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Let $K$ be a number field, let $phi in K(t)$ be a rational map of degree at least 2, and let $alpha, beta in K$. We show that if $alpha$ is not in the forward orbit of $beta$, then there is a positive proportion of primes ${mathfrak p}$ of $K$ such that $alpha mod {mathfrak p}$ is not in the forward orbit of $beta mod {mathfrak p}$. Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace $alpha$ by a hypersurface, such as the ramification locus of a morphism $phi : {mathbb P}^{n} to {mathbb P}^{n}$.
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Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is perfect and that $mathop{textrm{char}} k eq 2$. Assume that the Weierstrass points of $C$ are $K$-rational. Let $S = mathop{textrm{Spec}} R$. Let $mathcal{X}$ be the minimal proper regular model of $C$ over $S$. Let $mathop{textrm{Art}} (mathcal{X}/S)$ denote the Artin conductor of the $S$-scheme $mathcal{X}$ and let $ u (Delta)$ denote the minimal discriminant of $C$. We prove that $-mathop{textrm{Art}} (mathcal{X}/S) leq u (Delta)$. As a corollary, we obtain that the number of components of the special fiber of $mathcal{X}$ is bounded above by $ u(Delta)+1$.
We construct higher-dimensional Calabi-Yau varieties defined over a given number field with Zariski dense sets of rational points. We give two elementary constructions in arbitrary dimensions as well as another construction in dimension three which involves certain Calabi-Yau threefolds containing an Enriques surface. The constructions also show that potential density holds for (sufficiently) general members of the families.
This paper treats the dominant rational maps from the product of two very general curves to nonsingular projective surfaces. Combining the result by Bastianelli and Pirola, we prove that the product of two very general curves of genus $ggeq 7$ and $ggeq 3$ does not admit dominant rational maps of degree $> 1$ if the image surface is non-ruled. We also treat the case of the 2-symmetric product of a curve.
Let $S$ be the special fibre of a Shimura variety of Hodge type, with good reduction at a place above $p$. We give an alternative construction of the zip period map for $S$, which is used to define the Ekedahl-Oort strata of $S$. The method employed is local, $p$-adic, and group-theoretic in nature.
Let $p>3$ be a prime, and let $(frac{cdot}p)$ be the Legendre symbol. Let $binmathbb Z$ and $varepsilonin{pm 1}$. We mainly prove that $$left|left{N_p(a,b): 1<a<p text{and} left(frac apright)=varepsilonright}right|=frac{3-(frac{-1}p)}2,$$ where $N_p(a,b)$ is the number of positive integers $x<p/2$ with ${x^2+b}_p>{ax^2+b}_p$, and ${m}_p$ with $minmathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.
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