mathematics

Weil reciprocity law

Weil reciprocity law 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 Weil reciprocity law rather than just read about it. In short: In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e. rational functions on C, then f((g)) = g((f)) where the notation has this meaning: (h) is the divisor of the function h, or in other words the formal sum of its zeroes and poles counted with multiplicity; and a functi…

Key takeaways

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

Reference excerpt

In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e. rational functions on C, then

f((g)) = g((f)) where the notation has this meaning: (h) is the divisor of the function h, or in other words the formal sum of its zeroes and poles counted with multiplicity; and a function applied to a formal sum means the product (with multiplicities, poles counting as a negative multiplicity) of the values of the function at the points of the divisor. With this definition there must be the side-condition, that the divisors of f and g have disjoint support (which can be removed). In the case of the projective line, this can be proved by manipulations with the resultant of polynomials. To remove the condition of disjoint support, for each point P on C a local symbol

(f, g)P is defined, in such a way that the statement given is equivalent to saying that the product over all P of the local symbols is 1. When f and g both take the values 0 or ∞ at P, the definition is essentially in limiting or removable singularity terms, by considering (up to sign)

fagb with a and b such that the function has neither a zero nor a pole at P. This is achieved by taking a to be the multiplicity of g at P, and −b the multiplicity of f at P. The definition is then

(f, g)P = (−1)ab fagb. See for example Jean-Pierre Serre, Groupes algébriques et corps de classes, pp. 44–46, for this as a special case of a theory on mapping algebraic curves into commutative groups. There is a generalisation of Serge Lang to abelian varieties (Lang, Abelian Varieties).

References André Weil, Oeuvres Scientifiques I, p. 291 (in Lettre à Artin, a 1942 letter to Artin, explaining the 1940 Comptes Rendus note Sur les fonctions algébriques à corps de constantes finis) Griffiths, Phillip; Harris, Joseph (1994). Principles of Algebraic Geometry. Wiley Classics Library. New York, NY: John Wiley & Sons Ltd. pp. 242–3. ISBN 0-471-05059-8. Zbl 0836.14001. for a proof in the Riemann surface case Arbarello, E.; De Concini, C.; Kac, V.G. (1989). "The infinite wedge representation and the reciprocity law for algebraic curves". In Ehrenpreis, Leon; Gunning, Robert C. (eds.). Theta functions, Bowdoin 1987. (Proceedings of the 35th Summer Research Institute, Bowdoin Coll., Brunswick/ME July 6-24, 1987). Proceedings of Symposia in Pure Mathematics. Vol. 49. Providence, RI: American Mathematical Society. pp. 171–190. ISBN 0-8218-1483-4. Zbl 0699.22028. Serre, Jean-Pierre (1988). Algebraic groups and class fields. Graduate Texts in Mathematics. Vol. 117 (Translation of the French 2nd ed.). New York, etc.: Springer-Verlag. ISBN 3-540-96648-X. Zbl 0703.14001.

Worked examples

Example 1 — a first encounter with Weil reciprocity law

Start with the simplest possible case. Write down what Weil reciprocity law 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 Weil reciprocity law 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 Weil reciprocity law 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 Weil reciprocity law

In research: Weil reciprocity law 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 Weil reciprocity law 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: Weil reciprocity law is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic curves, Theorems in algebraic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Weil reciprocity law 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 Weil reciprocity law in 20 minutes

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

Frequently asked questions

What is Weil reciprocity law in simple terms?

Why does Weil reciprocity law 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 Weil reciprocity law?

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 Weil reciprocity law.