I just realized that the OP links to a YouTube video and to some slides, but the two don't match—they're two different talks by Buzzard.

For completeness, let me therefore mention some results by James Arthur, which are mentioned in the linked slides but not the linked YouTube video. On page 13 of Abelian Surfaces over totally real fields are Potentially Modular by George Boxer, Frank Calegari, Toby Gee, and Vincent Pilloni, there is the following remark.

It should be noted that we use Arthur’s multiplicity formula for the discrete spectrum of GSp₄, as announced in [Art04]. A proof of this (relying on Arthur’s work for symplectic and orthogonal groups in [Art13]) was given in [GT18], but this proof is only as unconditional as the results of [Art13] and [MW16a, MW16b]. In particular, it depends on cases of the twisted weighted fundamental lemma that were announced in [CL10], but whose proofs have not yet appeared, as well as on the references [A24], [A25], [A26] and [A27] in [Art13], which at the time of writing have not appeared publicly.

Arthur's (unavailable) references [A24] through [A27] are:

[A24] Endoscopy and singular invariant distributions, in preparation.

[A25] Duality, Endoscopy, and Hecke operators, in preparation.

[A26] A nontempered intertwining relation for $GL(N)$, in preparation.

[A27] Transfer factors and Whittaker models, in preparation.

I just realized that the OP links to a YouTube video and to some slides, but the two don't match—they're two different talks by Buzzard.

For completeness, let me therefore mention some results by James Arthur, which are mentioned in the linked slides but not the linked YouTube video. On page 13 of Abelian Surfaces over totally real fields are Potentially Modular by George Boxer, Frank Calegari, Toby Gee, and Vincent Pilloni, there is the following remark.

It should be noted that we use Arthur’s multiplicity formula for the discrete spectrum of GSp₄, as announced in [Art04]. A proof of this (relying on Arthur’s work for symplectic and orthogonal groups in [Art13]) was given in [GT18], but this proof is only as unconditional as the results of [Art13] and [MW16a, MW16b]. In particular, it depends on cases of the twisted weighted fundamental lemma that were announced in [CL10], but whose proofs have not yet appeared, as well as on the references [A24], [A25], [A26] and [A27] in [Art13], which at the time of writing have not appeared publicly.

Arthur's (unavailable) references [A24] through [A27] are:

[A24] Endoscopy and singular invariant distributions, in preparation.

[A25] Duality, Endoscopy, and Hecke operators, in preparation.

[A26] A nontempered intertwining relation for $GL(N)$, in preparation.

[A27] Transfer factors and Whittaker models, in preparation.

There are several results by James Arthur that fall into this category. (I didn't mention them previously because I thought they were already mentioned by Kevin Buzzard in the talk cited by the OP, but I belatedly realized that I was confusing that with a different talk by Buzzard.)

On page 13 of Abelian Surfaces over totally real fields are Potentially Modular by George Boxer, Frank Calegari, Toby Gee, and Vincent Pilloni, there is the following remark.

It should be noted that we use Arthur’s multiplicity formula for the discrete spectrum of GSp₄, as announced in [Art04]. A proof of this (relying on Arthur’s work for symplectic and orthogonal groups in [Art13]) was given in [GT18], but this proof is only as unconditional as the results of [Art13] and [MW16a, MW16b]. In particular, it depends on cases of the twisted weighted fundamental lemma that were announced in [CL10], but whose proofs have not yet appeared, as well as on the references [A24], [A25], [A26] and [A27] in [Art13], which at the time of writing have not appeared publicly.

Arthur's (unavailable) references [A24] through [A27] are:

[A24] Endoscopy and singular invariant distributions, in preparation.

[A25] Duality, Endoscopy, and Hecke operators, in preparation.

[A26] A nontempered intertwining relation for $GL(N)$, in preparation.

[A27] Transfer factors and Whittaker models, in preparation.

I just realized that the OP links to a YouTube video and to some slides, but the two don't match—they're two different talks by Buzzard.

For completeness, let me therefore mention some results by James Arthur, which are mentioned in the linked slides but not the linked YouTube video. On page 13 of Abelian Surfaces over totally real fields are Potentially Modular by George Boxer, Frank Calegari, Toby Gee, and Vincent Pilloni, there is the following remark.

It should be noted that we use Arthur’s multiplicity formula for the discrete spectrum of GSp₄, as announced in [Art04]. A proof of this (relying on Arthur’s work for symplectic and orthogonal groups in [Art13]) was given in [GT18], but this proof is only as unconditional as the results of [Art13] and [MW16a, MW16b]. In particular, it depends on cases of the twisted weighted fundamental lemma that were announced in [CL10], but whose proofs have not yet appeared, as well as on the references [A24], [A25], [A26] and [A27] in [Art13], which at the time of writing have not appeared publicly.

Arthur's (unavailable) references [A24] through [A27] are:

[A24] Endoscopy and singular invariant distributions, in preparation.

[A25] Duality, Endoscopy, and Hecke operators, in preparation.

[A26] A nontempered intertwining relation for $GL(N)$, in preparation.

[A27] Transfer factors and Whittaker models, in preparation.