Tarix: 17.03.2025
Saat: 9.30
Məkan:
Towards the Bloch-Kato conjecture for GSp6.
Waqar Ali Shah, University of California, Santa Barbara
Xülasə:
One of the central problems in number theory is the Birch and Swinnerton-Dyer conjecture, which asserts that the order of vanishing of the -function of a rational elliptic curve at the central value coincides with the rank of its Mordell-Weil group. A far-reaching generalization of this is the Bloch-Kato conjecture, which posits a similar relationship between the order of vanishing of the -functions associated with Galois representations and the dimension of their Selmer groups. In recent years, significant progress has been made in establishing new cases of this conjecture for automorphic Galois representations arising in the cohomology of Shimura varieties, most notably in the work of Skinner, Loeffler, and Zerbes for -Shimura varieties. A crucial ingredient in these developments is the construction of an Euler system, a powerful tool for studying the dimension of Selmer groups.
In this talk, we recall the Bloch-Kato conjecture in the setting of -Shimura varieties and present the construction of an Euler system using a novel method that overcomes a major obstacle. As a consequence, we obtain the first non-trivial result towards the Bloch-Kato conjecture in this setting.
Tarix: 11.03.2025
Saat: 10.30
Məkan: Rəqəmsal Tədqiqat Laboratoriyası, Bakı Dövlət Universiteti, Əsas Bina, 3-cü mərtəbə
Multiplicative functions $k$-additive on hexagonal
numbers
Elçin Həsənəlizadə, ADA Universiteti
Xülasə:
Characterization of the identity function using functional equations has been actively studied by many authors. In 1992, Claudia Spiro introduced the concept of additive uniqueness. A set $E\subseteq\mathbb{N}$ is called an {\it additive uniqueness set} of a set of arithmetic functions $\mathcal{F}$ if there is exactly one element $f\in\mathcal{F}$ which satisfies
\[f(m+n)=f(m)+f(n) \ \text{for all} \ m,n\in E.\]
For example, $\mathbb{N}$ and $\{1\}\cup2\mathbb{N}$ are trivially additive uniqueness sets of the set of multiplicative functions.
She proved that the set of primes is an additive uniqueness set of the collection of multiplicative functions. Many mathematicians have generalized her result. If the additive condition is replaced by multivariate Cauchy’s equation this gives rise to the notion of a k-additive uniqueness set. It is known that squares, cubes, triangular numbers, generalized pentagonal numbers, practical numbers, odious numbers are k-additive uniqueness sets for the set of multiplicative functions. In the present talk, we add another example to the growing list of k-additive uniqueness sets. Namely, we will show that for $k\ge3$ the set of all nonzero hexagonal numbers is a new k-additive uniqueness set for the collection of multiplicative functions. This extends B. Kim et al.’s work for $k = 2$. This is a joint work with Poo-Sung Park (Kyungnam University, Republic of Korea) and Emil Inochkin (ADA University, Azerbaijan)
Tarix: 26.12.2024
Saat: 10.30
Məkan: Rəqəmsal Tədqiqat Laboratoriyası, Bakı Dövlət Universiteti, Əsas Bina, 3-cü mərtəbə
A Discrete Mean Value of the Riemann Zeta Function and its Derivatives
Ertan Elma, MRL
Xülasə:
In this talk, we will discuss an estimate for a discrete mean value of the Riemann zeta function and its derivatives multiplied by Dirichlet polynomials. Assuming the Riemann Hypothesis, we obtain a lower bound for the \(2k\)-th moment of all the derivatives of the Riemann zeta function evaluated at its nontrivial zeros. This is based on a joint work with Kübra Benli and Nathan Ng.