Searcharxiv⌕ Search

arXiv subjects

Oliver Lorscheid

Publications and source records attributed to Oliver Lorscheid.

At least 55 records · Page 3Linked to original sources
arXiv2015

Schubert decompositions for quiver Grassmannians of tree modules

Let $Q$ be a quiver, $M$ a representation of $Q$ with an ordered basis $\cB$ and $\ue$ a dimension vector for $Q$. In this note we extend the methods of \cite{L12} to establish Schubert decompositions of quiver Grassmannians $\Gr_\ue(M)$ into affine spaces to the ramified case, i.e.\ the canonical morphism $F:T\to Q$ from the coefficient quiver $T$ of $M$ w.r.t.\ $\cB$ is not necessarily unramified. In particular, we determine the Euler characteristic of $\Gr_\ue(M)$ as the number of \emph{extremal successor closed subsets of $T_0$}, which extends the results of Cerulli Irelli (\cite{Cerulli11}) and Haupt (\cite{Haupt12}) (under certain additional assumptions on $\cB$).

math.RT↗
arXiv2013

A blueprinted view on $\mathbb F_1$-geometry

This overview paper has two parts. In the first part, we review the development of $\mathbb F_1$-geometry from the first mentioning by Jacques Tits in 1956 until the present day. We explain the main ideas around $\mathbb F_1$, embedded into the historical context, and give an impression of the multiple connections of $\mathbb F_1$-geometry to other areas of mathematics. In the second part, we review (and preview) the geometry of blueprints. Beyond the basic definitions of blueprints, blue schemes and projective geometry, this includes a theory of Chevalley groups over $\mathbb F_1$ together with their action on buildings over $\mathbb F_1$; computations of the Euler characteristic in terms of $\mathbb F_1$-rational points, which involve quiver Grassmannians; $K$-theory of blue schemes that reproduces the formula $K_i(\mathbb F_1)=π^{st}_i(S^0)$; models of the compactifications of $\Spec \mathbb Z$ and other arithmetic curves; and explanations about the connections to other approaches towards $\mathbb F_1$ like monoidal schemes after Deitmar, $B_1$-algebras after Lescot, $Λ$-schemes after Borger, relative schemes after Toën and Vaquié, log schemes after Kato and congruence schemes after Berkovich and Deitmar.

math.AG↗
arXiv2013

Torified varieties and their geometries over F_1

This paper invents the notion of torified varieties: A torification of a scheme is a decomposition of the scheme into split tori. A torified variety is a reduced scheme of finite type over $\Z$ that admits a torification. Toric varieties, split Chevalley schemes and flag varieties are examples of this type of scheme. Given a torified variety whose torification is compatible with an affine open covering, we construct a gadget in the sense of Connes-Consani and an object in the sense of Soulé and show that both are varieties over $\F_1$ in the corresponding notion. Since toric varieties and split Chevalley schemes satisfy the compatibility condition, we shed new light on all examples of varieties over $\F_1$ in the literature so far. Furthermore, we compare Connes-Consani's geometry, Soulé's geometry and Deitmar's geometry, and we discuss to what extent Chevalley groups can be realized as group objects over $\F_1$ in the given categories.

math.AG↗
arXiv2013

On Schubert decompositions of quiver Grassmannians

In this paper, we introduce Schubert decompositions for quiver Grassmannians and investigate example classes of quiver Grassmannians with a Schubert decomposition into affine spaces. The main theorem puts the cells of a Schubert decomposition into relation to the cells of a certain simpler quiver Grassmannian. This allows us to extend known examples of Schubert decompositions into affine spaces to a larger class of quiver Grassmannians. This includes exceptional representations of the Kronecker quiver as well as representations of forests with block matrices of the form $\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}$. Finally, we draw conclusions on the Euler characteristics and the cohomology of quiver Grassmannians.

math.AG↗
arXiv2012

Blueprints - towards absolute arithmetic?

One of the driving motivations to develop $\F_1$-geometry is the hope to translate Weil's proof of the Riemann hypothesis from positive characteristics to number fields, which might result in a proof of the classical Riemann hypothesis. The underlying idea is that the spectrum of $\Z$ should find an interpretation as a curve over $\F_1$, which has a completion $\bar{\Spec\Z}$ analogous to a curve over a finite field. The hope is that intersection theory for divisors on the arithmetic surface $\bar{\Spec\Z} \times \bar{\Spec\Z}$ will allow to mimic Weil's proof. It turns out that it is possible to define an object $\bar{\Spec\Z}$ from the viewpoint of blueprints that has certain properties, which come close to the properties of its analogs in positive characteristic. This shall be explained in the following note, which is a summary of a talk given at the Max Planck Institute in March, 2012.

math.AG↗
arXiv2012

Projective geometry for blueprints

In this note, we generalize the Proj-construction from usual schemes to blue schemes. This yields the definition of projective space and projective varieties over a blueprint. In particular, it is possible to descend closed subvarieties of a projective space to a canonical F_1-model. We discuss this explicitly in case of the Grassmannian Gr(2,4).

