# IMBM Number Theory SemiNars 2021

Welcome to IMBM (Istanbul Center for Mathematical Sciences) Number Theory Seminars. We expect audience from all levels including advanced undergraduate and graduate students.

All of the talks will be online via zoom. Please register to receive the zoom link. You can find the schedule below. All times given are GMT+03:00 (İstanbul time).

**Schedule:**

1) April 9, 5pm

Rachel Newton ( University of Reading)

2) April 30, 4pm

Ravi Ramakrishna (Cornell University)

3) May 21, 4pm

Álvaro Lozano-Robledo (University of Connecticut)

4) June 11, 5pm

Jennifer Balakrishnan (Boston University)

**June 11, 5pm: **

**Jennifer Balakrishnan**

**Title**: Quadratic Chabauty for modular curves

**Abstract:** We describe how p-adic height pairings can be used to determine the set of rational points on curves, in the spirit of Kim's nonabelian Chabauty program. In particular, we discuss what aspects of the quadratic Chabauty method can be made practical for certain modular curves. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.

Past Talks:

May 21, 4pm:

Title: Towards a classification of adelic Galois representations attached to elliptic curves over Q

Abstract: Let E be an elliptic curve defined over Q. The adelic Galois representation attached to E (this object will be defined during the talk) captures all sorts of interesting information about the arithmetic of the points on E(Qbar), including data about the torsion subgroup, isogenies, and other finer invariants of the curve and its isogeny class. In this talk, we will give a summary of recent results towards the classification (up to isomorphism) of the possible adelic Galois representations that arise from elliptic curves over Q, and present some recent results of the author and his collaborators (Garen Chiloyan, Harris Daniels, Jackson Morrow) in this area.

April 30, 4pm:

Ravi Ramakrishna (Cornell University)

Title: Tame Galois extensions of number fields

Abstract: The theory of unramified and tamely ramified p-extensions of number fields has a long history, going back to the works of Golod and Shafarevich, Koch and others. The wildly ramified setting is "easier" because we have the duality theorems of Potou-Tate. I will talk about recent work with F. Hajir and C. Maire in the tame setting, starting with some two very basic questions, one resolved, the other open.

First talk: April 9, at 5pm (Istanbul time)

Rachel Newton ( University of Reading)

Title: Diophantine equations and when to quit trying to solve them

Click here for the slides.

Abstract: The study of integer or rational solutions to polynomial equations with integer coefficients is one of the oldest areas of mathematics and remains a very active field of research. The most basic question we can ask about such an equation is whether its set of rational solutions is empty or not. This turns out to be a very hard question! I will discuss modern methods for proving that the set of rational solutions is empty, touching upon some recent work with Martin Bright concerning the wild part of the Brauer--Manin obstruction.