Rosa Winter (UniDistance Suisse)
Abstract:
Title: Many rational points on del Pezzo surfaces of low degree Abstract: Let $X$ be an algebraic variety over a number field $k$. In arithmetic geometry we are interested in the set $X(k)$ of $k$-rational points on $X$. Questions one might ask are, is $X(k)$ empty or not? And if it is not empty, how `large\' is $X(k)$? Del Pezzo surfaces are surfaces classified by their degree~$d$, which is an integer between 1 and 9 (for $d\\geq3$, these are the smooth surfaces of degree $d$ in $\\mathbb{P}^d$). The lower the degree, the more complex del Pezzo surfaces are. I will give an overview of different notions of `many\' rational points, and go over several results for rational points on del Pezzo surfaces of degree 1 and 2. I will then focus on work in progress joint with Julian Demeio and Sam Streeter on the so-called \\textsl{Hilbert property} for del Pezzo surfaces of degree 1. Spiegelgasse 5, Seminarraum 05.002
14:15 • Universität Basel
John Voigt (University of Sydney)
Explicit modularity of K3 surfaces with complex multiplication of large degree abstract
Abstract:
We consider the transcendental motive of three K3 surfaces conjecturedto have CM. Under this assumption, we match these to explicit algebraicHecke quasi-characters and CM abelian threefolds.
14:15 • EPF Lausanne, GA 3 21
Elisa Gorla (Neuchâtel)
Multivariate cryptography and the hardness of polynomial system solving abstract
Abstract:
Multivariate cryptography belongs to post-quantum cryptography, which is the branch of cryptography that is supposed to remain secure even in the presence of a quantum computer. After introducing public-key cryptography and motivating the need for studying post-quantum cryptography, I will discuss the role played by commutative algebra techniques in multivariate cryptography. The security of multivariate cryptographic primitive relies on the hardness of computing the solutions of multivariate polynomial systems over finite fields. Since we can compute the solutions of a polynomial system from its Gröbner basis, bounds on the complexity of Gröbner bases computations provide bounds on the security of the corresponding multivariate cryptographic primitives. In the talk, I will introduce and discuss some algebraic invariants which play a role in these security estimates and motivate their importance in this applied setting.