Results 1 to 10 of about 27 (27)
E8 and the average size of the 3‐Selmer group of the Jacobian of a pointed genus‐2 curve
Proceedings of the London Mathematical Society, Volume 122, Issue 5, Page 678-723, May 2021., 2021 Abstract We prove that the average size of the 3‐Selmer group of a genus‐2 curve with a marked Weierstrass point is 4. We accomplish this by studying rational and integral orbits in the representation associated to a stably Z/3Z‐graded simple Lie algebra of type E8.
Beth Romano, Jack A. Thorne
wiley +1 more source
The most efficient indifferentiable hashing to elliptic curves of j-invariant 1728
Journal of Mathematical Cryptology, 2022 This article makes an important contribution to solving the long-standing problem of whether all elliptic curves can be equipped with a hash function (indifferentiable from a random oracle) whose running time amounts to one exponentiation in the basic ...
Koshelev Dmitrii
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Arithmetic hyperbolicity and a stacky Chevalley–Weil theorem
Journal of the London Mathematical Society, Volume 103, Issue 3, Page 846-869, April 2021., 2021 Abstract We prove an analogue for algebraic stacks of Hermite–Minkowski's finiteness theorem from algebraic number theory, and establish a Chevalley–Weil type theorem for integral points on stacks. As an application of our results, we prove analogues of the Shafarevich conjecture for some surfaces of general type.
Ariyan Javanpeykar, Daniel Loughran
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RATIONAL CURVES ON CUBIC HYPERSURFACES OVER FINITE FIELDS
Mathematika, Volume 67, Issue 2, Page 366-387, April 2021., 2021 Abstract Given a smooth cubic hypersurface X over a finite field of characteristic greater than 3 and two generic points on X, we use a function field analogue of the Hardy–Littlewood circle method to obtain an asymptotic formula for the number of degree d k‐rational curves on X passing through those two points.
Adelina Mânzăţeanu
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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 $h\geq \max \{2,\dim (\Lambda ...
Alex Massarenti
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Point counting for foliations over number fields
Forum of Mathematics, Pi, 2022 Let${\mathbb M}$ be an affine variety equipped with a foliation, both defined over a number field ${\mathbb K}$. For an algebraic $V\subset {\mathbb M}$ over ${\mathbb K}$, write $\delta _{V}$ for the maximum of the degree and log-height of V.
Gal Binyamini
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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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Hensel minimality II: Mixed characteristic and a diophantine application
Forum of Mathematics, Sigma, 2023 In this paper, together with the preceding Part I [10], we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to [10] the treatment of the mixed characteristic case.
Raf Cluckers +3 more
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Nonspecial varieties and generalised Lang–Vojta conjectures
Forum of Mathematics, Sigma, 2021 We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called
Erwan Rousseau +2 more
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Brauer–Manin obstruction for Erdős–Straus surfaces
Bulletin of the London Mathematical Society, Volume 52, Issue 4, Page 746-761, August 2020., 2020 Abstract We study the failure of the integral Hasse principle and strong approximation for the Erdős–Straus conjecture using the Brauer–Manin obstruction.
Martin Bright, Daniel Loughran
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