Chandrashekhar Khare
UCLA
January 4, 2024
Modularity of Galois representations and the magic of small primes: In 1994 Andrew Wiles deduced Fermat's Last Theorem from his proof of the modularity of semistable elliptic curves over the rationals $\mathbb Q$.His proof of modularity of elliptic curves exploits accidents that are peculiar to small primes like 3 and 5, for instance GL$_2(\mathbb F_p)$ is solvable for $p \leq 3$ and the modular curve $X(p)$ is of genus 0 for $p \leq 5$. Our proof of Serre's modularity conjecture in joint work with Wintenberger also exploits accidents particular to residue characteristics 2 and 3, namely results of Tate and Serre that for $p \leq 3$ there are no irreducible representations defined over $\mathbb Q$ to GL$_2(\overline{\mathbb F_p})$ unramified outside $p$. The recent work of Newton-Thorne and Clozel-Newton-Thorne on symmetric power functoriality and non-abelian base change to totally real fields for newforms also exploits the magic of small primes. I will cover the older and more recent developments in my talk.
UCLA
January 4, 2024
Modularity of Galois representations and the magic of small primes: In 1994 Andrew Wiles deduced Fermat's Last Theorem from his proof of the modularity of semistable elliptic curves over the rationals $\mathbb Q$.His proof of modularity of elliptic curves exploits accidents that are peculiar to small primes like 3 and 5, for instance GL$_2(\mathbb F_p)$ is solvable for $p \leq 3$ and the modular curve $X(p)$ is of genus 0 for $p \leq 5$. Our proof of Serre's modularity conjecture in joint work with Wintenberger also exploits accidents particular to residue characteristics 2 and 3, namely results of Tate and Serre that for $p \leq 3$ there are no irreducible representations defined over $\mathbb Q$ to GL$_2(\overline{\mathbb F_p})$ unramified outside $p$. The recent work of Newton-Thorne and Clozel-Newton-Thorne on symmetric power functoriality and non-abelian base change to totally real fields for newforms also exploits the magic of small primes. I will cover the older and more recent developments in my talk.