Let [ ] be a [pfr ]-adic field, and consider the system F = (F1,…,FR) of diagonal equations

[formula here]

with coefficients in [ ]. It is an interesting problem in number theory to determine when such a system possesses a nontrivial [ ]-rational solution. In particular, we define Γ*(k, R, [ ]) to be the smallest natural number such that any system of R equations of degree k in N variables with coefficients in [ ] has a nontrivial [ ]-rational solution provided only that N[ges ]Γ*(k, R, [ ]). For example, when k = 1, ordinary linear algebra tells us that Γ*(1, R, [ ]) = R + 1 for any field [ ]. We also define Γ*(k, R) to be the smallest integer N such that Γ*(k, R, ℚp) [les ] N for all primes p.

Let [ ] be a [pfr ]-adic field, and consider the system F = (F1,…,FR) of diagonal equations

with coefficients in [ ]. It is an interesting problem in number theory to determine when such a system possesses a nontrivial [ ]-rational solution. In particular, we define Γ*(k, R, [ ]) to be the smallest natural number such that any system of R equations of degree k in N variables with coefficients in [ ] has a nontrivial [ ]-rational solution provided only that N[ges ]Γ*(k, R, [ ]). For example, when k = 1, ordinary linear algebra tells us that Γ*(1, R, [ ]) = R + 1 for any field [ ]. We also define Γ*(k, R) to be the smallest integer N such that Γ*(k, R, ℚp) [les ] N for all primes p.

A fundamental aspect of the study of Diophantine equations is that of determining when an equation has a local solution. Artin once conjectured (see the preface to [1]) that if k is a complete, discretely valued field with finite residue class field, then every homogeneous form of degree d in greater than d2 variables whose coefficients are integers of k has a nontrivial zero. In this paper, we consider the case of this conjecture in which k is a p-adic field. Although a counterexample due to Terjanian [16] proved Artin's conjecture false in this situation, Ax and Kochen [2] have shown when [k:Qp] = n is finite, that given d, there exists a number p(d, n) such that Artin's conjecture is true provided that p is larger than p(d, n). Unfortunately, the methods of Ax and Kochen do not lead to explicit estimates for p(d, n). Cohen [5] found a method which determines the possible cardinalities of the residue class fields of all p- adic fields for which Artin's conjecture is false, and Brown [3] has used this to bound p(d, 1), but this bound is so large that one feels that it must be possible to do better. Hence, it is still an interesting problem to obtain estimates on the size of p(d, n).

A fundamental aspect of the study of Diophantine equations is that of determining when an equation has a local solution. Artin once conjectured (see the preface to [1]) that if k is a complete, discretely valued field with finite residue class field, then every homogeneous form of degree d in greater than d2 variables whose coefficients are integers of k has a nontrivial zero. In this paper, we consider the case of this conjecture in which k is a p-adic field. Although a counterexample due to Terjanian [16] proved Artin's conjecture false in this situation, Ax and Kochen [2] have shown when [k:Qp] = n is finite, that given d, there exists a number p(d, n) such that Artin's conjecture is true provided that p is larger than p(d, n). Unfortunately, the methods of Ax and Kochen do not lead to explicit estimates for p(d, n). Cohen [5] found a method which determines the possible cardinalities of the residue class fields of all p-adic fields for which Artin's conjecture is false, and Brown [3] has used this to bound p(d, 1), but this bound is so large that one feels that it must be possible to do better. Hence, it is still an interesting problem to obtain estimates on the size of p(d, n).

Dans cet article, nous nous intéressons aux problèmes de structure galoisienne associés aux couples (E/L, a), où E désigne une courbe elliptique définie sur le corps de nombres L et a un endomorphisme de E. Si E est à multiplication complexe, nous notons k le corps quadratique imaginaire dont l'anneau des entiers décrit End(E) et L une extension de k. Ainsi a est-il selon les cas un entier algébrique de k ou un élément de ℤ.

Dans cet article, nous nous intéressons aux problèmes de structure galoisienne associés aux couples (E/L, a), où E désigne une courbe elliptique définie sur le corps de nombres L et a un endomorphisme de E. Si E est à multiplication complexe, nous notons k le corps quadratique imaginaire dont l'anneau des entiers décrit End(E) et L une extension de k. Ainsi a est-il selon les cas un entier algébrique de k ou un élément de ℤ.

