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The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about.

  • Drinfeld modules are like the function field analogue of CM elliptic curves. To see this, complexify an elliptic curve $E$ to get a torus $\mathbb{C} / \Lambda$. If $K$ is an imaginary quadratic field, those lattices $\Lambda$ such that $\mathcal{O}_K \Lambda \subseteq \Lambda$ correspond to elliptic curves with CM, meaning that there exists a map $\mathcal{O}_K \to \operatorname{End} E$ whose 'derivative' is the inclusion $\mathcal{O}_K \hookrightarrow \mathbb{C}$. Now pass to function fields. Take $X$ a curve over $\mathbb{F}_q$, with function field $K$, and put $C$ for the algebraic closure of the completion. Then we can define Drinfeld modules as an algebraic structure on a quotient $C / \Lambda$.

  • Shtukas are a 'generalisation' of Drinfeld modules. According to Wikipedia, they consist roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it. From Goss' "What is..." article, I gather that some analogy with differential operators is also involved in their conception.

  • Shtukas are used to give a correspondence between automorphic forms on $\operatorname{GL}_n(K)$, with $K$ a function field, and certain representations of absolute Galois groups. For each automorphic form, one somehow considers the $\ell$-adic cohomology of the stack of rank-$n$ shtukas with a certain level structure, and I presume this cohomology has an equivariant structure that gives rise to a representation.

While this gives me a comfortable overview, one thing I cannot put my finger on is why things work the way they work. I fail to get a grasp on the intuition behind a shtuka, and I especially fail to see why it makes sense to study them with an outlook on the Langlands program. This leads to the following questions.

Question 1. What is the intuition behind shtukas? What are they, even roughly speaking? Is there a number field analogue that I might be more comfortable with?

 

Question 2. How can I 'see' that shtukas should be of aid with the Langlands program? What did Drinfeld see when he started out? Why should I want to take the cohomology of the moduli stack? Were there preceding results pointing in the direction of this approach?

The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about.

  • Drinfeld modules are like the function field analogue of CM elliptic curves. To see this, complexify an elliptic curve $E$ to get a torus $\mathbb{C} / \Lambda$. If $K$ is an imaginary quadratic field, those lattices $\Lambda$ such that $\mathcal{O}_K \Lambda \subseteq \Lambda$ correspond to elliptic curves with CM, meaning that there exists a map $\mathcal{O}_K \to \operatorname{End} E$ whose 'derivative' is the inclusion $\mathcal{O}_K \hookrightarrow \mathbb{C}$. Now pass to function fields. Take $X$ a curve over $\mathbb{F}_q$, with function field $K$, and put $C$ for the algebraic closure of the completion. Then we can define Drinfeld modules as an algebraic structure on a quotient $C / \Lambda$.

  • Shtukas are a 'generalisation' of Drinfeld modules. According to Wikipedia, they consist roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it. From Goss' "What is..." article, I gather that some analogy with differential operators is also involved in their conception.

  • Shtukas are used to give a correspondence between automorphic forms on $\operatorname{GL}_n(K)$, with $K$ a function field, and certain representations of absolute Galois groups. For each automorphic form, one somehow considers the $\ell$-adic cohomology of the stack of rank-$n$ shtukas with a certain level structure, and I presume this cohomology has an equivariant structure that gives rise to a representation.

While this gives me a comfortable overview, one thing I cannot put my finger on is why things work the way they work. I fail to get a grasp on the intuition behind a shtuka, and I especially fail to see why it makes sense to study them with an outlook on the Langlands program. This leads to the following questions.

Question 1. What is the intuition behind shtukas? What are they, even roughly speaking? Is there a number field analogue that I might be more comfortable with?

 

Question 2. How can I 'see' that shtukas should be of aid with the Langlands program? What did Drinfeld see when he started out? Why should I want to take the cohomology of the moduli stack? Were there preceding results pointing in the direction of this approach?

The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about.

