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).Comment: 22 page

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 $\P^1$-bundles on $X$. This allows a purely geometric approach, which involves, amongst others, a classification of the $\P^1$-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 $\P^1$-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 $q+1$.Comment: 26 page

Scheme theoretic tropicalization

In this paper, we introduce ordered blueprints and ordered blue schemes, which serve as a common language for the different approaches to tropicalizations and which enhances tropical varieties with a schematic structure. As an abstract concept, we consider a tropicalization as a moduli problem about extensions of a given valuation $v:k\to T$ between ordered blueprints $k$ and $T$. If $T$ is idempotent, then we show that a generalization of the Giansiracusa bend relation leads to a representing object for the tropicalization, and that it has yet another interpretation in terms of a base change along $v$. We call such a representing object a scheme theoretic tropicalization. This theory recovers and improves other approaches to tropicalizations as we explain with care in the second part of this text. The Berkovich analytification and the Kajiwara-Payne tropicalization appear as rational point sets of a scheme theoretic tropicalization. The same holds true for its generalization by Foster and Ranganathan to higher rank valuations. The scheme theoretic Giansiracusa tropicalization can be recovered from the scheme theoretic tropicalizations in our sense. We obtain an improvement due to the resulting blueprint structure, which is sufficient to remember the Maclagan-Rinc\'on weights. The Macpherson analytification has an interpretation in terms of a scheme theoretic tropicalization, and we give an alternative approach to Macpherson's construction of tropicalizations. The Thuillier analytification and the Ulirsch tropicalization are rational point sets of a scheme theoretic tropicalization. Our approach yields a generalization to any, possibly nontrivial, valuation $v:k\to T$ with idempotent $T$ and enhances the tropicalization with a schematic structure.Comment: 66 pages; for information about the changes in this version of the paper, please cf. the paragraph "Differences to previous versions" in the introductio

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.Comment: Slightly revised and extended version as in print. 51 page