Hervé Jacquet
Encyclopedia
Hervé Jacquet is a French American
mathematician
born in France
in 1939, working in automorphic forms. He is considered one of the founders of the theory of automorphic representations and their associated L-functions, and his results play a central role in modern number theory
.
in 1959 and obtained his doctorat d'état under the direction of Roger Godement
in 1967. He held academic positions at the Centre National de la Recherche Scientifique
(1963–1969), the Institute for Advanced Study
in Princeton
(1967–1969), the University of Maryland at College Park (1969–1970), the Graduate Center
of the City University of New York
(1970–1974), and became a Professor at Columbia University
in 1974, becoming Professor Emeritus in 2007. He was elected corresponding member of the Académie des Sciences in 1980.
on
was an eclipsing event in the history of number theory. It presented a representation theory
of automorphic forms and their associated L−functions for the general linear group
, establishing among other things the Jacquet-Langlands correspondence which explains very precisely how automorphic forms for 
relate to those for quaternion algebras. Equally important was the book by Roger Godement and Hervé Jacquet, which defined, for the first time, the standard L-functions attached to automorphic representations of 
, now called Godement-Jacquet L-functions, and proved their basic, oft-used analytic properties. The papers with Shalika and the papers with Piatetski-Shapiro and Shalika pertain to L-functions of pairs, called the Rankin-Selberg L-functions, attached to representations of 
and 
, and the so called converse theorem, which are crucial to our understanding of automorphic forms. A basic ingredient of this effort was an elaboration of properties of Whittaker models and functions, which Jacquet had made contributions to since his thesis. The papers with Shalika also established the uniqueness of isobaric decompositions of automorphic forms on 
, thus providing evidence for certain conjectures of Langlands. In the mid-eighties, Jacquet forayed into a new territory in the field and created the relative trace formula in representation theory, an important tool in modern number theory, which vastly generalizes the Kuznetsov and Petersson formulae
from the classical setup. While the usual Selberg trace formula
, as well as its generalizations due to Arthur, consists in developing an expression for the integral of the kernel over the diagonal, the relative version integrates the kernel over other appropriate subgroups.
French American
French Americans or Franco-Americans are Americans of French or French Canadian descent. About 11.8 million U.S. residents are of this descent, and about 1.6 million speak French at home.An additional 450,000 U.S...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
born in France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
in 1939, working in automorphic forms. He is considered one of the founders of the theory of automorphic representations and their associated L-functions, and his results play a central role in modern number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
Career
Jacquet entered the École Normale SupérieureÉcole Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
in 1959 and obtained his doctorat d'état under the direction of Roger Godement
Roger Godement
Roger Godement is a French mathematician, known for his work in functional analysis, and also his expository books.He started as a student at the École normale supérieure in 1940, where he became a student of Henri Cartan...
in 1967. He held academic positions at the Centre National de la Recherche Scientifique
Centre national de la recherche scientifique
The National Center of Scientific Research is the largest governmental research organization in France and the largest fundamental science agency in Europe....
(1963–1969), the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
in Princeton
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
(1967–1969), the University of Maryland at College Park (1969–1970), the Graduate Center
CUNY Graduate Center
The Graduate Center of the City University of New York brings together graduate education, advanced research, and public programming to midtown Manhattan hosting 4,600 students, 33 doctoral programs, 7 master's programs, and 30 research centers and institutes...
of the City University of New York
City University of New York
The City University of New York is the public university system of New York City, with its administrative offices in Yorkville in Manhattan. It is the largest urban university in the United States, consisting of 23 institutions: 11 senior colleges, six community colleges, the William E...
(1970–1974), and became a Professor at Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
in 1974, becoming Professor Emeritus in 2007. He was elected corresponding member of the Académie des Sciences in 1980.
Mathematical work
The book by Hervé Jacquet and Robert LanglandsRobert Langlands
Robert Phelan Langlands is a mathematician, best known as the founder of the Langlands program. He is an emeritus professor at the Institute for Advanced Study...
on
was an eclipsing event in the history of number theory. It presented a representation theoryRepresentation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
of automorphic forms and their associated L−functions for the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
, establishing among other things the Jacquet-Langlands correspondence which explains very precisely how automorphic forms for relate to those for quaternion algebras. Equally important was the book by Roger Godement and Hervé Jacquet, which defined, for the first time, the standard L-functions attached to automorphic representations of , now called Godement-Jacquet L-functions, and proved their basic, oft-used analytic properties. The papers with Shalika and the papers with Piatetski-Shapiro and Shalika pertain to L-functions of pairs, called the Rankin-Selberg L-functions, attached to representations of and , and the so called converse theorem, which are crucial to our understanding of automorphic forms. A basic ingredient of this effort was an elaboration of properties of Whittaker models and functions, which Jacquet had made contributions to since his thesis. The papers with Shalika also established the uniqueness of isobaric decompositions of automorphic forms on , thus providing evidence for certain conjectures of Langlands. In the mid-eighties, Jacquet forayed into a new territory in the field and created the relative trace formula in representation theory, an important tool in modern number theory, which vastly generalizes the Kuznetsov and Petersson formulaePetersson trace formula
In analytic number theory, the Petersson trace formula is a kind of orthogonality relation between coefficients of a holomorphic modular form. It is a specialization of the more general Kuznetsov trace formula....
from the classical setup. While the usual Selberg trace formula
Selberg trace formula
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of G on the space L2 of square-integrable functions, where G is a Lie group and Γ a cofinite discrete group...
, as well as its generalizations due to Arthur, consists in developing an expression for the integral of the kernel over the diagonal, the relative version integrates the kernel over other appropriate subgroups.