Achilles and the tortoise

Hello there! These are notes for the fourth class of the course " Topics in algebra and number theory " held in Block 4 of the academic year 2017/18 at the University of Copenhagen . In the previous lecture we have struggled with non-Archimedean places of a number field \( K \) and of a function field \( F(t) \). In particular we have proved that for every number field \( K \) the maps \[ \begin{aligned} \Sigma_K^{\infty} &\to \operatorname{Spec}(\mathcal{O}_K) \setminus \{ \mathbf{0} \} \\ [\phi] &\mapsto \{ x \in \mathcal{O}_K \mid \phi(x) < 1 \} \end{aligned} \qquad \text{and} \qquad \begin{aligned}  \operatorname{Spec}(\mathcal{O}_K) \setminus \{ \mathbf{0} \} &\to \Sigma_K^{\infty} \\ \mathfrak{p} &\mapsto [\, \lvert \cdot \rvert_{\mathfrak{p}} \,] \end{aligned} \] are one the inverse of the other. Now we want to prove a similar statement for the Archimedean places \( \Sigma_{K,\infty} \). Recall that the Archimedean world is the world of \( \mat...

Hello there! These are notes for the third class of the course " Topics in algebra and number theory " held in Block 4 of the academic year 2017/18 at the University of Copenhagen . In the previous lecture we have defined the set of places \( \Sigma_K \) of a field \( K \) which can be divided into Archimedean and non-Archimedean places (and yes, the trivial place). Moreover we have seen as an example that if we have a Dedekind domain \( R \) and a prime ideal \( \mathfrak{p} \subseteq R \) we can define a non-Archimedean absolute value \( \lvert \cdot \rvert_{\mathfrak{p}} \) on the field \( K := \operatorname{Frac}(R) \) by setting \( \lvert x \rvert_{\mathfrak{p}} := c^{-\operatorname{ord}_{\mathfrak{p}}(x)} \) for some constant \( c \in \mathbb{R}_{> 1} \), where \( \operatorname{ord}_{\mathfrak{p}}(x) \) is the maximum power of \( \mathfrak{p} \) which divides \( x R \). In this lecture we will see that this construction is actually almost a bijection when \...

Hello there! These are notes for the second class of the course " Topics in algebra and number theory " held in Block 4 of the academic year 2017/18 at the University of Copenhagen . As we said last time, the p-adic numbers \( \mathbb{Q}_p \) are obtained with a process of completion analogous to the classical way in which we obtain the real numbers \( \mathbb{R} \) by "filling the holes" in the rational numbers \( \mathbb{Q} \). In order to obtain \( \mathbb{Q}_p \) we need to introduce a new notion of distance (and, thus, new "holes") on the rational numbers, which will be related to the prime \( p \). This can be done in every field as we see in the next section.

Hello everyone! These are notes for the first class of the course " Topics in algebra and number theory " held in Block 4 of the academic year 2017/18 at the University of Copenhagen . This will be a course on p-adic numbers, so in this first hour I tried to give a little motivation about why one should study these objects in relation to some fundamental questions in number theory.

Hello world! This will be a place to share some thoughts about mathematics, theater and every other (more boring) aspect of life! For example I will talk a lot about number theory, which has statements like $$ \pi(x) = \operatorname{Li} (x) + O \left(x \exp \left( -\frac{A(\log x)^\frac35}{(\log \log x)^\frac15} \right) \right) $$  or like $$ \frac{L^{(r)}(E,1)}{r!} = \frac{\#\mathrm{Sha}(E)\Omega_E R_E \prod_{p|N}c_p}{(\#E_{\mathrm{Tor}})^2} $$ and in general I like thinking about the second kind of statements more than the first. Stay tuned for more interesting things to happen!