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Local criteria for the unit equation and the asymptotic Fermat's Last Theorem
Nuno Freitas1, Alain Kraus2, Samir Siksek3
1Departament de Matemàtiques i Informàtica, Universitat de Barcelona, 08007 Barcelona, Spain.
Summary
This study establishes local criteria for the asymptotic Fermat's Last Theorem and unit equation non-existence over totally real number fields of odd degree. Specific conditions on ramification and splitting guarantee the theorem
Area of Science:
- Number Theory
- Algebraic Number Theory
Background:
- The asymptotic Fermat's Last Theorem (AFLT) investigates generalizations of Fermat's Last Theorem to number fields.
- The unit equation plays a crucial role in Diophantine equations and algebraic number theory.
Purpose of the Study:
- To establish purely local criteria for AFLT over totally real number fields of odd degree.
- To provide local conditions for the nonexistence of solutions to the unit equation over such fields.
Main Methods:
- Utilizing properties of totally real number fields.
- Applying techniques from local class field theory.
- Analyzing the structure of the unit group in number fields.
Main Results:
- Several local criteria for AFLT over totally real number fields of odd degree are proven.
- Local conditions for the nonexistence of solutions to the unit equation are established.
- A specific example demonstrates that if 2 is totally ramified and 3 splits completely in F, then AFLT holds over F.
Conclusions:
- Local properties of number fields can effectively determine the validity of AFLT and the unit equation.
- The findings contribute to understanding Diophantine equations in algebraic number theory.
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