School and Workshop

“Selected topics in Arithmetic Algebraic Geometry”

1. Brion Michel: Rational actions of algebraic groups

Abstract: Weil's regularization theorem asserts that for any rational action of an algebraic group G on a variety X, there exists a G-variety Y on which G acts regularly and which is equivariantly birational to X. This theorem and its refinements are key ingredients for classifying algebraic subgroups of groups of birational automorphisms, in classical work of Enriques and recent work of Blanc, Zimmermann, Fong and others. The lectures will first discuss the notions occuring in the regularization theorem, and some of its applications. We will then present a new proof of this theorem and further developments.

 Reference: Michel Brion, On models of algebraic group actions, arXiv:2202.04352.

 2. Cesnavicius Kestutis: Problems about torsors over regular rings

Abstract: Problems about torsors, such as the Bass--Quillen and the Grothendieck--Serre conjectures, are captivating in that they force one to understand the fine local structure of a smooth variety or, more generally, of a regular local ring. In my lectures, I will discuss this interplay, outline the intervening geometric arguments, such as versions of the Gabber--Quillen presentation lemma, and will overview the problems that remain open. In part, I will base my lectures on the survey article

 Reference: Kestutis Cesnavicius,Problems about torsors over regular rings (with an appendix by Yifei Zhao). arXiv:2201.06424.

3. Im Bo-Hae (F): Elliptic curves and their Mordell-Weil groups

Abstract: I will give an introductory talk on the group structure of an elliptic curve over a number field, and of its quadratic twists, and I will survey some of related results on the rank of elliptic curves and the quadratic twists, and then on some growth of torsion subgroups upon base field change. Then I will introduce congruent numbers and theta-congruent numbers which are related with positive rank of certain elliptic curves over Q.

 Reference: Silverman, Joseph H. The arithmetic of elliptic curves. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986. xii+400 pp.

 4. Tavares Ribeiro Floric: Stark units and L-series of Drinfeld and Anderson modules

Abstract:  The Stark units of a Drinfeld module have been introduced in view of log-algebraicity results (that is, the construction of an element whose exponential is integral, with the help of the L-series attached to the module). The notion has since proven to be useful to study the L-series itself; in particular, as a generalisation of Taelman’s class formula, we recover the L-series as the regulator of the lattice of Stark units. I will present in the lecture some techniques and recent results involving Stark units of Drinfeld or, more generally, Anderson modules.

 Reference: F. Tavares Ribeiro, On the Stark units of Drinfeld modules. Arithmetic and geometry over local fields—VIASM 2018, 281–324, Lecture Notes in Math., 2275, Springer, Cham, [2021].

 5. dos Santos Joao Pedro: Connections in Algebraic Geometry

Abstract: Connections in Geometry are essentially systems of linear partial differential equations on manifolds, varieties, or schemes. In this course, I shall present basic material on the subject and some more advanced techniques which will be disguised with simpler language. Much material will be formulated in the language of complex manifolds and a bit of ease with algebraic geometry will make the transposition of results simple. (On the other hand, some sheaf theory will be necessary in all cases.)

After introducing the fundamental objects and the "Linear Algebra" accompanying them, I shall discuss the Atiyah-Weil Theorem for the existence of connections on proper curves, which will serve as motivation to talk about the celebrated Narasimhan-Seshadri Theorem. Afterwards, I will consider the Gauss-Manin connection in its context of function fields as originally presented by Manin. This will serve to link the theory to the Hypergeometric Differential Equation.To end, I will discuss Grothendieck's interpretation of connections and his algebraic differential calculus.

Reference:

1. M. F. Atiyah, Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85 (1957), 181--207.
2. P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978.
3. Nicholas M. Katz,  Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin,  Publ. Math. IH\'ES,  No. 39, (1970), 175 -- 232.
4. S. Kobayashi and K. Nomizu,  Foundations of Differential Geometry, Vol. 1.  
5. Ju. I. Manin,  Rational Points of Algebraic Curves over Function Fields, Izv. Akad. Nauk SSSR. Ser. Mat. 27 (1963), 1395--1440; English translation, Amer. Math. Soc. Translations, (2), vol. 50, (1966), 189--234.
6. William M. Messing, On the nilpotence of the hypergeometric equation. J. Math. Kyoto Univ. 12(2), 369-383 (1972).
7. Joao Pedro dos Santos, ConnectionsinAlgebraicGeometry, Lectures at IMPA , Rio de Janeiro, Brazil, 2022. https://webusers.imj-prg.fr/~joao-pedro.dos-santos/Enseignement.html

 6. Ishii Shihoko (F):  Introduction to arc space in singularity theory

Abstract: In 1968, Nash introduced arc spaces and posed Nash's problem. The problem was solved in all dimensions in 2012 by the contribution of many people. This problem gives a new viewpoint to singularity theory. In the lecture, I introduce the basic properties of the arc space and show its applications to singularity theory.

7. Waldschmidt Michel: Representation of integers by binary forms

Abstract: Given a binary form, one is interested in knowing which integers it represents, and how many times a given integer is represented. An early example is the study of sums of two squares. For quadratic binary forms, such questions were essentially solved by E. Landau and P. Bernays (1912). For higher degree forms, the main result is due to C.L. Stewart and S. Y. Xiao (2019).

In joint works with É. Fouvry, we investigate similar questions for binary forms belonging to some families. Examples include the sequence of cyclotomic forms (also joint work with C. Levesque) and families of binary forms aX^d+bY^d.