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Hervé Jacquet is a French mathematician known for his significant contributions to the theory of automorphic forms and the representation theory of reductive groups. He is particularly noted for the Jacquet–Langlands correspondence (developed jointly with Robert Langlands) and the concept of the “Jacquet module,” both of which have played important roles in modern number theory.

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R. P. Langlands 2
Let H be the direct product of the n upper half planes, and let be the connected component of the identity of the group of all analytic automorphisms of Hn. … Let H be the direct product of the n upper half planes, and let be the connected component of the identity of the group of all analytic automorphisms of Hn. is the direct product of n subgroups G1, G2, . . .* Gq each of which is isomorphic to the group of all analytic automorphisms of the upper half plane H. Consider a discrete subgroup of such that the factor space 17\G is of finite measure. (The notations Hn, or will keep, unless otherwise stated, these meanings throughout this paper.) For the study of the groups of this type, it is important to investigate the case where is an irreducible subgroup of in the following sense. A discrete subgroup of is said to be irreducible if is not commensurable' with any direct product F' x F, where F' and F are respectively discrete subgroups of the partial products G' and G of =G, x G2 x ... x Ga with = G' x G, G' I {1}, G # {1}. The main purpose of this paper was originally to calculate the dimension of the space of cusp foruis for an irreducible group by means of Selberg's trace formula; however, for the sake of completeness, and in view of the fact that no proof has been published as yet for the results stated by Pyatetzki-Shapiro in [6], it has been found desirable to prove these results here, following the ideas indicated by Pyatetzki-Shapiro himself. Therefore, no new results will be found in our ?? 1-3 except for some supplementary results such as Theorem 1, and for Theorems 6 and 7, which are proved under an additional condition on the fundamental domain of (Assumption (F) in No. 11, ?3). Theorems 2, 3, 4, 5 are the restatements of the results of Pyatetzki-Shapiro; but, for the sake of simplicity, we state here the latter three theorems only for irreducible case. For these results, Pyatetzki-Shapiro has indicated in [6] only a sketch of proof for Theorem 3 as well as the implications between them. However, our proofs will be probably more or less the same as the ones he has. In ??4-5, we shall calculate the dimension of the space of cusp forms for an irreducible group under the assumption (F). The main result is
Since Hecke [13] had given a general theory of constructing Dirichlet series with Euler-product and functional equation out of elliptic modular forms of any level, several authors considered its generalization … Since Hecke [13] had given a general theory of constructing Dirichlet series with Euler-product and functional equation out of elliptic modular forms of any level, several authors considered its generalization for other types of automorphic forms. In the case of the Hilbert modular group of level one, Herrmann [14] succeeded in this problem; he has shown the necessity of considering not only the product of the upper half-planes but also the domain r consisting of the points (zl, * * , Zr) of the r-dimensional complex vector space Cr such that Im(z,) # 0, * ., Im(zr) # 0, and h distinct discontinuous groups commensurable to each other, h being the class number of the totally real number field in the problem. On the other hand, the unit-group of an order in an indefinite quaternion algebra over the rational number field Q yields a fuchsian group. In this case, Eichler [6] defined Hecke's operators as representations of algebraic correspondences, called modular correspondences, and proved a formula for the trace of the operators. The trace-formula was proved also by Selberg [22] in a more general formulation. Recently, Godement [9] has given a theory of zeta-functions attached to division algebras; namely, he has shown the possibility of applying the adele-idele method of Iwasawa-Tate [15, 28] to automorphic functions and forms with respect to the unit-group of a division algebra over Q. The case of non-holomorphic automorphic functions of this type has been investigated by Tamagawa [27]. In [25] I have treated cusp-forms with respect to the unit-group of an indefinite quaternion algebra over Q. Now, the purpose of Part I of the present paper is to develop an analogous theory for automorphic forms on the domain ; mentioned above with respect to the unit-groups of an indefinite quaternion algebra. Let F be a totally real algebraic number field of degree t, and D be a quaternion algebra over F. Denote by r the number of infinite prime spots of F unramified in D, and suppose that r > 0. Let D,1 and D-2 denote the product of r copies of the total matric algebra M2(R) over the real number field R and the product of t r copies of the division ring K of real quaternions, respectively. Then, D( Q R is isomorphic to D,1 x D12. Let o be a maximal order in D and I' be the group of units in o. Let y, and 72 237
The present paper is concerned with some equalities between zeta functions of quaternion algebras introduced in Godement [6], Shimura [11], Tamagawa [13]. Let A be a quaternion algebra over a … The present paper is concerned with some equalities between zeta functions of quaternion algebras introduced in Godement [6], Shimura [11], Tamagawa [13]. Let A be a quaternion algebra over a totally real algebraic number field 1? of degree m, and let D be an order in A; let S be the idele group of A, and U the group of units in S with respect to D; let p be a representation of ( a/f)*, f being an integral two-sided rD-ideal, and let ki (1 < i < m) be non-negative integers. These being given, we can speak of a space of automorphic functions associated with (p, {ki}) (cf. ? 2.2 and ? 2.5) and of a repre-sentation Z of the Hecke ring 9R(U, @) in this space. Let T(q) be the sum of all integral elements in 9R(U, @ of norm q and let C(s) be the Dirichlet. series defined by
In this paper, we will develop a theory of C-functions with characters in a division algebra. The ordinary C-function of a division algebra was introduced by K. Hey [4], and … In this paper, we will develop a theory of C-functions with characters in a division algebra. The ordinary C-function of a division algebra was introduced by K. Hey [4], and generalized by M. Eichler [1] to L-functions with abelian characters. The first attempt to generalize these theories to C-functions with non-abelian characters is due to H. Maass [5]. Later, R. Godement [2] gave a method to get the most general formulations on these matters. In this note, we will define a type of C-functions of a division algebra over an algebraic number field which are included in Godement's work as a special case, and for which one can develop the theory of Euler products. The latter theory has its own meaning as an application of the theory of spherical functions on p-adic algebraic groups. Here we have a generalization of Hecke's theory of so-called Heckeoperators. (Theorem 1-6). One can expect that there exist similar theories for other simple algebraic groups defined over p-adic number fields, and that there will be applications of these theories to non-commutative number theory. The author wishes to express his thanks to Professor Shimura for some valuable suggestions about the first part of this paper.
