Talk

A crucial problem in geometric quantization is to understand the relationship among quantum spaces associated to different polarizations. Two types of polarizations on toric varieties, Kähler and real, have been studied extensively. In this talk, I will introduce the quantum spaces associated to mixed toric polarizations and explore their relationships with those associated with Kähler polarizations.

We begin by briefly introducing the subconvexity problem for $L$-functions and the delta method, which has proven to be a powerful line of attack in this context. As an application, for a $SL(2,\mathbb{Z})$ form $f$, we obtain the sub-Weyl bound:
$$L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-\delta+\varepsilon}$$ for some explicit $\delta>0$, thereby crossing the Weyl barrier for the first time beyond $GL(1)$. The proof uses a refinement of the 'trivial' delta method.

Modular forms are central to modern number theory for many reasons, one of which being that they are a rich source of 2-dimensional Galois representations. But what information about the modular form is contained in the Galois representation? And how does one extract this information, for example about the Fourier coefficients of f?

Seminar webpage:  https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html

I would like to give an overview of the insights from the work by Swinnerton-Dyer and Klagsbrun--Mazur--Rubin, who established relations between Markov operators over countable state spaces and changes in Mordell-Weil rank of elliptic curves with respect to base change to cyclic Galois extensions over number fields. If time allows, we will use this insight to generalize the technique to understand changes in Mordell-Weil ranks of elliptic curves with respect to families of S3-cubic extensions over number fields with fixed quadratic resolvents.

Zoom link:

https://eu02web.zoom-x.de/j/66594302263?pwd=6XqRNiAADoXfsLNrwCIji5UZgyh2jG.1

Meeting ID: 665 9430 2263
Passcode: 740104

If X is a curve of genus at least two defined over the rational numbers, we know by Faltings's Theorem that the set X(Q) of rational points is finite, but how to systematically compute it is still an open problem. In 2005, Minhyong Kim proposed a new framework for studying rational (or S-integral) points on curves, called the Chabauty–Kim method. It aims to produce p-adic analytic functions on X(Q_p) containing the rational points X(Q) in their zero locus.

The syzygies of the residue field over a commutative local ring are important invariants of the ring. However, their module structure remains largely unexplored. In this talk, I'll discuss our results, focusing on the direct summands of the syzygies of the residue field. The talk consists of two parts.