mathematics

Martin Eichler

Martin Eichler 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 Martin Eichler rather than just read about it. In short: Martin Maximilian Emil Eichler (29 March 1912 – 7 October 1992) was a German mathematician. Eichler received his Ph.D. from the Martin Luther University of Halle-Wittenberg in 1936.

Key takeaways

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

Reference excerpt

Martin Maximilian Emil Eichler (29 March 1912 – 7 October 1992) was a German mathematician. Eichler received his Ph.D. from the Martin Luther University of Halle-Wittenberg in 1936. Eichler and Goro Shimura developed a method to construct elliptic curves from certain modular forms. The converse notion that every elliptic curve has a corresponding modular form would later be the key to the proof of Fermat's Last Theorem.

Selected publications Quadratische Formen und orthogonale Gruppen, Springer 1952, 1974 Lectures on Modular Correspondences. Tata Institute. 1955; pbk, 169 pages{{cite book}}: CS1 maint: postscript (link) Einführung in die Theorie der algebraischen Zahlen und Funktionen, Birkhäuser 1963; Eng. trans. 1966, Introduction to the theory of algebraic numbers and functions, in which a section on modular forms is added; pbk 2014 reprint of 1963 German original Projective varieties and modular forms 1971 (Riemann–Roch theorem); Eichler, M. (15 November 2006). 2006 edition. Springer. ISBN 9783540368694. with Don Zagier: The Theory of Jacobi forms, Birkhäuser 1985; Eichler, Martin; Zagier, Don (14 December 2013). 2013 edition. Springer. ISBN 9781468491623. Über die Einheiten der Divisionsalgebren, Mathem. Annalen 1937[link removed] Neuere Ergebnisse der Theorie der einfachen Algebren, Jahresbericht DMV 1937[link removed] Allgemeine Integration linearer partieller Differentialgleichungen von elliptischem Typ bei zwei Grundvariablen, Abh. Math. Sem. Univ. Hamburg 15 (1947), 179–210. MR 0029054 On the differential equation uxx + uyy + N(x)u = 0, Trans. Amer. Math. Soc. 65 (1949), 259–278 doi:10.1090/S0002-9947-1949-0029055-8 Zur Algebra der orthogonalen Gruppen Mathem. Zeitschrift 1950[link removed] Zahlentheorie der Quaternionenalgebren, Crelle J. vol. 195, 1955, with errata [1] Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenz-Zetafunktion, Archiv Math. vol. 5, 1954, pp. 355–366 (Ramanujan–Petersson conjecture) Eine Verallgemeinerung der Abelschen Integrale, Math. Zeitschrift vol. 67, 1957, pp. 267-298[link removed] Quadratische Formen und Modulfunktionen Acta Arithmetica vol. 4, 1958, pp. 217–239 Eine Vorbereitung auf den Riemann-Rochschen Satz für algebraische Funktionenkörper, Crelle J. 1964 Einige Anwendungen der Spurformel im Bereich der Modularkorrespondenzen[link removed], Mathem. Annalen 1967, (Eichler–Shimura theory) Eichler Eine Spurformel von Korrespondenzen von algebraischen Funktionenkörpern mit sich selber, Inv. Math. vol. 2, 1967[link removed] with corrections [link removed] The basis problem for modular forms and the traces of the Hecke operators, Springer, Lecture notes Math. vol.320, 1973, pp. 75–152

See also Eichler–Shimura congruence relation Eichler–Shimura isomorphism Eichler cohomology Eichler order Eichler's proof of the BCH theorem

References

External links O'Connor, John J.; Robertson, Edmund F., "Martin Eichler", MacTutor History of Mathematics Archive, University of St Andrews Martin Kneser, Martin Eichler (1912-1992), Acta Arithmetica vol. 65, 1993, pp. 293–296, Obituary (in German). Jürg Kramer, Leben und Werk von Martin Eichler, Elemente der Mathematik vol. 49, 1994, pp. 45–60.

Illustrations

Worked examples

Example 1 — a first encounter with Martin Eichler

Start with the simplest possible case. Write down what Martin Eichler 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 Martin Eichler 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 Martin Eichler 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 Martin Eichler

In research: Martin Eichler 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 Martin Eichler 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: Martin Eichler is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1912 births, 1992 deaths, 20th-century German mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Martin Eichler 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 Martin Eichler in 20 minutes

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

Frequently asked questions

What is Martin Eichler in simple terms?

Martin Maximilian Emil Eichler (29 March 1912 – 7 October 1992) was a German mathematician. Eichler received his Ph.D. from the Martin Luther University of Halle-Wittenberg in 1936.

Why does Martin Eichler 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 Martin Eichler?

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 Martin Eichler.