Local Langlands conjectures

Key takeaways

Local Langlands conjectures belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Local Langlands conjectures to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Local Langlands conjectures from memory before moving on to harder problems.

Reference excerpt

In mathematics, the local Langlands conjectures, introduced by Robert Langlands, are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group G {\displaystyle G} over a local field F {\displaystyle F} , and representations of the Langlands group of F {\displaystyle F} into the L {\displaystyle L} -group of G {\displaystyle G} . This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups.

Local Langlands conjectures for GL1 The local Langlands conjectures for GL 1 ⁡ ( K ) {\displaystyle \operatorname {GL} _{1}(K)} follow from (and are essentially equivalent to) local class field theory. More precisely, the Artin map gives an isomorphism from the group GL 1 ⁡ ( K ) = K × {\displaystyle \operatorname {GL} _{1}(K)=K^{\times }} to the abelianization of the Weil group. In particular, irreducible smooth representations of GL 1 ⁡ ( K ) {\displaystyle \operatorname {GL} _{1}(K)} are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL 1 ⁡ ( C ) {\displaystyle \operatorname {GL} _{1}(\mathbb {C} )} . This gives the Langlands correspondence between homomorphisms of the Weil group to GL 1 ⁡ ( C ) {\displaystyle \operatorname {GL} _{1}(\mathbb {C} )} and irreducible smooth representations of GL 1 ⁡ ( K ) {\displaystyle \operatorname {GL} _{1}(K)} .

Representations of the Weil group Representations of the Weil group do not quite correspond to irreducible smooth representations of general linear groups. To get a bijection, one has to slightly modify the notion of a representation of the Weil group, to something called a Weil–Deligne representation. This consists of a representation of the Weil group on a vector space V {\displaystyle V} together with a nilpotent endomorphism N {\displaystyle N} of V {\displaystyle V} such that w N w − 1 = ‖ w ‖ N {\displaystyle wNw^{-1}=\|w\|N} , or equivalently a representation of the Weil–Deligne group. In addition, the representation of the Weil group should have an open kernel and be (Frobenius) semisimple. For every Frobenius semisimple complex n {\displaystyle n} -dimensional Weil–Deligne representation ρ {\displaystyle \rho } of the Weil group of F {\displaystyle F} there is an L-function L ( s , ρ ) {\displaystyle L(s,\rho )} and a local ε-factor ε ( s , ρ , ψ ) {\displaystyle \varepsilon (s,\rho ,\psi )} (depending on a character ψ {\displaystyle \psi } of F {\displaystyle F} ).

Representations of GLn(F) The representations of GL n ⁡ ( F ) {\displaystyle \operatorname {GL} _{n}(F)} appearing in the local Langlands correspondence are smooth irreducible complex representations.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Local Langlands conjectures

Start with the simplest possible case. Write down what Local Langlands conjectures claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Local Langlands conjectures before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Local Langlands conjectures ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Local Langlands conjectures

In research: Local Langlands conjectures appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Local Langlands conjectures in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Local Langlands conjectures is common in secondary-school and first-year university syllabi. It links to neighbouring topics Automorphic forms, Class field theory, Conjectures, so understanding it makes those chapters shorter.
In everyday life: Look for Local Langlands conjectures outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Local Langlands conjectures in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Local Langlands conjectures means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Local Langlands conjectures out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Local Langlands conjectures in simple terms?

In mathematics, the local Langlands conjectures, introduced by Robert Langlands, are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group G {\displaystyle G} over a local field F {\displaystyle F} , and representations of t…

Why does Local Langlands conjectures matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Local Langlands conjectures?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Local Langlands conjectures.