v A CIMPA research school on Group Actions in Arithmetic and Geometry Gadjah Mada University Yogyakarta, Indonesia February 17th-28th 2020 Courses

Each course consists of 6 lecture hours and 3 hours of training sessions

Groups and symmetries in geometry

Program:

LECTURES 1 AND 2, AND TRAINING SESSION 1: KLEIN'S ERLANGEN PROGRAM, RELATING GEOMETRY AND SYMMETRY
Lecture 1. Some geometries and their groups of transformations
- $\mathbb{R}^2$ as vector space with inner product (euclidean plane) and its affine transformations (that is, the isometries, including translations),
- $\mathbb{R}^2$ as vector space with its affine transformations (including translations),
- the projective plane with its projective transformations.
Lecture 2. Klein's program.
Invariants in the euclidean line and plane, in the affine line and plane, and in the projective line and plane.

References:
John Stillwell, The four pillars of geometry. Undergraduate Texts in Mathematics. Springer, New York, 2005. pdf
Lectures notes by Bas Edixhoven

LECTURES 3, 4, 5, 6, AND TRAINING SESSIONS 2 AND 3.
The course on representation theory of finite groups will discuss the orthonormal basis of the space of functions on a finite group coming from matrix coefficients of the irreducible representations. We will admit that the same holds for the compact groups $\mathbf{SO}_3(\mathbb{R})$ and $\mathbf{SU}_2(\mathbb{R})$. We will also admit that the irreducible representations of $\mathbf{SU}_2(\mathbb{R})$ are $\operatorname{Symk}(\mathbb{C}^2)$, for $k\geq 0$. That gives, via the covering $\mathbf{SU}_2(\mathbb{R})\rightarrow\mathbf{SO}_3(\mathbb{R})$, a basis for $\operatorname{L}^2(\mathbf{SO}_3(\mathbb{R}))$. The transitive action of $\mathbf{SO}_3(\mathbb{R})$ on $S^2$ gives a bijection $\mathbf{SO}_3(\mathbb{R}) /\textrm{(stabilizer of north pole) }\rightarrow S^2$. That gives us an orthonormal basis for $\operatorname{L}^2(S^2)$. We will make all this explicit and show that we get the standard spherical harmonic functions that one also finds in textbooks on the quantum mechanics of the hydrogen atom, usually characterized in terms of differential equations. So we get an explanation for the quantum numbers for the hydrogen atom, purely from symmetry considerations.

References:
Bas Edixhoven's Notes

Plan of the lectures
Lecture 1: Bas Edixhoven
Some geometries and their groups of transformations.

Lecture 2: Bas Edixhoven
Klein's program.

Lecture 3: Bas Edixhoven
Sections 1 and 2 of Benthem's BSc thesis: Manifolds, Lie groups and Lie algebras.

Section 3 of Benthem's BSc thesis: Representations of SU(2) and SO(3).

Section 4 of Benthem's BSc thesis: The Peter-Weyl theorem.

Lecture 6: Bas Edixhoven
Section 5 of Benthem's BSc thesis: Spherical harmonic functions, hydrogen atom.

Exercise Exercises for the first training session

Exercises for the second training session

Exercises for the third training session

Coding theory
Elisa Lorenzo Garcia and Kiki Ariyanti Sugeng

Program:

Lecture 1. Error-correcting Codes:
repetition code, parity check code, Hamming code. Hamming distance. Linear codes: generator matrix, parity check matrix, dual code.

Lecture 2. Decoding and error probability:
Basic decoding, symmetric channel. Equivalent codes. Shannon Theorem. Weight enumerator.

Lecture 3. Codes constructions and bounds:
Punturing, restriction, extension, short, augmentation, direct sum, juxtaposition, product and concatenation of codes. Singleton, Griesmer, Plotkin, Hamming, Gilbert Varshamov and asymptotic bounds.

Lecture 4. Cyclic codes:
cyclic codes as ideals. Encoding cyclic codes. Parity check polynomial. BCH bound. Examples.

Lecture 5. Algebraic geometry codes and applications:
Codes on curves, Goppa codes, Reid-Solomon codes. The McEliece cryptosystem.

Lecture 6: Codes on graphs: Some graph theory. Cycle code and graph code of a graph

Course material

### Handouts

Lecture-Exercises 1 and 2

Lecture-Exercises 3

Lecture-Exercises 4 and 5

Codes on graph

The course will give handouts of the lectures and use the chapters 1, 2, 3, 4 and 8 of the book:
Codes, Cryptology and Curves with Computer Algebra,
by R. Pellikaan, X.-W. Wu, S. Bulygin and R. Jurrius.
Cambridge University Press, November 2017.

And chapter 18 of the book:
Introduction to Cryptography with Coding Theory,
by W. Trappe and L. C. Washington.
Pearson Prentice Hall, 2006.