Infinite families of curves are constructed of genus 2 and 3 over Q whose jacobians have high rank over Q. More precisely, if [Escr ] is an elliptic curve with rank at least r over Q, an infinite family of curves are constructed of genus 2 whose jacobians have rank at least r+4 over Q, and, under certain conditions, an infinite family of curves are constructed of genus 3 whose jacobians have rank at least 2r over Q. On specialisation, a family of curves are obtained of genus 2 whose jacobians have rank at least 27 and a family of curves are obtained of genus 3 whose jacobians have rank at least 26; one of these has rank at least 42.

Infinite families of curves are constructed of genus 2 and 3 over Q whose jacobians have high rank over Q. More precisely, if [Escr ] is an elliptic curve with rank at least r over Q, an infinite family of curves are constructed of genus 2 whose jacobians have rank at least r+4 over Q, and, under certain conditions, an infinite family of curves are constructed of genus 3 whose jacobians have rank at least 2r over Q. On specialisation, a family of curves are obtained of genus 2 whose jacobians have rank at least 27 and a family of curves are obtained of genus 3 whose jacobians have rank at least 26; one of these has rank at least 42.

Let [gfr ] be a semisimple Lie algebra over an algebraically closed field [ ] of characteristic zero and G be its adjoint group. The notion of a principal nilpotent pair is a double counterpart of the notion of a regular (= principal) nilpotent element in [gfr ]. Roughly speaking, a principal nilpotent pair e = (e1, e2) consists of two commuting elements in [gfr ] that can independently be contracted to the origin and such that their simultaneous centralizer has the minimal possible dimension, that is, rk[gfr ]. The definition and the basic results are due to V. Ginzburg [3]. He showed that the theory of principal nilpotent pairs yields a refinement of well-known results by B. Kostant on regular nilpotent elements in [gfr ] and has interesting applications to representation theory. In particular, he proved that the number of G-orbits of principal nilpotent pairs is finite and gave a classification for [gfr ] = [sfr ][lfr ]([ ]). Trying to achieve a greater generality, Ginzburg also introduced a wider class of distinguished nilpotent pairs and, again, classified them for [sfr ][lfr ]([ ]). (The precise definitions for all notions related to nilpotent pairs are found in §1.)

Let [gfr ] be a semisimple Lie algebra over an algebraically closed field [ ] of characteristic zero and G be its adjoint group. The notion of a principal nilpotent pair is a double counterpart of the notion of a regular (= principal) nilpotent element in [gfr ]. Roughly speaking, a principal nilpotent pair e = (e1, e2) consists of two commuting elements in [gfr ] that can independently be contracted to the origin and such that their simultaneous centralizer has the minimal possible dimension, that is, rk[gfr ]. The definition and the basic results are due to V. Ginzburg [3]. He showed that the theory of principal nilpotent pairs yields a refinement of well-known results by B. Kostant on regular nilpotent elements in [gfr ] and has interesting applications to representation theory. In particular, he proved that the number of G-orbits of principal nilpotent pairs is finite and gave a classification for [gfr ] = [sfr ][lfr ]([ ]). Trying to achieve a greater generality, Ginzburg also introduced a wider class of distinguished nilpotent pairs and, again, classified them for [sfr ][lfr ]([ ]). (The precise definitions for all notions related to nilpotent pairs are found in §1.)

Let R be a commutative Noetherian ring. Let [Pscr ](R) (respectively, [Iscr ](R)) be the category of all finite R-modules of finite projective (respectively, injective) dimension. Sharp [9] constructed a category equivalence between [Iscr ](R) and [Pscr ](R) for certain Cohen–Macaulay local rings R. Thus many properties about finite modules of finite projective dimension can be connected with those of finite injective dimension through this equivalence.

Let R be a commutative Noetherian ring. Let [Pscr ](R) (respectively, [Iscr ](R)) be the category of all finite R-modules of finite projective (respectively, injective) dimension. Sharp [9] constructed a category equivalence between [Iscr ](R) and [Pscr ](R) for certain Cohen–Macaulay local rings R. Thus many properties about finite modules of finite projective dimension can be connected with those of finite injective dimension through this equivalence.

We consider the Dipper–James q-Schur algebra [Sscr ]q(n, r)k, defined over a field k and with parameter q ≠ 0. An understanding of the representation theory of this algebra is of considerable interest in the representation theory of finite groups of Lie type and quantum groups; see, for example, [6] and [11]. It is known that [Sscr ]q(n, r)k is a semisimple algebra if q is not a root of unity. Much more interesting is the case when [Sscr ]q(n, r)k is not semisimple. Then we have a corresponding decomposition matrix which records the multiplicities of the simple modules in certain ‘standard modules’ (or ‘Weyl modules’). A major unsolved problem is the explicit determination of these decomposition matrices.

