John Carlos Baez on Nostr: 2024-01-22 https://www.youtube.com/watch?v=V8gOqDDnPN4 Galois representations and ...
2024-01-22
https://www.youtube.com/watch?v=V8gOqDDnPN4
Galois representations and motives: trying to understand the Langlands program, or at least the modularity theorem. Galois representations from torsion points on elliptic curves.
Consider the 5-torsion points of the Gaussian elliptic curve. These 25 points form a discrete subvariety of the elliptic curve, which is a projective variety, but we can treat it as an affine variety, and since it's discrete it is the spectrum of a special commutative Frobenius algebra. Since the 5-torsion points form a group, this algebra also becomes a bicommutative Hopf algebra. But these 25 points are also naturally a 2-dimensional vector space over the finite field 𝔽₅, and because the Gaussian elliptic curve is defined over ℚ, this vector space gives a representation of the absolute Galois group of ℚ.
Working with other elliptic curves and other primes ℓ, we get a map from the moduli stack of elliptic curves to the moduli stack of representations of the absolute Galois group of ℚ on 2d vector spaces over 𝔽_ℓ . This map of moduli stacks really arises, in a contravariant way, from a map of theories in the doctrine of 2-rigs: namely, a map from the theory of 2d vector spaces over 𝔽_ℓ to the theory of elliptic curves! (A "theory in the doctrine of 2-rigs" is just a 2-rig.)
We would like to describe the theory of elliptic curves as the theory of a formal group together with extra relations which force our formal group to be an elliptic curve.
For more details see:
Nigel Boston, The proof of Fermat's last theorem, https://www.researchgate.net/profile/Thong-Nguyen-Quang-Do/post/Are-there-other-pieces-of-information-about-Victory-Road-to-FLT/attachment/5b0ab419b53d2f63c3ce5da8/AS%3A630893421006850%401527428121602/download/FLT+Boston.pdf
We would like to understand Chapter 4, and especially Theorem 4.28.
(5/n)
https://www.youtube.com/watch?v=V8gOqDDnPN4
Galois representations and motives: trying to understand the Langlands program, or at least the modularity theorem. Galois representations from torsion points on elliptic curves.
Consider the 5-torsion points of the Gaussian elliptic curve. These 25 points form a discrete subvariety of the elliptic curve, which is a projective variety, but we can treat it as an affine variety, and since it's discrete it is the spectrum of a special commutative Frobenius algebra. Since the 5-torsion points form a group, this algebra also becomes a bicommutative Hopf algebra. But these 25 points are also naturally a 2-dimensional vector space over the finite field 𝔽₅, and because the Gaussian elliptic curve is defined over ℚ, this vector space gives a representation of the absolute Galois group of ℚ.
Working with other elliptic curves and other primes ℓ, we get a map from the moduli stack of elliptic curves to the moduli stack of representations of the absolute Galois group of ℚ on 2d vector spaces over 𝔽_ℓ . This map of moduli stacks really arises, in a contravariant way, from a map of theories in the doctrine of 2-rigs: namely, a map from the theory of 2d vector spaces over 𝔽_ℓ to the theory of elliptic curves! (A "theory in the doctrine of 2-rigs" is just a 2-rig.)
We would like to describe the theory of elliptic curves as the theory of a formal group together with extra relations which force our formal group to be an elliptic curve.
For more details see:
Nigel Boston, The proof of Fermat's last theorem, https://www.researchgate.net/profile/Thong-Nguyen-Quang-Do/post/Are-there-other-pieces-of-information-about-Victory-Road-to-FLT/attachment/5b0ab419b53d2f63c3ce5da8/AS%3A630893421006850%401527428121602/download/FLT+Boston.pdf
We would like to understand Chapter 4, and especially Theorem 4.28.
(5/n)