Training sessions
Duration: 3 hours
The students will be assigned exercises of the given sections of the above books.
Student will be asked to present their solution.
Finite fields and number theory
Francesco Pappalardi and Michel Waldschmidt

Syllabus:
• History: Gauss fields
• Fields with $p$ elements with $p$ prime.
• Finite fields: existence, unicity, structure, explicit construction.
• Cyclotomic polynomials.
• Frobenius automorphisms. Galois's Theory of finite fields.
• Error correcting codes.
• Construction of Irreducible polynomials over finite fields
• Factorization of polynomials over finite fields
• Permutation Polynomials/Chebicev Polynomials
• Exponential sums over finite fields
• The discrete logarithm

• Training session

Tutorial 1
Solutions with GP/PARI
Tutorial 2
Solutions to Tutorial 2

References:

Notes by Michel Waldschmidt
Notes for Michel Waldschmidt's third lecture

Lidl, Rudolf; Niederreiter, Harald, Finite fields,
Paperback reprint of the hardback 2nd edition 1996. (English) Zbl 1139.11053
Encyclopedia of Mathematics and Its Applications 20. Cambridge: Cambridge University Press (ISBN 978-0-521-06567-2/pbk). xiv, 755 p. (2008).
Representation theory of finite groups
Laura Geatti and René Schoof

The goal of the course is to give a quick self-contained presentation of the representation theory of finite groups. This theory is important in its own sake, and also hints at the more general representation theory of compact Lie groups.

Program:

Representations and irreducible representations, operations with representations (direct sums and tensor products), degree of a representation. The group algebra $\mathbb{C}[G]$ of a finite group $G$, Schur's Lemma, complete reducibility of representations (Maschke's theorem). Characters and class functions of a finite groups, properties of characters, inner products of characters, character table, orthogonality relations. Induced characters and Frobenius reciprocity.

Training sessions:
The training sessions will be devoted to working out some explicit examples: abelian groups, symmetric groups Sn and alternating groups An for small n, dihedral groups, and the quaternion group. In particular we will compute the character table of several groups of small order.

Notes and exercises
Notes on the symmetric group
Exercises 1

Solutions of the first set of exercises

Exercises 2

The irreducible representations of $A_5$.

References
Finite Groups: An Introduction, J.P. Serre
www.amazon.com/Finite-Groups-Introduction-Jean- Pierre-Serre/dp/1571463275
Galois theory and profinite groups
Peter Stevenhagen and Indah Wijayanti

Program:

Lecture 1:
Historical introduction: describing the roots polynomial equations, real and imaginary roots, formulae for the roots of low degree polynomial, Galois' approach: symmetries of roots of polynomials. Abstract field extensions and their symmetries.

Lecture 2:
Separability and normality of field extensions. Finite Galois theory. The fundamental theorem of Galois theory. Explicit examples of Galois extensions.

Lecture 3:
Cyclotomic extensions, radical extensions, solvable extensions. Solvable groups and the insolubility of the general quintic.

Lecture 4:
Infinite Galois extensions. Profinite groups and their topology. The fundamental theorem of infinite Galois theory. Profinite integers.

Lecture 5:
Reduction modulo primes. Computation of Galois groups.

Lecture 6:
Absolute Galois groups. The Kronecker-Weber-theorem. Galois representations.

Training sessions

Training session 1:
basic exercises in field theory and Galois theory.

Training session 2:
explicit computation in finite and infinite extensions.

Training session 3:
the exercises for this part will have a more arithmetic flavor, and interact with the Waldschmidt-Pappalardi course. Computer-assisted exercises may be included.

Notes

Notes on Galois Theory by Peter Stevenhagen

Slides of lecture 2

Slides of lecture 3

Galois Theory Exercises
Modular forms
Marusia Rebolledo and Valerio Talamanca

Program:

Lecture 1: Valerio Talamanca
Crash course on complex analysis

Lecture 2: Valerio Talamanca
Modular group and its action on the upper half plane: description of the action, description of the quotient, cusps

Lecture 3: Valerio Talamanca
Modular functions and modular forms: definition, function on lattices, modular forms, zeroes of modular forms, examples

Lecture 4: Marusia Rebolledo
The space $\operatorname{M}_k(\mathbf{SL}_2(\mathbb{Z}))$: algebraic structure, q-development, computation of the dimension of $\operatorname{M}_k(\mathbf{SL}_2(\mathbb{Z}))$

Lecture 5 Marusia Rebolledo:
Application: Eisenstein series, Delta, arithmetic formulas; theta functions and sums of squares

Lecture 6: Marusia Rebolledo
Hecke operators: as correspondences, as double cosets ; Peterson product ; diagonalization

Training sessions:
The training session will be devoted to working out some explicit examples.
Notes

Slides of lecture 1

Slides of lecture 2

Training session 1

Training session 2
References:

Lecture notes in complex analysis
by Prof. M.Thamban Nair

Chapter VII of Serre's book A course in arithmetic.

Bruinier, J.H., van der Geer, G., Harder, G., Zagier, D.
The 1-2-3 of Modular Forms Lectures at a Summer School in Nordfjordeid, Norway, Editors: Ranestad, Kristian (Ed.)

Don Zagier Elliptic Modular Forms and Their Applications

Further references:
H. Cohen, F. Strömberg
Modular Forms a Classical approach Graduate Studeis in Mathematics AMS

F. Diamond and J. Im
Modular forms and modular curves Seminar on Fermat's last theorem: 1993-994, the Fields Institute for Research in the Mathematical Sciences, Toronto, Ontario, Canada, 1995.

F. Diamond and J. Shurman
A First Course in Modular Forms Springer Science & Business Media, March 2006.

J. S. Milne
Modular functions and modular forms University of Michigan lecture notes, 1997.

K. A. Ribet and W. A. Stein
Lectures on modular forms and Hecke operators