To any k-dimensional subspace of $\mathbb {Q}^n$ one can naturally associate a point in the Grassmannian $\mathrm {Gr}_{n,k}(\mathbb {R})$ and two shapes of lattices of rank k and $n-k$, respectively. These lattices originate by intersecting the k-dimensional subspace and its orthogonal with the lattice $\mathbb {Z}^n$. Using unipotent dynamics, we prove simultaneous equidistribution of all of these objects under congruence conditions when $(k,n) \neq (2,4)$.

We extend field patching to the setting of Berkovich analytic geometry and use it to prove a local–global principle over function fields of analytic curves with respect to completions. In the context of quadratic forms, we combine it with sufficient conditions for local isotropy over a Berkovich curve to obtain applications on the $u$-invariant. The patching method we adapt was introduced by Harbater and Hartmann [Patching over fields, Israel J. Math. 176 (2010), 61–107] and further developed by these two authors and Krashen [Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), 231–263]. The results presented in this paper generalize those of Harbater, Hartmann, and Krashen [Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), 231–263] on the local–global principle and quadratic forms.

We show that over any field $F$ of characteristic 2 and 2-rank $n$, there exist $2^{n}$ bilinear $n$-fold Pfister forms that have no slot in common. This answers a question of Becher [‘Triple linkage’, Ann.$K$-Theory, to appear] in the negative. We provide an analogous result also for quadratic Pfister forms.

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

Subconvexity bounds on the critical line are proved for general Epstein zeta-functions of $k$-ary quadratic forms. This is related to sup-norm bounds for unitary Eisenstein series on $\text{GL}(k)$ associated with the maximal parabolic of type $(k-1,1)$, and the exact sup-norm exponent is determined to be $(k-2)/8$ for $k\geqslant 4$. In particular, if $k$ is odd, this exponent is not in $\frac{1}{4}\mathbb{Z}$, which is relevant in the context of Sarnak’s purity conjecture and shows that it can in general not directly be generalized to Eisenstein series.

Let $M$ be an irreducible holomorphic symplectic (hyperkähler) manifold. If $b_{2}(M)\geqslant 5$, we construct a deformation $M^{\prime }$ of $M$ which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real $(1,1)$-classes is hyperbolic. If $b_{2}(M)\geqslant 14$, similarly, we construct a deformation which admits a parabolic automorphism (and many other automorphisms as well).

We study the set $D(M,N)$ of all possible mapping degrees from $M$ to $N$ when $M$ and $N$ are quasitoric $4$-manifolds. In some of the cases, we completely describe this set. Our results rely on Theorems proved by Duan and Wang and the sets of integers obtained are interesting from the number theoretical point of view, for example those representable as the sum of two squares $D(\mathbb{C}P^{2}\sharp \mathbb{C}P^{2},\mathbb{C}P^{2})$ or the sum of three squares $D(\mathbb{C}P^{2}\sharp \mathbb{C}P^{2}\sharp \mathbb{C}P^{2},\mathbb{C}P^{2})$. In addition to the general results about the mapping degrees between quasitoric 4-manifolds, we establish connections between Duan and Wang’s approach, quadratic forms, number theory and lattices.

In terms of class field theory we give a necessary and sufficient condition for an integer to be representable by the quadratic form $x^{2}+xy+ny^{2}$ ($n\in \mathbb{N}$ arbitrary) under extra conditions $x\equiv 1\;\text{mod}\;m$, $y\equiv 0\;\text{mod}\;m$ on the variables. We also give some examples where their extended ring class numbers are less than or equal to $3$.

For non-singular intersections of pairs of quadrics in 11 or more variables, we prove an asymptotic for the number of rational points in an expanding box.

In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in anyHecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic $\mathbb{Z}\left[ \text{ }\!\!\lambda\!\!\text{ } \right]$-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic $\mathbb{Z}\left[ \text{ }\!\!\lambda\!\!\text{ } \right]$-binary quadratic forms.

We compute the total Ij -cohomology of a projective bundle over a smooth scheme.

Let $p$ be a prime, and let ${{\zeta }_{p}}$ be a primitive $p$-th root of unity. The lattices in Craig's family are $(p\,-\,1)$-dimensional and are geometrical representations of the integral $\mathbb{Z}[{{\zeta }_{p}}]$-ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}$, where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p\,-\,1$ where $149\,\le \,p\,\le \,3001$, Craig's lattices are the densest packings known. Motivated by this, we construct $(p\,-\,1)(q\,-\,1)$-dimensional lattices from the integral $\mathbb{Z}[{{\zeta }_{pq}}]$-ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}{{\left\langle 1\,-\,{{\zeta }_{q}} \right\rangle }^{j}}$, where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.

