mathematics

Jacquet–Langlands correspondence

Jacquet–Langlands correspondence is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Jacquet–Langlands correspondence rather than just read about it. In short: In mathematics, the Jacquet–Langlands correspondence is a correspondence between automorphic forms on GL2 and its twisted forms, proved by Jacquet and Langlands (1970, section 16) in their book Automorphic Forms on GL(2) using the Selberg trace formula. It was one of the first examples of the Langlands philosophy that maps between L-groups should induce maps between automorphic representations.

Key takeaways

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

Reference excerpt

In mathematics, the Jacquet–Langlands correspondence is a correspondence between automorphic forms on GL2 and its twisted forms, proved by Jacquet and Langlands (1970, section 16) in their book Automorphic Forms on GL(2) using the Selberg trace formula. It was one of the first examples of the Langlands philosophy that maps between L-groups should induce maps between automorphic representations. There are generalized versions of the Jacquet–Langlands correspondence relating automorphic representations of GLr(D) and GLdr(F), where D is a division algebra of degree d2 over the local or global field F. Suppose that G is an inner twist of the algebraic group GL2, in other words the multiplicative group of a quaternion algebra. The Jacquet–Langlands correspondence is bijection between

Automorphic representations of G of dimension greater than 1 Cuspidal automorphic representations of GL2 that are square integrable (modulo the center) at each ramified place of G. Corresponding representations have the same local components at all unramified places of G. Rogawski (1983) and Deligne, Kazhdan & Vignéras (1984) extended the Jacquet–Langlands correspondence to division algebras of higher dimension.

References Deligne, Pierre; Kazhdan, David; Vignéras, M.-F. (1984), "Représentations des algèbres centrales simples p-adiques", Représentations des groupes réductifs sur un corps local, Travaux en Cours, Paris: Hermann, pp. 33–117, ISBN 978-2-7056-5989-9, MR 0771672 Henniart, Guy (2006), "On the local Langlands and Jacquet-Langlands correspondences", in Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis; et al. (eds.), International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, pp. 1171–1182, ISBN 978-3-03719-022-7, MR 2275640, archived from the original on 2012-03-15, retrieved 2011-07-01 Jacquet, H.; Langlands, Robert P. (1970), Automorphic Forms on GL(2), Lecture Notes in Mathematics, vol. 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, ISBN 978-3-540-04903-6, MR 0401654 Rogawski, Jonathan D. (1983), "Representations of GL(n) and division algebras over a p-adic field", Duke Mathematical Journal, 50 (1): 161–196, doi:10.1215/s0012-7094-83-05006-8, ISSN 0012-7094, MR 0700135

Worked examples

Example 1 — a first encounter with Jacquet–Langlands correspondence

Start with the simplest possible case. Write down what Jacquet–Langlands correspondence 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 Jacquet–Langlands correspondence 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 Jacquet–Langlands correspondence 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 Jacquet–Langlands correspondence

In research: Jacquet–Langlands correspondence 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 Jacquet–Langlands correspondence 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: Jacquet–Langlands correspondence is common in secondary-school and first-year university syllabi. It links to neighbouring topics Automorphic forms, Theorems in harmonic analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Jacquet–Langlands correspondence 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.

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How to study Jacquet–Langlands correspondence in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Jacquet–Langlands correspondence 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 Jacquet–Langlands correspondence out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Jacquet–Langlands correspondence in simple terms?

Why does Jacquet–Langlands correspondence 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 Jacquet–Langlands correspondence?

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 Jacquet–Langlands correspondence.