arXiv subjects
Publications and source records attributed to Eva Bayer-Fluckiger.
We determine the Hodge endomorphism algebras of non-projective complex K3 surfaces (and more generally, hyperk\"ahler manifolds). We show that they are either totally real fields or number fields generated by Salem numbers. This is unlike the projective case, where the endomorphism fields are either totally real or CM. We also develop precise existence criteria and explore the relations to number theory and dynamics.
Following Valloni, we study complex projective K3 surfaces having complex multiplication by rings of integers.
Every Salem numbers of degree 4,6,8,12,14 or 16 is the dynamical degree of an automorphism of a non-projective K3 surface. We define a notion of signature of an automorphism, and use it to give a necessary and sufficient condition for Salem numbers of degree 10 and 18 to be realized as the dynamical degree of such an automorphism. The first part of the paper contains results on isometries of lattices.
Let X be a complex projective K3 surface, and let T(X) be its transcendental lattice; the characteristic polynomials of the isometries of T(X) induced by automorphisms of X are powers of cyclotomic polynomials. Which powers of cyclotomic polynomials occur ? The aim of this note is to answer this question, as well as related ones, and give an alternative approach to some results of Kondo, Machida, Oguiso, Vorontsov, Xiao and Zhang; this leads to questions and results concerning orthogonal groups of lattices.
Let $E$ be a totally real number field of degree $d$ and let $m \geqslant 3$ be an integer. We show that if $md \leqslant 21$ then there exists an $(m-2)$-dimensional family of complex projective $K3$ surfaces with real multiplication by $E$. Analogous results are proved for CM number fields and also for all known higher-dimensional hyperk\"ahler manifolds.
We establish an explicit upper bound for the Euclidean minimum of a number field which depends, in a precise manner, only on its discriminant and the number of real and complex embeddings. Such bounds were shown to exist by Davenport and Swinnerton-Dyer. In the case of totally real fields, an optimal bound was conjectured by Minkowski and it is proved for fields of small degree. In this note we develop methods of McMullen in the case of mixed signature in order to get explicit bounds for the Euclidean minimum.
Let $k$ be a global field and let $L_0$,...,$L_m$ be finite separable field extensions of $k$. In this paper, we are interested in the Hasse principle for the multinorm equation $\underset{i=0}{\overset{m}{\prod}}N_{L_i/k}(t_i)=c$. Under the assumption that $L_0$ is a cyclic extension, we give an explicit description of the Brauer-Manin obstruction to the Hasse principle. We also give a complete criterion for the Hasse principle for multinorm equations to hold when $L_0$ is a meta-cyclic extension.
The aim of this paper is to give necessary and sufficient conditions for an integral polynomial to be the characteristic polynomial of a semi-simple isometry of some even unimodular lattice of given signature. This result has applications applications to automorphisms of K3 surfaces; in particular, we show that every Salem number of degree 4, 6, 8,1 2, 14 or 16 is the dynamical degree of an automorphism of a non-projective K3 surface.
We prove some new cases of the Grothendieck-Serre conjecture for classical groups. This is based on a new construction of the Gersten-Witt complex for Witt groups of Azumaya algebras with involution on regular semilocal rings, with explicit second residue maps; the complex is shown to be exact when the ring is of dimension $\le 2$ (or $\le 4$, with additional hypotheses on the algebra with involution). Note that we do not assume that the ring contains a field.
We give necessary and sufficient conditions for an integer to be the signature of a (4q-1)-knot in the (4q+1)-sphere with a given square-free Alexander polynomial.
Let $k$ be a field, and let $L$ be an étale k-algebra of finite rank. If $a$ is a nonzero element in $k$, let $X_a$ be the affine variety defined by the norm equation $N_{L/k}(x) = a$. Assuming that $L$ has at least one factor that is a cyclic field extension of $k$, we give a combinatorial description of the unramified Brauer group of $X_a$.
The aim of this paper is to revisit the question of local-global principles for embeddings of étale algebras with involution into central simple algebras with involution over global fields of characteristic not 2. A necessary and sufficient condition is given in [BLP 18]. In the present paper, we give a simpler description of the obstruction group. It is also shown that if the etale algebra is a product of pairwise linearly disjoint field extensions, then the Hasse principle holds, and that if an embedding exists after an odd degree extension, then it also exists over the global field itself. An appendix gives a generalization of this later result, in the framework of a question of Burt Totaro.
We give necessary and sufficient conditions for an integral polynomial without linear factors to be the characteristic polynomial of an isometry of some even, unimodular lattice of given signature. This gives rise to Hasse principle questions, which we answer in a more general setting. As an application, we prove a Hasse principle for signatures of knots.
In this paper we introduce a method to obtain algebraic information using arithmetic one in the study of tori and their principal homogeneous spaces. In particular, using some results of the authors with Tingyu Lee, we determine the unramified Brauer groups of some norm one tori, and their torsors.
The aim of this note is to give a formula expressing the trace form associated with the 27 lines of a cubic surface.
The aim of this note is to revisit the question of local-global principles for embeddings of etale algebras with involution into central simple algebras with involution over global fields of characteristic not 2. A necessary and sufficient condition is given in a joint paper with Tingyu Lee and Raman Parimala. In the present paper, we give a simpler description of the obstruction group. It is also shown that if the etale algebra is a product of pairwise independent field extensions, then the Hasse principle holds.
We characterize the irreducible polynomials that occur as a characteristic polynomial of an automorphism of an even unimodular lattice of given signature, generalizing a theorem of Gross and McMullen. As part of the proof, we give a general criterion in terms of Witt groups for a bilinear form equipped with an action of a group G over a discretely valued field to contain a unimodular G-stable lattice.
Let $R$ be a semilocal Dedekind domain with fraction field $F$. We show that two hereditary $R$-orders in central simple $F$-algebras which become isomorphic after tensoring with $F$ and with some faithfully flat étale $R$-algebra are isomorphic. On the other hand, this fails for hereditary orders with involution. The latter stands in contrast to a result of the first two authors, who proved this statement for hermitian forms over hereditary $R$-orders with involution. The results can be restated by means of étale cohomology and can be seen as variations of the Grothendieck--Serre conjecture on principal homogeneous bundles of reductive group schemes. Connections with Bruhat--Tits theory are also discussed.