The u-invariant of p-adic function fields

We show that a system of r quadratic forms over a 𝔭-adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax–Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630] requires the characteristic to be large in terms of the degree of the field over ℚp. The proofs use a 𝔭-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry.

We obtain new results on linear spaces on the intersection of two quadratic forms defined over a non-dyadic p-adic field \({{\Bbb Q}_p}\). One of our main tools is a recent result of Parimala and Suresh on isotropy of quadratic forms over functions fields over \({{\Bbb Q}_p}\). As a corollary we also get new bounds for the number of variables necessary to always find a non-trivial p-adic zero of a system of quadratic forms.

A system of quadratic forms is associated to every generalised quadratic form over a division algebra with involution of the first kind in characteristic two. It is shown that this system determines the isotropy behaviour and the isometry class of generalised quadratic forms. An application of this construction to the Witt index of generalised quadratic forms is also given.

We investigate the ramification of embeddings of local lattices or quadratic forms over the ring of -adic numbers. It is proved that every primitive embedding decomposes uniquely into an orthogonal sum of minimal indecomposable embeddings, and all such embeddings are constructed for -elementary lattices. Ramification theory enables us to find the number of orbits of representations for forms and, in particular, for numbers by other quadratic forms over , and to calculate the local multipliers in the weight formula for representations of a form by a genus of quadratic forms.

This book grew out of seminar held at the University of Paris 7 during the academic year 1985-86. The aim of the seminar was to give an exposition of the theory of the Metaplectic Representation (or Weil Representation) over a p-adic field. The book begins with the algebraic theory of symplectic and unitary spaces and a general presentation of metaplectic representations. It continues with exposes on the recent work of Kudla (Howe Conjecture and induction) and of Howe (proof of the conjecture in the unramified case, representations of low rank). These lecture notes contain several original results. The book assumes some background in geometry and arithmetic (symplectic forms, quadratic forms, reductive groups, etc.), and with the theory of reductive groups over a p-adic field. It is written for researchers in p-adic reductive groups, including number theorists with an interest in the role played by the Weil Representation and -series in the theory of automorphic forms.

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).

We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ℚ p , enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of T p : they are simple and the nonzero ones lie in the unit circle of the unramified extension of ℚ p , (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of T p in the field ℚ p , (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of T n is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.

Hilbertʼs Tenth Problem for rational function fields over p-adic fields

In this note (cf. sections 3, 4) I shall give an axiomatization of those fields (of characteristic ≠ 2) which have a theory of quadratic forms like the -adic numbers or like the real numbers. This leads then, for instance, to a generalization of the well-known theorems on -adic forms to a wider class of fields, including non-local ones. The main purpose of the exercise is, however, to separate out the roles of the arithmetic in the underlying field, on the one hand, which solely enters into the verification of the axioms, and of the ordinary algebra of quadratic forms on the other hand. The resulting clarification of the structure of the theory is of interest even in the known -adic case.

A celebrated theorem of Wedderburn says that a finite associative division ring is a finite (commutative) field. An elegant proof of this, due to E. Artin [3, p. 72], reduces the problem to showing the reduced (or generic) norm has a nontrivial zero, and then applies the theorem (conjectured by Artin and proved by Chevalley) that a polynomial with coefficients in a finite field and zero constant term has a nontrivial zero in that field if its degree is less than the number of variables. The Wedderburn theorem was generalized by A. A. Albert to finite strictly power-associative division rings of characteristic # 2 in [1, p. 301 ] and [2, p. i1 ] (see also [5 ]). The proof reduced the powerassociative case to the associative case by using the properties of Jordan division algebras. It is the purpose of this note to show that Artin's method carries over directly to the power-associative case using the generic norm introduced by N. Jacobson [4]. By avoiding Jordan algebras one is able to extend Albert's result to the characteristic 2 case. The usual definition of a division ring requires that left and right multiplications by a nonzero element be bijective. If the ring is finite it is necessarily an algebra, over a finite field 4, which is algebraic (even finite-dimensional) and without zero divisors. Also, as we shall see later, the algebra necessarily has a unit element. We wish to consider something slightly more general. For our purposes we say an algebraic power-associative algebra is a division algebra if it is unital and no element xO 0 is a zero divisor in the subalgebra (D[x] it generates (since 4<[x] is finite-dimensional, this is equivalent to saying x is invertible in b [x]). Our conditions are less restrictive, since the Jordan algebra of a quadratic form can be a division algebra in our sense with zero divisors. With this definition we have

