Arbitrary Base Field Research Articles

The slice spectral sequence for a motivic analogue of the connective K(1)$K(1)$‐local sphere

AbstractWe compute the 2‐adic effective slice spectral sequence (ESSS) for the motivic stable homotopy groups of , a motivic analogue of the connective ‐local sphere over prime fields of characteristic not two. Together with the analogous computation over algebraically closed fields, this yields information about the motivic ‐local sphere over arbitrary base fields of characteristic not two. To compute the spectral sequence, we prove several results that may be of independent interest. We describe the ‐differentials in the slice spectral sequence in terms of the motivic Steenrod operations over general base fields, building on analogous results of Ananyevskiy, Röndigs, and Østvær for the very effective cover of Hermitian K‐theory. We also explicitly describe the coefficients of certain motivic Eilenberg–MacLane spectra and compute the ESSS for the very effective cover of Hermitian K‐theory over prime fields.

An improved error term for counting D4$D_4$‐quartic fields

AbstractWe prove that the number of quartic fields with discriminant whose Galois closure is equals , improving the error term in a well‐known result of Cohen, Diaz y Diaz, and Olivier. We prove an analogous result for counting quartic dihedral extensions over an arbitrary base field.

On the Arithmetic Fundamental Lemma conjecture over a general $p$-adic field

We prove the Arithmetic Fundamental Lemma conjecture over a general p -adic field with odd residue cardinality q\geq \dim V . Our strategy is similar to the one used by the second author in his proof of the AFL over \mathbb{Q}_p , but only requires the modularity of divisor generating series on the Shimura variety (as opposed to its integral model). The resulting increase in flexibility allows us to work over an arbitrary base field. To carry out our strategy, we also generalize results of Howard (2012) on CM cycle intersection and of Ehlen–Sankaran (2018) on Green function comparison from \mathbb{Q} to general totally real base fields.

On the structure of graded 3-Leibniz algebras

– We study the structure of a “3-Leibniz algebra” T graded by an arbitrary abelian group G, which is considered of arbitrary dimension and over an arbitrary base field . We show that T is of the form with a linear subspace of the homogeneous component associated to the unit element 1 in G, and every Ij is a well described graded ideal of T, satisfying if In the case of T being of maximal length, we characterize the gr-simplicity of the algebra in terms of connections in the support of the grading.

Leibniz algebras and graphs

We consider a Leibniz algebra over an arbitrary base field , being the ideal generated by the products . This ideal has a fundamental role in the study presented in our paper. A basis of is called multiplicative if for any we have that for some . We associate an adequate graph to relative to . By arguing on this graph we show that decomposes as a direct sum of ideals, each one being associated to one connected component of . Also the minimality of and the division property of are characterized in terms of the weak symmetry of the defined subgraphs and .

The Chow $t$-structure on the $\infty$-category of motivic spectra

We define the Chow $t$-structure on the $\infty$-category of motivic spectra $\mathcal{SH}(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $\mathcal{SH}(k)^{c\heartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $\mathcal{SH}(k)^{\mathrm{cell},c\heartsuit}$ as the category of even graded $\mathrm{MU}_{2*}\mathrm{MU}$-comodules. Furthermore, we show that the $\infty$-category of modules over the Chow truncated sphere spectrum $\mathbbm{1}_{c=0}$ is algebraic. Our results generalize the ones in Gheorghe--Wang--Xu in three aspects: to integral results; to all base fields other than just $\mathbb{C}$ to the entire $\infty$-category of motivic spectra $\mathcal{SH}(k)$, rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of ($p$-completed) spheres over an arbitrary base field $k$ using the Postnikov--Whitehead tower associated to the Chow $t$-structure and the motivic Adams spectral sequences over $k$.

Algorithms for checking zero-dimensional complete intersections

Given a 0-dimensional affine K-algebra R=K[x1,…,xn]∕I, where I is an ideal in a polynomial ring K[x1,…,xn] over a field K, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether R is a complete intersection at a maximal ideal, whether R is locally a complete intersection, and whether R is a strict complete intersection. These algorithms are based on Wiebe’s characterization of 0-dimensional local complete intersections via the 0-th Fitting ideal of the maximal ideal. They allow us to detect which generators of I form a regular sequence resp. a strict regular sequence, and they work over an arbitrary base field K. Using degree filtered border bases, we can detect strict complete intersections in certain families of 0-dimensional ideals.

On Strassen's Rank Additivity for Small Three-way Tensors

We address the problem of the additivity of the tensor rank. That is, for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples were proposed by Shitov. The latter are not very explicit, and they are only known to exist asymptotically for very large tensor spaces. In this article we prove that for some small three-way tensors the additivity holds. For instance, if the rank of one of the tensors is at most 6, then the additivity holds. Or, if one of the tensors lives in ${\mathbb C}^k\otimes {\mathbb C}^3\otimes {\mathbb C}^3$ for any $k$, then the additivity also holds. More generally, if one of the tensors is concise and its rank is at most 2 more than the dimension of one of the linear spaces, then additivity holds. In addition we also treat some cases of the additivity of the border rank of such tensors. In particular, we show that the additivity of the border rank holds if the direct sum tensor is contained in ${\mathbb C}^4\otimes {\mathbb C}^4\otimes {\mathbb C}^4$. Some of our results are valid over an arbitrary base field.

