On hermitian trace forms over hilbertian fields

The structure of analytic τ-sheaves

LetK~$\tilde K$be a fixed algebraic closure of an infinite fieldK. We consider an absolutely integral curveΓinPKn$\mathbb{P}_{K}^{n}$withn≥ 2. The curveΓK~$\it\Gamma_{\tilde{K}}$should have only finitely many inflection points, finitely many double tangents, and there exists no point inPK~n$\mathbb{P}_{\tilde{K}}^{n}$through which infinitely many tangents toΓK~$\it\Gamma_{\tilde{K}}$go. In addition there exists a prime numberqsuch thatΓK~$\it\Gamma_{\tilde{K}}$has a cusp of multiplicityqand the multiplicities of all other points ofΓK~$\it\Gamma_{\tilde{K}}$are at mostq. Under these assumptions, we construct a non-empty Zariski-open subsetOofPK~n$\mathbb{P}_{\tilde{K}}^{n}$such that ifn≥ 3, the projection from each pointo∈O(K) birationally mapsΓonto an absolutely integral curveΓ′inPKn−1$\mathbb{P}_{K}^{n-1}$with the same properties asΓ(keepingqunchanged). Ifn= 2, then the projection from eacho∈O(K) mapsΓontoPK1$\mathbb{P}_{K}^{1}$and leads to astabilizingelementtof the function fieldFofΓoverK. The latter means thatF/K(t) is a finite separable extension whose Galois closureF^${\hat F}$is regular overK.

We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields and their behavior under tame base change. For a semiabelian variety, this behavior is governed by a finite sequence of (a priori) real numbers between 0 and 1, called jumps. The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions and generalize Raynaud’s description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence that decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.

The generalized Rédei-matrix for function fields

Some enumeration results on trace forms of low radical over finite fields of characteristic two

A 3-fold Pfister form is associated to every involution of the second kind on a central simple algebra of degree 3. This quadratic form is associated to the restriction of the reduced trace quadratic form to the space of symmetric elements; it is shown to classify involutions up to conjugation. Subfields with dihedral Galois group in central simple algebras of arbitrary odd degree with involution of the second kind are investigated. A complete set of cohomological invariants for algebras of degree 3 with involution of the second kind is given.

We defined the quadratic Hermitian forms on a space of even dimension over a finite field; they are constructed from classical Hermitian forms. We give the exact value of exponential sums associated to the quadratic Hermitian forms. On the other hand, we define with the space of quadratic Hermitian forms a new family of subcodes of the Reed-Muller code of order two, and we give the value of the parameters of these codes and of the associated trace codes.

We study the parameters of codes spanned by quadratic and Hermitian forms in projective space. The resulting codes have high minimum distance and few distinct weights. An example is a new ternary [91.9,54] code.

Given an Azumaya algebra with involution $$(A,\sigma )$$ over a commutative ring R and some auxiliary data, we construct an 8-periodic chain complex involving the Witt groups of $$(A,\sigma )$$ and other algebras with involution, and prove it is exact when R is semilocal. When R is a field, this recovers an 8-periodic exact sequence of Witt groups of Grenier-Boley and Mahmoudi, which in turn generalizes exact sequences of Parimala–Sridharan–Suresh and Lewis. We apply this result in several ways: We establish the Grothendieck–Serre conjecture on principal homogeneous bundles and the local purity conjecture for certain outer forms of $${\mathbf {GL}}_n$$ and $${\mathbf {Sp}}_{2n}$$ , provided some assumptions on R. We show that a 1-hermitian form over a quadratic étale or quaternion Azumaya algebra over a semilocal ring R is isotropic if and only if its trace (a quadratic form over R) is isotropic, generalizing a result of Jacobson. We also apply it to characterize the kernel of the restriction map $$W(R)\rightarrow W(S)$$ when R is a (non-semilocal) 2-dimensional regular domain and S is a quadratic étale R-algebra, generalizing a theorem of Pfister. In the process, we establish many fundamental results concerning Azumaya algebras with involution and hermitian forms over them.

Quadratic forms and Hermitian forms

For each positive integer n n , let g Z ( n ) g_\mathbb {Z}(n) be the smallest integer such that if an integral quadratic form in n n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z ( n ) g_\mathbb {Z}(n) squares of integral linear forms. We show that every positive definite integral quadratic form is equivalent to what we call a balanced Hermite–Korkin–Zolotarev-reduced form and use it to show that the growth of g Z ( n ) g_\mathbb {Z}(n) is at most an exponential of n \sqrt {n} . Our result improves the best known upper bound on g Z ( n ) g_\mathbb {Z}(n) which is on the order of an exponential of n n . We also define an analogous number g O ∗ ( n ) g_{\mathcal O}^*(n) for writing Hermitian forms over the ring of integers O \mathcal O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1 1 , we show that the growth of g O ∗ ( n ) g_{\mathcal O}^*(n) is at most an exponential of n \sqrt {n} . We also improve on results of both Conway and Sloane and Kim and Oh on s s -integrable lattices.

In these lectures we give an introduction to the reduction theory of binary forms starting with quadratic forms with real coefficients, Hermitian forms, and then define the Julia quadratic for any degree n binary form. A survey of a reduction algorithm over ${\mathbb{Z}}$ is described based on recent work of Cremona and Stoll.

The Kobayashi indicatrix (infinitesimal unit ball) of a domain in IE n is known to be a biholomorphic invariant. In particular, if a domain is biholomorphic to a ball, then the indicatrix is the ball. Until the recent deep results of Lempert [4], it was not known to what extent the indicatrix characterizes the domain. Sibony had shown earlier that the indicatrix of any pseudoconvex circular domain is the domain itself [10]; hence the indicatrix determines such domains up to biholomorphism. This is not true in general. Lempert has shown that, for smoothly bounded strictly convex domains, three invariants determine the domain up to biholomorphism and a linear change of variables at a base point. These invariants consist of not only the Kobayashi indicatrix but also a quadratic form and a hermitian form on a certain vector bundle. In two dimensions, there is an open set in a Frechet space, corresponding to domains not biholomorphic to the ball but with indicatrix the ball. The result of this paper is that, for ellipsoids, if the Kobayashi indicatrix is the ball then the ellipsoid is biholomorphic to the ball. This is proved in the following equivalent form: if an ellipsoid symmetric about 0 is not biholomorphic to the ball, then its infinitesimal Kobayashi metric at 0 is not hermitian. In this language, our result answers a question posed by Reiffen in [7] about the differential-geometric nature of the Carath6odory metric for ellipsoids. By two results of Lempert ([3] and [2]), for strictly convex domains the Carathe6dory and Kobayashi metrics coincide and determine a Finsler metric. One now sees that, for ellipsoids, the metric is not hermitian and hence, by a theorem of Reiffen [7], not Riemannian. We prove that the infinitesimal Kobayashi metric at 0 is not hermitian by proving that the parallelogram law fails to hold. An ellipsoid ~ may be written (via a linear change of coordinates) as )-], ] zj] 2 ..~ ,~j Re (z 2) < , 0 < Aj < 1. By a

Hermitian forms and locally toroidal regular polytopes

For any field K, a “K-curve” is an elliptic curve E defined over some finite separable extension of K such that every Galois conjugate of E is isogenous with E over a separable closure of K. A K-curve must either have complex multiplication (CM) or be isogenous to a collection of K-curves parametrized by a K-rational point on some modular curve X* (N). We give the first proof of this result, obtained in 1993 but not heretofore published. We also indicate some of the reasons for interest in K-curves, and in particular in Q-curves, and give some computational examples and open questions.