Holmstrom Arithmetic schemes

Try to understand and gather examples of arithmetic schemes.

Source of examples: Can I get notes from the summer school at EPFL 2012 on del Pezzo and K3 surfaces, by Anthony Várilly-Alvarado?

http://mathoverflow.net/questions/97086/does-smooth-and-proper-over-mathbb-z-imply-rational

Title: Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects Authors: Reynald Lercier, Christophe Ritzenthaler Categories: math.NT Number Theory (math.AG Algebraic Geometry) Comments: 46 pages ; related programs are available on the web pages of the authors MSC: 14Q05, 13A50, 14H10, 14H37 Abstract: We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We focus on genus 3 hyperelliptic curves. Both geometric and arithmetic aspects are considered.

Arithmetic moduli of elliptic curves, by Katz and Mazur

Deninger: l-adic Lefschetz numbers of arithmetic schemes

Asked Jakob Scholbach about whether a scheme over Z which is nonflat must lie over a finite set of primes. Answer: The following is e.g. in Hartshorne: let X be reduced. A map $X \to Spec Z$ is flat iff every irreducible component is dominant over Z, i.e., contains the generic point Spec Q. Actually I was erroneously forgetting the irreducible component (or connected component, since I only look at regular schemes at these places). So, if it is not flat (but reduced and irred.) it cannot be dominant, hence must be a $\sqcup \mathrm{Spec} F_{p_i}$ - scheme (finite disjoint union of some finite fields). The reducedness is nowhere a problem for me, mainly since $M(X) \cong M(X_{red})$ by localization, i.e., the motive does not see whether the scheme is reduced or not.

Understand relation to log geometry

Shuji Saito research summary. Check also his preprints and publications pages for new preprints, in Sep 2009 he mentions some in preparation.

David Holmes once mentioned a paper related to hyperelliptic genus two curves, and Tate’s algorithm, in which they do blowups over Spec Z.

http://mathoverflow.net/questions/9576/smooth-proper-scheme-over-z about smooth proper schemes without sections.

http://mathoverflow.net/questions/20913/families-of-sheaves-on-arithmetic-varieties

http://mathoverflow.net/questions/77644/are-there-nonobvious-cases-where-equations-have-finitely-many-algebraic-integer-s

http://mathoverflow.net/questions/22798/is-the-group-of-integer-points-on-a-finite-type-group-scheme-over-z-finitely-pres

http://mathoverflow.net/questions/77546/schemes-over-with-a-graded-existence-over-sub1-sub

All answers of Liu: http://mathoverflow.net/users/3485/qing-liu

Toric varieties

See for example the review of Maillot: Un théorème de Bernstein-Koushnirenko arithmétique (the article is not available online yet, maybe later on Science Direct), for examples of how toric varieties can be viewed as schemes over Z. In the setting of this article, the canonical metric on the Hermitian bundle is not smooth in general, but Maillot overcomes this problem. See also G´eom´etrie d’Arakelov des vari´et´es toriques et fibr´es en droites int´egrables by same author.

Maillot considers cellular arithmetic schemes in Progr Math 171 article, such as toric schemes, flag schemes, and Grassmann schemes over Z. See review if article not available.

arXiv:0910.2349 Non-liftable Calabi-Yau spaces from arXiv Front: math.AG by Slawomir Cynk, Matthias Schuett We construct many new non-liftable three-dimensional Calabi-Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi-Yau space over some number field as introduced by the first author and D. van Straten.

arXiv:0910.2589 Explicit Kummer surface theory for arbitrary characteristic from arXiv Front: math.AG by Jan Steffen Müller We explicitly find an equation and a projective embedding of the Kummer surface associated to the Jacobian of a curve of genus 2 given by an equation of the form y^2 + h(x)y = f(x) over an arbitrary ground field as well as several maps that can be used to perform arithmetic on it. This extends earlier work by Flynn and has applications, for instance, in computations of canonical heights for genus 2 Jacobians and in cryptography.

arXiv:0907.0298 Elliptic Surfaces from arXiv Front: math.NT by Matthias Schuett, Tetsuji Shioda This survey paper concerns elliptic surfaces with section. We give a detailed overview of the theory including many examples. Emphasis is placed on rational elliptic surfaces and elliptic K3 surfaces. To this end, we particularly review the theory of Mordell-Weil lattices and address arithmetic questions.

arXiv:1002.2142 A Lefschetz fixed point formula for singular arithmetic schemes with smooth generic fibres from arXiv Front: math.AG by Shun Tang In this article, we consider singular equivariant arithmetic schemes whose generic fibres are smooth. For such schemes, we prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. This formula is an analog, in the arithmetic case, of the Lefschetz formula proved by R. W. Thomason. In particular, our result implies a fixed point formula which was conjectured by V. Maillot and D. Rössler.

arXiv:1003.2451 The Langlands-Kottwitz approach for some simple Shimura Varieties from arXiv Front: math.AG by Peter Scholze We show how the Langlands-Kottwitz method can be used to determine the semisimple local factors of the Hasse-Weil zeta-function of certain Shimura varieties. On the way, we prove a conjecture of Haines and Kottwitz in this special case.

http://mathoverflow.net/questions/40486/comparing-cohomology-over-mathbb-c-and-over-mathbb-f-q

nLab page on Arithmetic schemes