math.AG↗
arXiv2012

Sheaves and $K$-theory for $\mathbb{F}_1$-schemes

This paper is devoted to the open problem in $\mathbb{F}_1$-geometry of developing $K$-theory for $\mathbb{F}_1$-schemes. We provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for $\mathcal{M}_0$-schemes and $\mathbb{F}_1$-schemes in the sense of Connes and Consani. A wide range of results hopefully lies the background for further developments of the algebraic geometry over $\mathbb{F}_1$. Special attention is paid to two aspects particular to $\mathbb{F}_1$-geometry, namely, normal morphisms and locally projective sheaves, which occur when we adopt Quillen's Q-construction to a definition of $G$-theory and $K$-theory for $\mathbb{F}_1$-schemes. A comparison with Waldhausen's $S_{\bullet}$-construction yields the ring structure of $K$-theory. In particular, we generalize Deitmar's $K$-theory of monoids and show that $K_*(\Spec\mathbb{F}_1)$ realizes the stable homotopy of the spheres as a ring spectrum.

math.KT↗
arXiv2012

The geometry of blueprints. Part I: Algebraic background and scheme theory

In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp.\ congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and $\mathbb{F}_1$-schemes (after Kato, Deitmar and Connes-Consani). Beside this unification, the category of blueprints contains new interesting objects as "improved" cyclotomic field extensions $\mathbb{F}_{1^n}$ of $\mathbb{F}_1$ and "archimedean valuation rings". It also yields a notion of semiring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Tits' idea of Chevalley groups over $\mathbb{F}_1$, congruence schemes, sheaf cohomology, $K$-theory and a unified view on analytic geometry over $\mathbb{F}_1$, adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.

math.AG↗
arXiv2012

The geometry of blueprints. Part II: Tits-Weyl models of algebraic groups

This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called $\mathbb{F}_1$, \emph{the field with one element}. Based on Part I of The geometry of blueprints, we introduce the class of \emph{Tits morphisms} between blue schemes. The resulting \emph{Tits category} $\textup{Sch}_\mathcal{T}$ comes together with a base extension to (semiring) schemes and the so-called \emph{Weyl extension} to sets. We prove for $\mathcal{G}$ in a wide class of Chevalley groups---which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups---that a linear representation of $\mathcal{G}$ defines a model $G$ in $\textup{Sch}_\mathcal{T}$ whose Weyl extension is the Weyl group $W$ of $\mathcal{G}$. We call such models \emph{Tits-Weyl models}. The potential of Tits-Weyl models lies in \textit{(a)} their intrinsic definition that is given by a linear representation; \textit{(b)} the (yet to be formulated) unified approach towards thick and thin geometries; and \textit{(c)} the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry.

math.AG↗
arXiv2011

Toroidal automorphic forms, Waldspurger periods and double Dirichlet series

The space of toroidal automorphic forms was introduced by Zagier in the 1970s: a GL_2-automorphic form is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems (amongst others) from the fact that an Eisenstein series of weight s is toroidal for a given torus precisely if s is a non-trivial zero of the zeta function of the quadratic field corresponding to the torus. In this paper, we study the structure of the space of toroidal automorphic forms for an arbitrary number field F. We prove that it decomposes into a space spanned by all derivatives up to order n-1 of an Eisenstein series of weight s and class group character omega precisely if s is a zero of order n of the L-series corresponding to omega at s, and a space consisting of exactly those cusp forms the central value of whose L-series is zero. The proofs are based on an identity of Hecke for toroidal integrals of Eisenstein series and a result of Waldspurger about toroidal integrals of cusp forms combined with non-vanishing results for twists of L-series proven by the method of double Dirichlet series.

math.NT↗
arXiv2010

Automorphic forms for elliptic function fields

Let $F$ be the function field of an elliptic curve $X$ over $\F_q$. In this paper, we calculate explicit formulas for unramified Hecke operators acting on automorphic forms over $F$. We determine these formulas in the language of the graph of an Hecke operator, for which we use its interpretation in terms of $¶^1$-bundles on $X$. This allows a purely geometric approach, which involves, amongst others, a classification of the $¶^1$-bundles on $X$. We apply the computed formulas to calculate the dimension of the space of unramified cusp forms and the support of a cusp form. We show that a cuspidal Hecke eigenform does not vanish in the trivial $¶^1$-bundle. Further, we determine the space of unramified $F'$-toroidal automorphic forms where $F'$ is the quadratic constant field extension of $F$. It does not contain non-trivial cusp forms. An investigation of zeros of certain Hecke $L$-series leads to the conclusion that the space of unramified toroidal automorphic forms is spanned by the Eisenstein series $E(\blanc,s)$ where $s+1/2$ is a zero of the zeta function of $X$---with one possible exception in the case that $q$ is even and the class number $h$ equals $q+1$.