We review some recent results on the mathematical foundations of a quantum theory over a scalar field that is a quadratic extension of the non-Archimedean field of [Formula: see text]-adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, [Formula: see text]-adic string, Class. Quantum Grav. 4 (1987) L83–L87] — that space, below a suitably small scale, does not behave as a continuum and, accordingly, should be modeled as a totally disconnected metrizable topological space, ruled by a metric satisfying the strong triangle inequality. The first step of our construction is a suitable definition of a [Formula: see text]-adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a [Formula: see text]-adic Hilbert space — we consider an algebraic definition of physical states in [Formula: see text]-adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a [Formula: see text]-adic counterpart of the POVMs associated with a standard quantum system over the complex numbers. Interestingly, it turns out that the typical convex geometry of the space of states of a standard quantum system is replaced, in the [Formula: see text]-adic setting, with an affine geometry; therefore, a symmetry transformation of a [Formula: see text]-adic quantum system may be defined as a map preserving this affine geometry. We argue that, as a consequence, the group of all symmetry transformations of a [Formula: see text]-adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers.

LetQ i, i=1, …,t, be real nondegenerate indefinite quadratic forms ind variables. We investigate under what conditions the closure of the set $$\left\{ {\left( {Q_1 \left( {\bar x} \right),...,Q_t \left( {\bar x} \right)} \right):\bar x \in \mathbb{Z}^d - \left\{ {\bar 0} \right\}} \right\}$$ contains (0, …, 0). As a corollary, we deduce several results on the magnitude of the set Δ ofg ∈ GL(d, ℝ) such that the closure of the set $$\left\{ {\left( {Q_1 \left( {g\bar x} \right),...,Q_t \left( {g\bar x} \right)} \right):\bar x \in \mathbb{Z}^d - \left\{ {\bar 0} \right\}} \right\}$$ contains (0, …, 0). Special cases are described when, depending on the mutual position of the hypersurfaces {Q i=0},i=1, …,t, the set Δ has full Haar measure or measure zero and Hausdorff dimensiond 2−(d−2)/2.

We study the parabolically induced complex representations of the unitary group in 5 variables,U(5), defined over a p-adic field.Let F be a p-adic field. Let E:F be a field extension of degree two. U(5) has three proper standard Levi subgroups, the minimal Levi subgroup M 0 ≅E * ×E * ×E 1 and the two maximal Levi subgroups M 1 ≅GL(2,E)×E 1 and M 2 ≅E * ×U(3).We consider representations induced from M 0 , representations induced from non-cuspidal, not fully-induced representations of M 1 and M 2 and representations induced from cuspidal representations of M 1 .We determine the points and lines of reducibility and the irreducible subquotients of these representations. Further we describe - except several particular cases - the unitary dual in terms of Langlands quotients.

On the scaled trace forms and the transfer of a number field extension

We consider the affine real variety \(V=F^{-1}(0)\) in \(\mathbf{R}^n\) where \(F\) is given by two quadratic forms. We announce several recent results, obtained in collaboration with various coauthors, about the topology of the generic \(V\) and of its deformations \(V_t=F^{-1}(t)\), especially those that are smooth. This work started many years ago [16] with the case where the two quadratic forms are simultaneously diagonalizable and with some restrictions. Now we have the result without restrictions and including the non-diagonalizable case so we have a complete topological description of \(V\) in all generic cases. We have also results about the topology of the intersection of \(V\) with a half-space \(\{x_i\ge 0\}\) and the topology of the various deformations \(V_t\). However, in these cases the results are for the moment not as complete as those obtained for the topology of \(V\). Proofs are referred to published articles or only outlined when complete proofs are still to appear.