Introduction

Andrew Wiles, the renowned mathematician, achieved fame in 1995 when he provided a proof for Fermat's Last Theorem, one of the most famous unsolved problems in mathematics for over three centuries. However, Wiles' contributions to mathematics extend far beyond this singular achievement. Since his proof of Fermat's Last Theorem, Wiles has continued to explore and contribute to several significant areas in number theory and algebraic geometry. This article delves into the post-Fermat work of Andrew Wiles, highlighting his notable endeavors and scholarly outputs.

1. The Modularity of Galois Representations

One of Wiles' primary areas of post-Fermat research has been the modularity of Galois representations. This field intersects deeply with number theory and algebraic geometry. In 1995, the proof of Fermat's Last Theorem hinged on proving the modularity of semistable elliptic curves, and Wiles continued this line of inquiry with several important papers.

1.1 Collaboration with Chris Skinner

Wiles worked with Christopher Skinner, a number theorist at Princeton University, on further questions related to the modularity of Galois representations. Their 126-page paper tackled the cases of residually reducible representations and representations of totally real fields. Their research contributes to a deeper understanding of the modularity conjecture and has wide-ranging implications for other areas of number theory. This work has been widely cited and utilized by other mathematicians in the field.

1.2 Paper on Rational Points and Residual Reducibility

Wiles and his colleagues Mirela Ciperiani wrote a paper investigating the existence of rational points over solvable extensions. This research is crucial for understanding the arithmetic properties of algebraic varieties, providing insights that are valuable for mathematicians studying Diophantine equations and related questions.

1.3 Collaboration with Andrew Snowden

Wiles also collaborated with Andrew Snowden, a mathematician known for his work in combinatorial algebraic geometry. Together, they explored questions regarding the modularity of Galois representations, contributing to the ongoing discussion and expanding the scope of possible techniques and methods used in this area.

2. Birch and Swinnerton-Dyer Conjecture and Iwasawa Theory

Beyond research in modularity, Wiles has also pursued significant work on major conjectures in number theory. His early collaboration with John Coates in proving a major result on the Birch and Swinnerton-Dyer conjecture and his joint work with Barry Mazur on the Main Conjecture of Iwasawa Theory exemplify his commitment to solving some of the most challenging problems in mathematics.

2.1 The Birch and Swinnerton-Dyer Conjecture

Wiles' early paper with John Coates on the Birch and Swinnerton-Dyer conjecture represents his initial contributions to this area. This conjecture is highly significant for understanding the arithmetic of elliptic curves, and Wiles' work here laid important groundwork, although a full proof remains challenging. Wiles has continued to explore related questions, contributing to the broader efforts in this field.

2.2 Main Conjecture of Iwasawa Theory

Wiles' collaboration with Barry Mazur on the Main Conjecture of Iwasawa Theory, which connects the arithmetic of elliptic curves with the theory of $p$-adic $L$-functions, is another major contribution. This conjecture plays a critical role in the modern theory of modular forms and has far-reaching implications for the study of algebraic number fields.

Conclusion

Andrew Wiles' mathematical journey is not limited to the proof of Fermat's Last Theorem. His contributions to the modularity of Galois representations and his work on the Birch and Swinnerton-Dyer conjecture, as well as Iwasawa Theory, demonstrate his continued excellence in the field of number theory. Through his extensive research and collaborations, Wiles has advanced our understanding of complex algebraic structures and continues to influence mathematical scholarship.