# Number Theory

# Number Theory

Number Theory is a branch of pure mathematics devoted innitialy to the study of prime numbers as well as the properties of objects made out of integers or defined as generalizations of such. Number theory problems are for example the search of integer solutions of diophantine equations, or in general varieties, or over some ring or field where the equation or a variety is defined. The latter is considered to be Arithmetic Geometry which is nowadays one of the most interesting branches of Number Theory. Number Theory (or Arithmetic Geometry) has a deep interaction with different areas like algebra, analysis, algebraic geometry, topology, dynamical systems and many others. As is well-known, this area was considered by Gauss as “the Queen of mathematics’’ and usually a Field Medalist in almost every ICM is a mathematician which contributed to the field.

Number Theory is well consolidated in Barcelona, thanks to the influence of professor Pilar Bayer, professor Pascual Llorente (during his stay in Barcelona) and their descendants. Nowadays, there are groups in UB, UAB and UPC working in different aspects of Arithmetic Geometry and Number Theory. All these research groups are unified under STNB “Seminari de Teoria de Nombres de Barcelona’’.The latest activities are available on the STNB website.

Research Lines

##### Galois theory, Galois representations and inverse Galois problem.

**Keywords: **Representation. Absolute Galois group. Hopf Galois. Differential Galois theory. Serre conjecture. Fontaine-Mazur conjecture. Automorphic forms.

Pilar Bayer (UB)

Teresa Crespo (UB)

Luis Dieulefait (UB)

Anna Rio (UPC)

Montse Vela (UPC)

Núria Vila (UB)

##### Arithmetic properties of elliptic curves and classical modular curves

**Keywords:**BSD conjecture. Selmer group. Modular cusp forms. Automorphism of modular curves. Parametrization of curves. Jacobians of modular curves.

Francesc Bars (UAB)

Jordi Quer (UPC)

Joan Carles Lario (UPC)

Victor Rotger (UPC)

Xavier Xarles (UAB)

##### Arithmetic properties of global fields and curves and applications

**Keywords: **Curves of lower genus. Twists. Field of moduli. Rational points. Montes algorithm. Iwasawa theory. P-adic methods. Valuations theory .

Luis Dieulefait (UB)

Francesc Bars (UAB)

Jorge Jiménez(UPC)

Jordi Quer (UPC)

Xavier Guitart (UPC)

Jordi Guàrdia (UPC)

Joan Carles Lario (UPC)

Enric Nart (UAB)

Joaquim Roé (UAB)

Victor Rotger (UPC)

Xavier Xarles (UAB)

##### Arithmetic properties of Shimura curves and Shimura varieties

**Keywords: **

Montse Alsina (UPC)

Pilar Bayer (UB)

Xavier Guitart (UPC)

Victor Rotger (UPC)

Links to research groups

Researchers in the Barcelona area, and more generally in Catalonia, group themselves into smaller research groups, following not only academic but also administrative criteria, like location or research grants.

Groups with BGSMath faculty

Research group | Contact person |

Number Theory at UB | Luis Dieulefait |

Number Theory at UPC | Jordi Quer, Joan Carles Lario |

Algebraic and Arithmetic Geometry | Enric Nart |

Other groups in Catalonia

Research group | Contact person |

Cryptography and graphs (Universitat de Lleida) | Josep Maria Miret Biosca |