math.NT↗
arXiv2010

Graphs of Hecke operators

Let $X$ be a curve over $\F_q$ with function field $F$. In this paper, we define a graph for each Hecke operator with fixed ramification. A priori, these graphs can be seen as a convenient language to organize formulas for the action of Hecke operators on automorphic forms. However, they will prove to be a powerful tool for explicit calculations and proofs of finite dimensionality results. We develop a structure theory for certain graphs $G_x$ of unramified Hecke operators, which is of a similar vein to Serre's theory of quotients of Bruhat Tits trees. To be precise, $G_x$ is locally a quotient of a Bruhat Tits tree and has finitely many components. An interpretation of $G_x$ in terms of rank 2 bundles on $X$ and methods from reduction theory show that $G_x$ is the union of finitely many cusps, which are infinite subgraphs of a simple nature, and a nucleus, which is a finite subgraph that depends heavily on the arithmetics of $F$. We describe how one recovers unramified automorphic forms as functions on the graphs $G_x$. In the exemplary cases of the cuspidal and the toroidal condition, we show how a linear condition on functions on $G_x$ leads to a finite dimensionality result. In particular, we re-obtain the finite-dimensionality of the space of unramified cusp forms and the space of unramified toroidal automorphic forms. In an Appendix, we calculate a variety of examples of graphs over rational function fields.

math.NT↗
arXiv2010

Toroidal automorphic forms for function fields

The space of toroidal automorphic forms was introduced by Zagier in 1979. Let $F$ be a global field. An automorphic form on $\GL(2)$ is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems from the fact (amongst others) that an Eisenstein series of weight $s$ is toroidal if $s$ is a non-trivial zero of the zeta function, and thus a connection with the Riemann hypothesis is established. In this paper, we concentrate on the function field case. We show the following results. The $(n-1)$-th derivative of a non-trivial Eisenstein series of weight $s$ and Hecke character $χ$ is toroidal if and only if $L(χ,s+1/2)$ vanishes in $s$ to order at least $n$ (for the "only if"-part we assume that the characteristic of $F$ is odd). There are no non-trivial toroidal residues of Eisenstein series. The dimension of the space of derivatives of unramified Eisenstein series equals $h(g-1)+1$ if the characterisitc is not 2; in characteristic 2, the dimension is bounded from below by this number. Here $g$ is the genus and $h$ is the class number of $F$. The space of toroidal automorphic forms is an admissible representation and every irreducible subquotient is tempered.

math.NT↗
arXiv2010

Functional equations for zeta functions of $\mathbb{F}_1$-schemes

For a scheme $X$ whose $\mathbb F_q$-rational points are counted by a polynomial $N(q)=\sum a_iq^i$, the $\mathbb{F}_1$-zeta function is defined as $ζ(s)=\prod(s-i)^{-a_i}$. Define $χ=N(1)$. In this paper we show that if $X$ is a smooth projective scheme, then its $\mathbb{F}_1$-zeta function satisfies the functional equation $ζ(n-s) = (-1)^χζ(s)$. We further show that the $\mathbb{F}_1$-zeta function $ζ(s)$ of a split reductive group scheme $G$ of rank $r$ with $N$ positive roots satisfies the functional equation $ζ(r+N-s) = (-1)^χ( ζ(s) )^{(-1)^r}$.

math.NT↗
arXiv2009

Mapping F_1-land:An overview of geometries over the field with one element

This paper gives an overview of the various approaches towards F_1-geometry. In a first part, we review all known theories in literature so far, which are: Deitmar's F_1-schemes, Toën and Vaquié's F_1-schemes, Haran's F-schemes, Durov's generalized schemes, Soulé's varieties over F_1 as well as his and Connes-Consani's variations of this theory, Connes and Consani's F_1-schemes, the author's torified varieties and Borger's Lambda-schemes. In a second part, we will tie up these different theories by describing functors between the different F_1-geometries, which partly rely on the work of others, partly describe work in progress and partly gain new insights in the field. This leads to a commutative diagram of F_1-geometries and functors between them that connects all the reviewed theories. We conclude the paper by reviewing the second author's constructions that lead to realization of Tits' idea about Chevalley groups over F_1.

math.AG↗
arXiv2009

Algebraic groups over the field with one element

Remarks in a paper by Jacques Tits from 1956 led to a philosophy how a theory of split reductive groups over $\F_1$, the so-called field with one element, should look like. Namely, every split reductive group over $\Z$ should descend to $\F_1$, and its group of $\F_1$-rational points should be its Weyl group. We connect the notion of a torified variety to the notion of $\F_1$-schemes as introduced by Connes and Consani. This yields models of toric varieties, Schubert varieties and split reductive groups as $\Fun$-schemes. We endow the class of $\F_1$-schemes with two classes of morphisms, one leading to a satisfying notion of $\F_1$-rational points, the other leading to the notion of an algebraic group over $\F_1$ such that every split reductive group is defined as an algebraic group over $\F_1$. Furthermore, we show that certain combinatorics that are expected from parabolic subgroups of $\GL(n)$ and Grassmann varieties are realized in this theory.

math.AG↗
arXiv2008

Toroidal automorphic forms for some function fields

Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on GL_2 is toroidal if all its right translates integrate to zero over all nonsplit tori in GL_2, and an Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the global function fields of class number one and genus g zero or one, and with a rational place. The space has dimension g and is spanned by the expected Eisenstein series. We deduce an "automorphic" proof for the Riemann hypothesis for the zeta function of those curves.

math.NT↗