COMPETENCIES



1. To apply the main methods for the study of arithmetic functions.

2. To relate different problems of number theory with arithmetic functions.

3. To know the problem of factorization in the rings of integers of number fields.

4. To know the basic facts about elliptic curves, the operation between its points and some of its properties and applications.

5. To know what are the main problems of additive number theory and its relations to other problems.



LEARNING OUTCOMES



1. To know how to deduce the laws of decomposition of primes in abelian extensions of the field of rational numbers.

2. To know how to apply the methods of algebraic number theory in the resolution of diophantine equations.

3. To be able to recognize problems of number theory whose solution depends on an elliptic curve.

4. To know how to calculate the rank and the torsion of the group of rational points of an elliptic curve in simple cases.

5. To know how to find estimates for different measures of algebraic numbers: means and measures of Mahler.

1. ARITHMETIC FUNCTIONS: Dirichlet products and means. Distribution of prime numbers: Theorem of Chebyshev. The Prime number theorem. Its elementary proof. Its analytical proof. Characters and Theorem of Dirichlet.



2. NUMBER FIELDS AND RINGS OF INTEGERS: Integral extensions of rings. Dedekind rings. Unique factorization of ideals. Laws of decomposition of primes.



3. ELLIPTIC CURVES: The group law on a cubic. Rational points. Torsion points. Theorem of Mordell-Weil. Computation of the rank.



4. ADDITIVE THEORY OF NUMBERS: Sums of squares. Partitions. Jacobi functions. The problem of Waring.

The theoretical content will be exposed in master classes following basic references that appear in the Bibliography. These master classes will be complemented by problem classes (classroom practices) in which students will apply the knowledge acquired in the theoretical lectures in order to solve problems. In the seminar sessions, exercises and representative examples will be considered. These will have been given to the students in advance, for them to have enough time to work out the solutions. Students must participate actively in the seminar sessions, and discussion of the solutions will be encouraged. Individual work on theory and problems might be proposed to the students, with the support of the lecturer, if needed, during the seminar sessions.

There will be a final writing exam. To pass the subject it will be enough to pass the writing exam and follow the activities in class. If the student decides to go to the final exam, the final mark will be the weighted average of the following activities, with the indicated weights:



20%, for other types of exercises, either individual or in groups, and written or with oral exposition, developed during the course.



80%, the final written exam (but, in any case, a minimum of four points out of 10 will be necessary to pass the subject)



In the event that the sanitary conditions prevent the realization of a face-to-face evaluation, a non-face-to-face evaluation will be activated, of which the students will be informed promptly.

Basic bibliography

P. SAMUEL, Théorie algèbrique des nombres, Hermann, Paris, 1967.

I. STEWART, D. TALL, Algebraic Number Theory, Chapman&Hall, 1987.

In-depth bibliography

S. LANG, Algebraic Number Theory,1994.
R. LONG, Algebraic Number Theory, Marcel Dekker,1977.
D.A. MARCUS, Number Fields, Springer,1977.
T. ONO, An Introduction to Algebraic Number Theory, Plenum,1990.