We consider the Dipper–James q-Schur algebra [Sscr ]q(n, r)k, defined over a field k and with parameter q ≠ 0. An understanding of the representation theory of this algebra is of considerable interest in the representation theory of finite groups of Lie type and quantum groups; see, for example, [6] and [11]. It is known that [Sscr ]q(n, r)k is a semisimple algebra if q is not a root of unity. Much more interesting is the case when [Sscr ]q(n, r)k is not semisimple. Then we have a corresponding decomposition matrix which records the multiplicities of the simple modules in certain ‘standard modules’ (or ‘Weyl modules’). A major unsolved problem is the explicit determination of these decomposition matrices.

Symplectic groups are well known as the groups of isometries of a vector space with a non-singular bilinear alternating form. These notions can be extended by replacing the vector space by a module over a ring R, but if R is non-commutative, it will also have to have an involution. We shall here be concerned with symplectic groups over free associative algebras (with a suitably defined involution). It is known that the general linear group GLn over the free algebra is generated by the set of all elementary and diagonal matrices (see [1, Proposition 2.8.2, p. 124]). Our object here is to prove that the symplectic group over the free algebra is generated by the set of all elementary symplectic matrices. For the lowest order this result was obtained in [4]; the general case is rather more involved. It makes use of the notion of transduction (see [1, 2.4, p. 105]). When there is only a single variable over a field, the free algebra reduces to the polynomial ring and the weak algorithm becomes the familiar division algorithm. In that case the result has been proved in [3, Anhang 5].

Symplectic groups are well known as the groups of isometries of a vector space with a non-singular bilinear alternating form. These notions can be extended by replacing the vector space by a module over a ring R, but if R is non-commutative, it will also have to have an involution. We shall here be concerned with symplectic groups over free associative algebras (with a suitably defined involution). It is known that the general linear group GLn over the free algebra is generated by the set of all elementary and diagonal matrices (see [1, Proposition 2.8.2, p. 124]). Our object here is to prove that the symplectic group over the free algebra is generated by the set of all elementary symplectic matrices. For the lowest order this result was obtained in [4]; the general case is rather more involved. It makes use of the notion of transduction (see [1, 2.4, p. 105]). When there is only a single variable over a field, the free algebra reduces to the polynomial ring and the weak algorithm becomes the familiar division algorithm. In that case the result has been proved in [3, Anhang 5].

Let F be a non-Archimedean local field and G be the group of F-points of a connected reductive group defined over F. Let M be an F-Levi subgroup of G and P = MN be a parabolic subgroup with Levi decomposition P = MN. Jacquet, or truncated, restriction gives a functor from the category of smooth representations of G to that of M. The main result describes this functor in terms of homomorphisms and localizations of Hecke algebras attached to certain compact open subgroups of G and M. This leads to new and straightforward proofs of some fundamental results. The first computes the smooth dual of a Jacquet module of a smooth representation of G, generalizing the corresponding result for admissible representations due to Harish-Chandra and Casselman. The second identifies the co-adjoint of the Jacquet functor relative to P as the induction functor relative to the M-opposite of P, an unpublished result of J.-N. Bernstein.

Let F be a non-Archimedean local field and G be the group of F-points of a connected reductive group defined over F. Let M be an F-Levi subgroup of G and P = MN be a parabolic subgroup with Levi decomposition P = MN. Jacquet, or truncated, restriction gives a functor from the category of smooth representations of G to that of M. The main result describes this functor in terms of homomorphisms and localizations of Hecke algebras attached to certain compact open subgroups of G and M. This leads to new and straightforward proofs of some fundamental results. The first computes the smooth dual of a Jacquet module of a smooth representation of G, generalizing the corresponding result for admissible representations due to Harish-Chandra and Casselman. The second identifies the co-adjoint of the Jacquet functor relative to P as the induction functor relative to the M-opposite of P, an unpublished result of J.-N. Bernstein.

The paper obtains the limiting behaviour of the expectations of the zeros of holomorphic sections of the canonical line bundles over a tower of covers of a compact complex manifold.

The paper obtains the limiting behaviour of the expectations of the zeros of holomorphic sections of the canonical line bundles over a tower of covers of a compact complex manifold.