  • Drinfeld modules are like the function field analogue of CM elliptic curves. To see this, complexify an elliptic curve $E$ to get a torus $\mathbb{C} / \Lambda$. If $K$ is an imaginary quadratic field, those lattices $\Lambda$ such that $\mathcal{O}_K \Lambda \subseteq \Lambda$ correspond to elliptic curves with CM, meaning that there exists a map $\mathcal{O}_K \to \operatorname{End} E$ whose 'derivative' is the inclusion $\mathcal{O}_K \hookrightarrow \mathbb{C}$. Now pass to function fields. Take $X$ a curve over $\mathbb{F}_q$, with function field $K$, and put $C$ for the algebraic closure of the completion. Then we can define Drinfeld modules as an algebraic structure on a quotient $C / \Lambda$.

  • Shtukas are a 'generalisation' of Drinfeld modules. According to Wikipedia, they consist roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it. From Goss' "What is..." article, I gather that some analogy with differential operators is also involved in their conception.

  • Shtukas are used to give a correspondence between automorphic forms on $\operatorname{GL}_n(K)$, with $K$ a function field, and certain representations of absolute Galois groups. For each automorphic form, one somehow considers the $\ell$-adic cohomology of the stack of rank-$n$ shtukas with a certain level structure, and I presume this cohomology has an equivariant structure that gives rise to a representation.

While this gives me a comfortable overview, one thing I cannot put my finger on is why things work the way they work. I fail to get a grasp on the intuition behind a shtuka, and I especially fail to see why it makes sense to study them with an outlook on the Langlands program. This leads to the following questions.

Question 1. What is the intuition behind shtukas? What are they, even roughly speaking? Is there a number field analogue that I might be more comfortable with?

Question 2. How can I 'see' that shtukas should be of aid with the Langlands program? What did Drinfeld see when he started out? Why should I want to take the cohomology of the moduli stack? Were there preceding results pointing in the direction of this approach?

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How can I see the relation between shtukas and the Langlands conjecture?

The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about.

  • Drinfeld modules are like the function field analogue of CM elliptic curves. To see this, complexify an elliptic curve $E$ to get a torus $\mathbb{C} / \Lambda$. If $K$ is an imaginary quadratic field, those lattices $\Lambda$ such that $\mathcal{O}_K \Lambda \subseteq \Lambda$ correspond to elliptic curves with CM, meaning that there exists a map $\mathcal{O}_K \to \operatorname{End} E$ whose 'derivative' is the inclusion $\mathcal{O}_K \hookrightarrow \mathbb{C}$. Now pass to function fields. Take $X$ a curve over $\mathbb{F}_q$, with function field $K$, and put $C$ for the algebraic closure of the completion. Then we can define Drinfeld modules as an algebraic structure on a quotient $C / \Lambda$.

  • Shtukas are a 'generalisation' of Drinfeld modules. According to Wikipedia, they consist roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it. From Goss' "What is..." article, I gather that some analogy with differential operators is also involved in their conception.

  • Shtukas are used to give a correspondence between automorphic forms on $\operatorname{GL}_n(K)$, with $K$ a function field, and certain representations of absolute Galois groups. For each automorphic form, one somehow considers the $\ell$-adic cohomology of the stack of rank-$n$ shtukas with a certain level structure, and I presume this cohomology has an equivariant structure that gives rise to a representation.

While this gives me a comfortable overview, one thing I cannot put my finger on is why things work the way they work. I fail to get a grasp on the intuition behind a shtuka, and I especially fail to see why it makes sense to study them with an outlook on the Langlands program. This leads to the following questions.

Question 1. What is the intuition behind shtukas? What are they, even roughly speaking? Is there a number field analogue that I might be more comfortable with?

Question 2. How can I 'see' that shtukas should be of aid with the Langlands program? What did Drinfeld see when he started out? Why should I want to take the cohomology of the moduli stack? Were there preceding results pointing in the direction of this approach?