Results 11 to 20 of about 667 (60)
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
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
wiley +1 more source
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
wiley +1 more source
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
wiley +1 more source
How to generate all integral triangles containing a given angle
International Journal of Mathematics and Mathematical Sciences, Volume 24, Issue 8, Page 569-572, 2000., 2000 We present an elementary prescription based on the rational secant method for generating all the integral triangles containing a given angle of rational cosine. This is a direct generalization of the ancient problem of finding all the Pythagorean triples. As an example, we discuss a specific equation studied by Diophantus of Alexandria, which turns out
Nelson Petulante, Ifeoma Kaja
wiley +1 more source
$E_{6}$ AND THE ARITHMETIC OF A FAMILY OF NON-HYPERELLIPTIC CURVES OF GENUS 3
Forum of Mathematics, Pi, 2015 We study the arithmetic of a family of non-hyperelliptic curves of genus 3 over the field $\mathbb{Q}$ of rational numbers. These curves are the nearby fibers of the semi-universal deformation of a simple singularity of type $E_{6}$. We show that average
JACK A. THORNE
doaj +1 more source
Sums and differences of four k-th powers [PDF]
, 2009 We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent results by ...
C. Hooley +11 more
core +2 more sources
NON-ARCHIMEDEAN YOMDIN–GROMOV PARAMETRIZATIONS AND POINTS OF BOUNDED HEIGHT
Forum of Mathematics, Pi, 2015 We prove an analog of the Yomdin–Gromov lemma for $p$-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected ...
RAF CLUCKERS +2 more
doaj +1 more source
The set of non-squares in a number field is diophantine
, 2007 Fix a number field k. We prove that k* - k*^2 is diophantine over k. This is deduced from a theorem that for a nonconstant separable polynomial P(x) in k[x], there are at most finitely many a in k* modulo squares such that there is a Brauer-Manin ...
Poonen, Bjorn
core +1 more source
Rational torsion points on Jacobians of modular curves
, 2015 Let $p$ be a prime greater than 3. Consider the modular curve $X_0(3p)$ over $\mathbb{Q}$ and its Jacobian variety $J_0(3p)$ over $\mathbb{Q}$. Let $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ be the group of rational torsion points on $J_0(3p)$ and the ...
Yoo, Hwajong
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