Genus theory and ε-conjectures on p-class groups

We suspect that the “genus part” of the class number of a number field K may be an obstruction for an “easy and unconditional proof” of the classical p-rank ε-conjecture for p-class groups and, a fortiori, for a proof of the “strong ε-conjecture”: #(CℓK⊗Zp)≪d,p,ε(|DK|)ε for all K of degree d. We analyze the weight of genus theory in this inequality by means of an infinite family of degree p cyclic fields with many ramified primes, then we prove the p-rank ε-conjecture: #(CℓK⊗Fp)≪d,p,ε(|DK|)ε, for d=p and the family of degree p cyclic extensions (Theorem 2.5) then sketch the case of arbitrary base fields. The possible obstruction for the strong form, in the degree p cyclic case, is the order of magnitude of the set of “exceptional” p-classes given by a well-known non-predictible algorithm, but largely controlled thanks to recent density results due to Koymans–Pagano. Then we compare the ε-conjectures with some p-adic conjectures, of Brauer–Siegel type, about the torsion group TK of the Galois group of the maximal abelian p-ramified pro-p-extension of totally real number fields K. We give numerical computations with the corresponding PARI/GP programs.

On derived categories of arithmetic toric varieties

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type $A$. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.

Finiteness results for K3 surfaces over arbitrary fields

Over an algebraically closed field, various finiteness results are known regarding the automorphism group of a K3 surface and the action of the automorphisms on the Picard lattice. We formulate and prove versions of these results over arbitrary base fields, and give examples illustrating how behaviour can differ from the algebraically closed case.

On the Cayley-Bacharach Property

The Cayley-Bacharach Property (CBP), which has been classically stated as a property of a finite set of points in an affine or projective space, is extended to arbitrary 0-dimensional affine algebras over arbitrary base fields. We present characterizations and explicit algorithms for checking the CBP directly, via the canonical module, and in combination with the property of being a locally Gorenstein ring. Moreover, we characterize strict Gorenstein rings by the CBP and the symmetry of their affine Hilbert function, as well as by the strict CBP and the last difference of their affine Hilbert function.

Maximal linear spaces contained in the base loci of pencils of quadrics

The geometry of the Fano scheme of maximal linear spaces contained in the base locus of a pencil of quadrics has been studied by algebraic geometers when the base field is algebraically closed. In this paper, we work over an arbitrary base field of characteristic not equal to 2 and show how these Fano schemes are related to the Jacobians of hyperelliptic curves. In particular, if $B$ is the base locus of a generic pencil of quadrics in $\bbp^{2n+1}$, and $F$ is the Fano variety of $n - 1$ planes contained in $B$, then $F$ is a component of a disconnected commutative algebraic group $G = \picz(C) \dcup F \dcup \pico(C) \dcup F'$, where $C$ is the hyperelliptic curve defined by the discriminant form of the pencil. In the second half of this paper, we study regular pencils of quadrics, where the hyperelliptic curve defined by the discriminant is singular.

Bilinear maps and graphs

Let V be a linear space of arbitrary dimension and over an arbitrary base field F, endowed with a bilinear map f:V×V→V. A basis B={vi}i∈I of V is an f-basis if for any i,j∈I we have that f(vi,vj)∈Fvk for some k∈I. We associate to any triplet (V,f,B) an adequate graph (V,E). By arguing on this graph we show that V decomposes as a direct sum of strongly f-invariant linear subspaces, each one being associated to one connected component of (V,E). Also the B-semisimplicity and the B-simplicity of V are characterized in terms of the associated graph.

An explicit Waldspurger formula for Hilbert modular forms

We describe a construction of preimages for the Shimura map on Hilbert modular forms and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted L L -functions. Our construction is inspired by that of Gross and applies to any nontrivial level and arbitrary base field, subject to certain conditions on the Atkin-Lehner eigenvalues and on the weight.

N-Algebras admitting a multiplicative basis

Let [Formula: see text] be an [Formula: see text]-algebra of arbitrary dimension and over an arbitrary base field [Formula: see text]. A basis [Formula: see text] of [Formula: see text] is said to be multiplicative if for any [Formula: see text], we have either [Formula: see text] or [Formula: see text] for some (unique) [Formula: see text]. If [Formula: see text], we are dealing with algebras admitting a multiplicative basis while if [Formula: see text] we are speaking about triple systems with multiplicative bases. We show that if [Formula: see text] admits a multiplicative basis then it decomposes as the orthogonal direct sum [Formula: see text] of well-described ideals admitting each one a multiplicative basis. Also, the minimality of [Formula: see text] is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is by means of the family of its minimal ideals.

A proof of the integral identity conjecture, II

In this note, using Cluckers–Loeser's theory of motivic integration, we prove the integral identity conjecture with framework a localized Grothendieck ring of varieties over an arbitrary base field of characteristic zero.

Parameter Spaces for Algebraic Equivalence

Abstract A cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth integral scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families of cycles parameterized by curves, or by abelian varieties. In this article, we extend these results to arbitrary base fields. The strengthening of these results turns out to be a key step in our work elsewhere extending Murre’s results on algebraic representatives for varieties over algebraically closed fields to arbitrary perfect fields.

Classification of Nilpotent Malcev Algebras of Small Dimensions Over Arbitrary Fields of Characteristic not 2

The paper is devoted to classify all of the nilpotent Malcev algebras of dimension ≤ 6 over an arbitrary base field 𝔽 of characteristic not 2. We also classify all 7-dimensional non-Lie nilpotent Malcev algebras which are not metabelian over any field of characteristic not 2.

Group actions on algebraic stacks via butterflies

We introduce an explicit method for studying actions of a group stack G on an algebraic stack X. As an example, we study in detail the case where X=P(n0,⋯,nr) is a weighted projective stack over an arbitrary base S. To this end, we give an explicit description of the group stack of automorphisms of P(n0,⋯,nr), the weighted projective general linear 2-groupPGL(n0,⋯,nr). As an application, we use a result of Colliot-Thélène to show that for every linear algebraic group G over an arbitrary base field k (assumed to be reductive if char(k)>0) such that Pic(G)=0, every action of G on P(n0,⋯,nr) lifts to a linear action of G on Ar+1.