Martin Maximilian Emil Eichler (29 March 1912 – 7 October 1992) was a German
number theorist
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
.
Eichler received his Ph.D. from the
Martin Luther University
Martin Luther University of Halle-Wittenberg (german: Martin-Luther-Universität Halle-Wittenberg), also referred to as MLU, is a public, research-oriented university in the cities of Halle and Wittenberg and the largest and oldest university in ...
of Halle-Wittenberg in 1936.
Eichler and
Goro Shimura
was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multip ...
developed a method to construct
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s from certain
modular forms
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
. 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
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have bee ...
.
Selected publications
* ''Quadratische Formen und orthogonale Gruppen'', Springer 1952,
1974
*
* ''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
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It ...
);
* with
Don Zagier
Don Bernard Zagier (born 29 June 1951) is an American-German mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany. He was a professor at the ''Col ...
: ''The Theory of Jacobi forms'', Birkhäuser 1985;
''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 errat
*''Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenz-Zetafunktion'', Archiv Math. vol. 5, 1954, pp. 355–366 (
Ramanujan–Petersson conjecture In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients of the cusp form of weight
:\Delta(z)= \sum_\tau(n)q^n=q\prod_\left (1-q^n \right)^ = q-24q^2+252q^3- 1472q^4 + 4830q^5- ...
)
''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, 1967with correction
* ''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
In number theory, the Eichler–Shimura congruence relation expresses the local ''L''-function of a modular curve at a prime ''p'' in terms of the eigenvalues of Hecke operators. It was introduced by and generalized by . Roughly speaking, it say ...
*
Eichler–Shimura isomorphism
In mathematics, Eichler cohomology (also called parabolic cohomology or cuspidal cohomology) is a cohomology theory for Fuchsian groups, introduced by , that is a variation of group cohomology analogous to the image of the cohomology with compact ...
*
Eichler cohomology Several people are named Eichler:
* August W. Eichler (1839–1887), German botanist
* Caroline Eichler (1808/9–1843), German inventor, first woman to be awarded a patent (for her leg prosthesis)
* Eunice Eichler (1932–2017), New Zealand Salva ...
*
Eichler order In mathematics, an Eichler order, named after Martin Eichler, is an order of a quaternion algebra that is the intersection of two maximal order
In mathematics, an order in the sense of ring theory is a subring \mathcal of a ring A, such that
#' ...
*
Eichler's proof of the CBH theorem
References
External links
*
* Martin Kneser
''Martin Eichler (1912-1992)'' Acta Arithmetica
''Acta Arithmetica'' is a scientific journal of mathematics publishing papers on number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers a ...
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.
1912 births
1992 deaths
20th-century German mathematicians
Number theorists
Academic staff of the University of Münster
{{Germany-mathematician-stub