twists, whereas the second family is constructed with a combination of elliptic and hyperelliptic curves.
This talk is based on a very recent joint work with Kalyan Banerjee and Arkabrata Ghosh, which is accepted for publication in Research in Number Theory.
A Hilbert cube of dimension d is the set of integers H(a0;a1,...,ad) = {a0 + {0,a1} + ··· + {0,ad} = a0 + ε1a1 + ... + εdad : εi ∈{0,1}}. Brown, Erdős and Freedman asked whether the maximal dimension of a Hilbert cube in the set S = {n² : n ∈ N} of integer squares is absolutely bounded or not. Dietmann and Elsholtz proved that if H(a0;a1,...,ad) ⊂ S ∩ [0, N], then d ≤ 7 log log N for all sufficiently large values of N. Here we prove that there exist at least ≫ N^(1/8) Hilbert cubes H(a0;a1,a2,a3) with a0, a1, a2, a3 ∈ [0, N] in the set of squares. Moreover, we prove that for each i,j ∈ {0,1,2,3} with i<j, the set {ai/aj : H(a0;a1,a2,a3) ⊂ S} is dense in the set of positive real numbers (in the Euclidean topology). The talk is based on a joint work with Andrew Bremner and Christian Elsholtz.
I will give a brief introduction to well-rounded lattices and to their utility in (post-quantum) security. We will see how the lattice theta series naturally arises in these contexts and discuss its connections to well-rounded lattices.
A (positive definite and integral) quadratic form f is called irrecoverable (from its subforms) if there is a quadratic form F that represents all proper subforms except for f itself, and such a quadratic form F is called an isolation of f. In this talk, we present recent advances on irrecoverable quadratic forms and discuss their possible generalizations.
In this talk, I will discuss the Kaplansky radical of a field, an object introduced by Irving Kaplansky in the 1960s to characterize fields with a unique non-split quaternion algebra. I will focus on the case of function fields of curves, and in particular on arithmetic curves, where I will show how the Kaplansky radical can be connected to the reduction of the curve.
I will present joint work with Maxim Gerspach on lower-order terms in Selberg's central limit theorem. In particular, we compute precise asymptotic formulas for the third moment of both the real and imaginary parts of the logarithm of the Riemann zeta function. Our results are conditional on the Riemann Hypothesis, Hejhal's triple correlation, and a new conjecture that describes the interaction between prime powers and Montgomery's pair correlation function. To support this conjecture, which we refer to as the "twisted" pair correlation conjecture, we prove it unconditionally in a limited range and under the Hardy-Littlewood conjecture in a larger range.
Abstract: Let f be a function on the natural numbers, periodic with period N>1, with f(n) taking values -1 or 1 for 0 < n < N, and f(N) = 0. In the 1950s, Erdős conjectured that for such a function f, the infinite series ∑ₙ f(n)/n is non-zero, whenever it converges. This is in the same spirit as the non-vanishing of L(1,χ) for non-principal real Dirichlet characters χ, which is an essential step in proving Dirichlet's theorem on the infinitude of primes in arithmetic progressions. In this talk, we discuss the nuances of Erdos's conjecture and present a new approach to it, which leads to its resolution in `almost all' cases. More specifically, we describe joint work with Abhishek Bharadwaj and Ram Murty in which we settle Erdős's conjecture, except for the case when N is a perfect square.
The Weil height captures the arithmetic complexity of algebraic numbers and gives a partial ordering on numbers with bounded degree. It is an important theme to characterize sets of algebraic numbers such that their height is bounded below. Such sets are said to have the Bogomolov property. In this talk, we will discuss Bogomolov property over infinite extensions for algebraic numbers and also for points on elliptic curves over infinite extensions. This is joint work with Sushant Kala.