We construct new invariants of quadratic forms over commutative rings, using ideas from Topology. More precisely, we define a hermitian analog of the Bott class with target algebraic K-theory, based on the classification of Clifford modules. These invariants of quadratic forms go beyond the classical invariants defined via the Clifford algebra. An appendix by J.-P. Serre, of independent interest, describes the “square root” of the Bott class in the general framework of lambda rings.

Many statistical applications require establishingcentral limit theorems for sums/integrals $S_T(h)=\int_{t \in I_T} h (X_t) {\rm d}t$ or for quadratic forms $Q_T(h)=\int_{t,s \in I_T} \hat{b}(t-s) h (X_t, X_s) {\rm d}s {\rm d}t$ , where X t is a stationaryprocess. A particularly important case is that of Appellpolynomials h(X t ) = P m (X t ), h(X t ,X s ) = P m , n (X t ,X s ), since the “Appell expansion rank" determines typically thetype of central limit theorem satisfied by the functionalsS T (h), Q T (h). We review and extend here to multidimensional indices, along lines conjectured in [F. Avram and M.S. Taqqu, Lect. Notes Statist.187 (2006) 259–286], a functionalanalysis approach to this problem proposed by [Avram and Brown, Proc. Amer.Math. Soc.107 (1989) 687–695] based on the method of cumulants and on integrabilityassumptions in the spectral domain; several applications arepresented as well.

In this appendix to [4], we state a periodicity conjecture using form parameters. This conjecture contains as particular case the fundamental theorem in hermitian K-theory proved in [6] but also some results of Bak [1], Barge and Lannes [2], Sharpe [8], for lower hermitian K-groups. The interest of this conjecture lies also in the parallel use of hermitian forms and quadratic forms with form parameters introduced by Bak.

Let $K$ be a number field, and let $F$ be a symmetric bilinear form in $2N$ variables over $K$. Let $Z$ be a subspace of ${{K}^{N}}$. A classical theorem of Witt states that the bilinear space $\left( Z,\,F \right)$ can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights of $F$ and $Z$. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan-Dieudonné theorem. Namely, we show that every isometry $~\sigma$ of a regular bilinear space $\left( Z,\,F \right)$ can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights of $F$, $Z$, and $~\sigma$.

We consider quadratic diophantine equations of the shape

$$ Q(x_1, \ldots, x_s) = 0 \qquad (1)$$

for a polynomial $Q(X_1, \ldots, X_s) \in \mathbf{Z}[X_1, \ldots, X_s]$ of degree $2$. Let $H$ be an upper bound for the absolute values of the coefficients of $Q$, and assume that the homogeneous quadratic part of $Q$ is non-singular. We prove, for all $s \ge 3$, the existence of a polynomial bound $\Lambda_s(H)$ with the following property: if equation (1) has a solution $\mathbf{x} \in \mathbf{Z}^s$ at all, then it has one satisfying

$$ |x_i| \le \Lambda_s(H) \quad (1 \le i \le s). $$

For $s = 3$ and $s = 4$ no polynomial bounds $\Lambda_s(H)$ were previously known, and for $s \ge 5$ we have been able to improve existing bounds quite significantly.

Under regularity assumptions, we establish a sharp largedeviation principle for Hermitian quadratic forms ofstationary Gaussian processes. Our result is similar tothe well-known Bahadur-Rao theorem [2] on the samplemean. We also provide several examples of applicationsuch as the sharp large deviation properties ofthe Neyman-Pearson likelihood ratio test, of the sum of squares,of the Yule-Walkerestimator of the parameter of a stable autoregressive Gaussian process,and finally of the empirical spectral repartition function.

The splitting pattern of a quadratic form $q$ over a field $k$ consists of all distinct Witt indices that occur for $q$ over extension fields of $k$. In small dimensions, the complete list of splitting patterns of quadratic forms is known. We show that all splitting patterns of quadratic forms of dimension at most nine can be realized by trace forms.

As it had been recognized by Liouville, Hermite, Mordell and others, the number of non-negative integer solutions of the equation in the title is strongly related to the class number of quadratic forms with discriminant —n. The purpose of this note is to point out a deeper relation which makes it possible to derive a reasonable upper bound for the number of solutions.