Time：March 5/12/16/19/26/30, April 2/6/9, 2026
Venue：E13-128
Course: Research lectures on Geometry - Introduction to topological adelic curves
Speaker：Antoine Sedillot
Abstract: In this series of lectures, we will deal with several frame-works used to study the arithmetic of fields of global nature. Starting with the classical Diophantine geometry of function fields and global fields, we will move on to the formalism of adelic curves, recently introduced by Chen and Moriwaki [CM19] (Lecture 1). Roughly speaking, a proper adelic curve is the data of a field together with a family of absolutes values parametrised by a measure space satisfying a product formula. Adelic curves are very flexible objects that appear in a wide range of situations and a lot of the usual tools and notions of classical Arakelov geometry have a counterpart on them [CM21, CM24].
The goal of this mini-course is to introduce a variation of the formalism of adelic curves, where we replace the measure parameter space by a topological space. Our principal motivation is the analogy between Diophantine approximation and valuation theory [Voj87] (Lecture 2). The price to pay is that we need to enlarge the set of absolute values on a field, yielding the notion of pseudo-absolute values [Séd25a] (Lecture 3). Although their definition is elementary, they carry some meaningful geometric information and the space of pseudo-absolute values on a field behaves as an "analytic Zariski-Riemann space" (Lecture 4).
Once the local theory is settled, we will devote the remaining lectures introducing topolog- ical adelic curves [Séd25b]. This consists in three parts. In the first one, we will define them and give basic constructions and examples (Lecture 5). In the second one, we will study the notion of adelic vector bundles, which encodes the intrinsic geometry of the topological adelic curve (Lectures 6-7). Finally, we will give the construction of height functions (Lectures 8-9).
We will try to be as self-contained as possible but nonetheless assume some familiarity with basic algebraic geometry.
Lecture 1：Introduction to topological adelic curves I
Time：13:30-15:55, March 5, 2026
Abstract:
– Heights over global fields (product formula, Weil height).
– Adelic structure on Q(T) and Arakelov geometry on the arithmetic surface.
– Definition of adelic curves.
– Globally valued fields.
Lecture 2: Diophantine approximation and Nevanlinna theory
Time：13:30-15:55, March 12, 2026
Abstract:
– Reminder on Diophantine approximation (Roth theorem).
– Reminder on Nevanlinna theory (counting/proximity function, First and Second main theorems).
– Description of the analogy.
– Previous results: Gubler's M-fields.
– Motivation for topological adelic curves.
Lecture 3: Local theory: pseudo-absolute values (1)
Time：9:50-12:15, March 16, 2026
Abstract:
– Reminder on valuation rings
– Definition and examples of pseudo-absolute values.
– Extension of pseudo-absolute values. Maybe a few words about the completion process.
– Pseudo-norms.
Lecture 4: Local theory: pseudo-absolute values (2)
Time：13:30-15:55, March 19, 2026
Abstract:
– Reminder on the Berkovich spectrum of a Banach ring.
– Space of pseudo-absolute values: compactness and examples.
– Integral structures: definition and example in Nevanlinna theory.
– Zariksi-Riemann spaces.
Lecture 5: Global theory: topological adelic curves (1)
Time：13:30-15:55, March 26, 2026
Abstract:
– Definitions, height functions, Zariski-Rimeann spaces, construction and examples.
– Families of topological adelic curves: main focus on the Nevanlinna theoretic example.
– Algebraic coverings of topological adelic curves.
Lecture 6-7: Global theory: topological adelic curves (2)
Time：9:50-12:15, March 30 & 13:30-15:55, April 2, 2026
Abstract:
– Motivation: Euclidean lattices.
– Pseudo-norm families: definition and correspondence with metrised vector bundles on the Zariski-Riemann space.
– Adelic vector bundles on a topological curve: definition, Arakelov degree and example in Nevanlinna theory.
– Harder-Narasimhan filtrations on (families of) topological adelic curves.
Lecture 8-9: Global theory: topological adelic curves (3)
Time：9:50-12:15, April 6 & 13:30-15:55, April 9, 2026
Abstract:
– Reminder on metrics on Berkovich spaces.
– Pseudo-metrics on a line bundle.
– Pseudo-metric families. We will robably only focus on the simpler case over an integral topological adelic curve with the idea of Nevanlinna theory in mind.
– Adelic line bundles.
– Height of closed points: general construction and Nevanlinna theoretic example (generalisation of Gubler's result).
– Heights of subvarieties: sketch of the construction and Nevanlinna theoretic example (generalisation of Gubler's result).
