Martin Eichler Explained

Martin Eichler
Birth Date:	29 March 1912
Nationality:	German
Field:	Number theory and Mathematics

Martin Maximilian Emil Eichler (29 March 1912  - 7 October 1992) was a German number theorist.

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.^{[1] [2]}

Selected publications

• Quadratische Formen und orthogonale Gruppen, Springer 1952,^[3] 1974
• Book: Lectures on Modular Correspondences. Tata Institute. 1955.

  pbk, 169 pages

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• 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); Book: 2006 edition. 9783540368694. Eichler. M.. 15 November 2006.
• with Don Zagier: The Theory of Jacobi forms, Birkhäuser 1985; Book: 2013 edition. 9781468491623. Eichler. Martin. Zagier. Don. 14 December 2013.
• Über die Einheiten der Divisionsalgebren, Mathem. Annalen 1937
• Neuere Ergebnisse der Theorie der einfachen Algebren, Jahresbericht DMV 1937
• Allgemeine Integration linearer partieller Differentialgleichungen von elliptischem Typ bei zwei Grundvariablen, Abh. Math. Sem. Univ. Hamburg 15 (1947), 179–210.
• On the differential equation u_xx + u_yy + N(x)u = 0, Trans. Amer. Math. Soc. 65 (1949), 259–278
• Zur Algebra der orthogonalen Gruppen Mathem. Zeitschrift 1950
• Zahlentheorie der Quaternionenalgebren, Crelle J. vol. 195, 1955, with errata https://web.archive.org/web/20151222003559/http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=260614
• 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
• 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, Mathem. Annalen 1967, (Eichler–Shimura theory)
• Eichler Eine Spurformel von Korrespondenzen von algebraischen Funktionenkörpern mit sich selber, Inv. Math. vol. 2, 1967 with corrections http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=166721
• 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 CBH theorem

External links

• • 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.

Notes and References

1. Web site: Modular Arithmetic: Driven by Inherent Beauty and Human Curiosity . Institute for Advanced Study . Richard Taylor . 2012 . 2014-12-22 .
2. Book: 978-0465050741. Love and Math: The Heart of Hidden Reality. registration. Edward Frenkel. Edward Frenkel. 90.
3. Review: Quadratische Formen und orthogonale Gruppen by Martin Eichler. Whaples, G.. Bull. Amer. Math. Soc.. 61. 1955. 589–593. 10.1090/S0002-9904-1955-10001-8. free.