Diophantine approximation and deformation

We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the exponents of the power series, with more generic curves giving lower exponents. If we transport Vojta's conjecture on height inequality to finite characteristic by modifying it by adding suitable deformation theoretic condition, then we see that the numbers giving rise to general curves approach Roth's bound. We also prove a hierarchy of exponent bounds for approximation by algebraic quantities of bounded degree

Relativity of arithmetic as a fundamental symmetry of physics

Arithmetic operations can be defined in various ways, even if one assumes commutativity and associativity of addition and multiplication, and distributivity of multiplication with respect to addition. In consequence, whenever one encounters `plus' or `times' one has certain freedom of interpreting this operation. This leads to some freedom in definitions of derivatives, integrals and, thus, practically all equations occurring in natural sciences. A change of realization of arithmetic, without altering the remaining structures of a given equation, plays the same role as a symmetry transformation. An appropriate construction of arithmetic turns out to be particularly important for dynamical systems in fractal space-times. Simple examples from classical and quantum, relativistic and nonrelativistic physics are discussed, including the eigenvalue problem for a quantum harmonic oscillator. It is explained why the change of arithmetic is not equivalent to the usual change of variables, and why it may have implications for the Bell theorem

Special arithmetic of flavor

We revisit the classification of rank-1 4d N= 2 QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-N\ue9ron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs (\u3b5, F 1e) where E is a relatively minimal, rational elliptic surface with section, and F 1e a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on (\u3b5, F 1e) equivalent to the \u201csafely irrelevant conjecture\u201d. The Mordell-Weil group of E (with the N\ue9ron-Tate pairing) contains a canonical root system arising from ( 121)-curves in special position in the N\ue9ron-Severi group. This canonical system is identified with the roots of the flavor group F: the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al

Good reduction of Fano threefolds and sextic surfaces

We investigate versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings, for other classes of varieties. We first obtain analogues for certain Fano threefolds. We use these results to prove the Shafarevich conjecture for smooth sextic surfaces, which appears to be the first non-trivial result in the literature on the arithmetic of such surfaces. Moreover, we exhibit certain moduli stacks of Fano varieties which are not hyperbolic, which allows us to show that the analogue of the Shafarevich conjecture does not always hold for Fano varieties. Our results also provide new examples for which the conjectures of Campana and Lang-Vojta hold.Comment: 22 pages. Minor change

Hilbert's Tenth Problem in Coq (Extended Version)

We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem -- in our case by a Minsky machine -- is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer. Additionally, we prove the reverse direction and show that every Diophantine relation is recognisable by $\mu$-recursive functions and give a certified compiler from $\mu$-recursive functions to Minsky machines.Comment: submitted to LMC