This is a seminar for advanced Bachelor students and Master students, providing an introduction to selected topics in modern number theory.

It is a general principle in mathematics that a problem defined over a given field (or any other algebraic object) may become more accessible by passing to a suitable extension field. For instance, the rational numbers Q have the (analytical) drawback that they are not complete with respect to the "ordinary" (also known as "archimedean") absolute value, so one passes to the field of real numbers, which is the completion of Q with respect to that absolute value. Moreover, any prime number p gives rise to an absolute value on Q (these are called "non-archimedean"), and the corresponding completion is called the field of p-adic numbers. By a theorem of Ostrowski, there are essentially no further absolute values on Q. The real and p-adic numbers belong to the class of local fields, which basically refers to the fact that they pay respect to one single absolute value. However, in order to solve our initial problem, it may be necessary to consider all these absolute values at once! So one could try to patch those completions together cleverly, in the hope that the resulting global object (i.e., involving all absolute values) has more convenient properties than Q. Indeed, this process leads to the adele ring A of Q, which is also a locally compact space (in contrast to Q), so that we may perform integration and harmonic analysis on A.

In the first part of the seminar, we shall introduce the adele ring as well as the idele group (which is the unit group of the adele ring, but equipped with a finer topology) of a number field and study their properties. Depending on the prerequisites of the participants, we may also cover the basics about number fields and local fields beforehand. Anyway, this discussion will culminate in the first highlights of our seminar: proofs of the finiteness of class number and of Dirichlet's unit theorem. Both are statements about number fields that may be established by adelic means.

The second part of the seminar will be devoted to one of the major applications of adeles/ideles, namely, either to global class field theory (which aims to describe the Galois group of the maximal abelian extension of a number field) or to Tate's thesis (providing a conceptional method to derive meromorphic continuations of certain zeta and L-functions). The choice of topic depends on the interests and prerequisites of the participants.

There will be a preliminary meeting taking place on Friday, January 28th, between 12-13 o'clock (sharp) via Zoom. Meeting-ID: 637 9871 3793. Password: 235711. If you would like to participate in the seminar, then feel free to write me an e-mail: lukas.prader "at" ur.de

Moreover, a detailed description of the contents of the seminar will soon (or at least before January 28th) be available here: https://sites.google.com/view/lukas-prader/teaching

GRIPS page of the seminar: https://elearning.uni-regensburg.de/course/view.php?id=54374

• Organisational meeting/distribution of topics: The preliminary meeting takes place on Friday,
  January 28th, between 12-13 o'clock (sharp) via Zoom. Meeting-ID: 637 9871 3793; Password:
  235711
• If you would like to participate in the seminar, then feel free to write me an e-mail:
  lukas.prader "at" ur.de
• Registration for course work/examination/ECTS: FlexNow

• Presentation: Giving a seminar talk of roughly 90 minutes

• Detailed written report of the seminar talk