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We construct del Pezzo surfaces of degree $4$ violating the Hasse principle explained by the Brauer-Manin obstruction. Using these del Pezzo surfaces, we show that there are algebraic families of $K 3$ surfaces violating the Hasse principle explained by the Brauer-Manin obstruction. Various examples are given.

The arithmetic of curves defined by iteration

Wade Hindes (2015)

Acta Arithmetica

We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...

The arithmetic of Fermat curves.

William G. McCallum (1992)

Mathematische Annalen

The Brauer group of torsors and its arithmetic applications

David Harari, Alexei N. Skorobogatov (2003)

Annales de l'Institut Fourier

Let $X$ be an algebraic variety defined over a field $k$ of characteristic $0$ , and let $Y$ be an $X$ -torsor under a torus. We compute the Brauer group of $Y$ . In the case of a number field $k$ we deduce results concerning the arithmetic of $X$ .

The canonical height and integral points on elliptic curves.

J.H. Silverman, M. Hindry (1988)

Inventiones mathematicae

The density of rational and integral points on algebraic varieties

M. L. Brown (1992)

Annales scientifiques de l'École Normale Supérieure

The field of definition of the Mordell-Weil group of an elliptic curve over a function field

Masato Kuwata (1990)

Compositio Mathematica

The groups of points on abelian varieties over finite fields

Sergey Rybakov (2010)

Open Mathematics

Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f A(1 − t).

The Hasse zeta function of a K3 surface related to the number of words of weight 5 in the Melas codes.

C. Peters, M. van der Vlugt, J. Top (1992)

Journal für die reine und angewandte Mathematik

The integral points on elliptic curves $y^{2} = x^{3} + (36 n^{2} - 9) x - 2 (36 n^{2} - 5)$

Hai Yang, Ruiqin Fu (2013)

Czechoslovak Mathematical Journal

Let $n$ be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if $n > 1$ and both $6 n^{2} - 1$ and $12 n^{2} + 1$ are odd primes, then the general elliptic curve $y^{2} = x^{3} + (36 n^{2} - 9) x - 2 (36 n^{2} - 5)$ has only the integral point $(x, y) = (2, 0)$ . By this result we can get that the above elliptic curve has only the trivial integral point for $n = 3, 13, 17$ etc. Thus it can be seen that the elliptic curve $y^{2} = x^{3} + 27 x - 62$ really is an unusual elliptic curve which has large integral points.

The Modell-Weil groups of unirational quasi-elliptic surfaces in characteristic 3.

Hiroyuki Ito (1992)

Mathematische Zeitschrift

The modified diagonal cycle on the triple product of a pointed curve

Benedict H. Gross, Chad Schoen (1995)

Annales de l'institut Fourier

Let $X$ be a curve over a field $k$ with a rational point $e$ . We define a canonical cycle $Δ_{e} \in Z^{2} (X^{3})_{hom}$ . Suppose that $k$ is a number field and that $X$ has semi-stable reduction over the integers of $k$ with fiber components non-singular. We construct a regular model of $X^{3}$ and show that the height pairing $⟨ τ_{*} (Δ_{e}), τ_{*}^{'} (Δ_{e}) ⟩$ is well defined where $τ$ and $τ^{'}$ are correspondences. The paper ends with a brief discussion of heights and $L$ -functions in the case that $X$ is a modular curve.

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Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication.

The absolute Galois group of a pseudo real closed field

The algebraic groups leading to the Roth inequalities

The Algebraic nature of the elementary theory of PRC fields.

The arithmetic of certain del Pezzo surfaces and K3 surfaces