Let G be a connected semisimple Lie group and K a maximal compact subgroup of G.We shall show in this paper that G has a discrete series (see [4 (d), … Let G be a connected semisimple Lie group and K a maximal compact subgroup of G.We shall show in this paper that G has a discrete series (see [4 (d), w 5]) ff and only if it has a compact Cartan subgroup B. Let Ea denote the set of all equivalence classes of irreducible unitary representations of G, wMch are square-integrable.For any ~o E ~a, let | denote the character, X~ the infinitesimal character and d(r the formal degree (see [4 (d), w 3]) of co.Then it is known [4 (d), w 5] that the distribution
In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these … In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these investigations is a general relation which can be considered as a generalization of the so-called Poisson summation formula (in one or more dimensions). This relation we here refer to as the trace-formula.
Deux Th\'eor\'emes d'Arithm\'etique Claude CHEVALLEY $1\backslash ousT$ nous proposons de d\'emontrer deux th\'eor\'emes de la th\'eorie des corps de nombres alg\'ebriques, qui n'ont l'un avec l'autre d'aucun autre rapport que … Deux Th\'eor\'emes d'Arithm\'etique Claude CHEVALLEY $1\backslash ousT$ nous proposons de d\'emontrer deux th\'eor\'emes de la th\'eorie des corps de nombres alg\'ebriques, qui n'ont l'un avec l'autre d'aucun autre rapport que celui d'\^etre tous deux utilis\'es par A. Weil dans son m\'emoire " Sur la th\'eorie du corps de classes '', qui parait dans le m\'eme num\'ero de ce journal.I Soit $K$ un corps de nombres alg\'ebriques de degr\'e fini sur le corps des lationnels.Un $g\check{r}oupe$ multiplicatif $E$ d'\'el ments $\neq 0$ de A sera dit $a\backslash $ engendrement fini si $E$ poss\'ede un ensemble fini de g\'en\'erateurs.C'est ainsi que le groupe des unit $s$ de $K$ est un groupe \'a engendrement fini; plus g\'en\'eralement, si on se donne un nombre fini $d^{)}id$ ' aux premiers $\mathfrak{p}1'\cdots$ , $\mathfrak{p}_{h}$ de $K$ , le $gro_{\llcorner}^{\prime}|p_{\vee}^{a}$ des \'el\'ements $x\neq 0$ de $Kte$ ls que l'id\'eal frac- tionnairc principal engendr\'e par $x$ soit un produit de puissances $(d' ex-$ posants positifs, nuls ou n\'egatifs) de $\mathfrak{p}_{1},$ $\cdots,$ $\mathfrak{p}_{h}$ est un groupe \'a engendre- ment fini, comme il est bien connu.Nous nous $p^{lO}posons$ de $d_{\vee}^{1}\prime mo\iota 1trer$ le th\'eor\'eme suivant: Th or\'eme 1. Soienl If un corps de nombres $al_{3^{\prime r}}\ovalbox{\tt\small REJECT}\ell^{\prime}briques$ de $deg/\acute{e}fni$ , et $E$ un $ sous-gronp\iota$ a $engendrem\iota^{J}/\iota t$ fini $du$ groupe $multiplica[lf$ des \'el\'emenls $\neq 0$ de K. Soit $m$ un en.$ier>0$ , et soit $b$ un entier rationnel quelconque.$1l$ existe a $r_{ors}$ un $e$ ntier ralionnel $a$ , premier \'a $b$ , qui jouit de la propri\'et\'e suivante: tout \'el\'e $ne'\iota tx$ de $E$ qui est $\equiv 1(mod a)$ est puissance m-i\'eme d'un $\sqrt[\prime]{}\acute{e}me'\iota t$ de $E$ .La condition $\cdot x\equiv 1(mod a)$ doit s'interpr\^eter au sens des congruences multiplicatives de H. Hasse; elle signifie que $x-1$ est de la $fo\iota$ me $ay_{\sim}\alpha^{-1}$ o\'u $y$ et $\alpha$ sont des entiers de $K$ et $\sim\sigma$ est premier \'a $a$ dans l'anneau des entiers de $K$ .