Results 21 to 30 of about 3,799,599 (344)
Quartic and Quintic Hypersurfaces with Dense Rational Points
Forum of Mathematics, Sigma, 2023 Let $X_4\subset \mathbb {P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field k. We show that if either $X_4$ contains a linear subspace $\Lambda $ of dimension $h\geq \max \{2,\dim (\Lambda ...
Alex Massarenti
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Rational points on a certain genus 2 curve
Comptes Rendus. Mathématique, 2023 We give a correct proof to the fact that all rational points on the curve \[ y^2=(x^2+1)(x^2+3)(x^2+7) \] are $\pm \infty $ and $(\pm 1,\,\pm 8)$. This corrects previous works of Cohen [3] and Duquesne [4, 5].
Nguyen, Xuan Tho
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Rational Singularities and Rational Points [PDF]
Pure and Applied Mathematics Quarterly, 2008 If $X$ is a projective, geometrically irreducible variety defined over a finite field $\F_q$, such that it is smooth and its Chow group of 0-cycles fulfills base change, i.e. $CH_0(X\times_{\F_q}\bar{\F_q(X)})=\Q$, then the second author's theorem asserts that its number of rational points satisfies $|X(\F_q)| \equiv 1$ modulo $q$. If $X$ is not smooth,
Blickle, Manuel, Esnault, Hélène
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On rational points in CFT moduli spaces
Journal of High Energy Physics, 2021 Motivated by the search for rational points in moduli spaces of two-dimensional conformal field theories, we investigate how points with enhanced symmetry algebras are distributed there.
Nathan Benjamin +3 more
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Uniform Bound for the Number of Rational Points on a Pencil of Curves [PDF]
International mathematics research notices, 2019 Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell conjecture, a theorem of Faltings.
V. Dimitrov, Ziyang Gao, P. Habegger
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Primitive rational points on expanding horocycles in products of the modular surface with the torus [PDF]
Ergodic Theory and Dynamical Systems, 2019 We prove effective equidistribution of primitive rational points and of primitive rational points defined by monomials along long horocycle orbits in products of the torus and the modular surface.
M. Einsiedler, M. Luethi, N. Shah
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Frobenian multiplicative functions and rational points in fibrations [PDF]
Journal of the European Mathematical Society (Print), 2019 We consider the problem of counting the number of varieties in a family over $\mathbb{Q}$ with a rational point. We obtain lower bounds for this counting problem for some families over $\mathbb{P}^1$, even if the Hasse principle fails.
D. Loughran, Lilian Matthiesen
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Rational points in the moduli space of genus two [PDF]
, 2019 We build a database of genus 2 curves defined over $\mathbb Q$ which contains all curves with minimal absolute height $h \leq 5$, all curves with moduli height $\mathfrak h \leq 20$, and all curves with extra automorphisms in standard form $y^2=f(x^2 ...
Tanush Shaska +5 more
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Families of polynomials of every degree with no rational preperiodic points
Comptes Rendus. Mathématique, 2021 Let $K$ be a number field. Given a polynomial $f(x)\in K[x]$ of degree $d\ge 2$, it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $[K:\mathbb{Q}]$.
Sadek, Mohammad
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Heights on stacks and a generalized Batyrev–Manin–Malle conjecture
Forum of Mathematics, Sigma, 2023 We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties.
Jordan S. Ellenberg +2 more
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