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The arithmetic of certain del Pezzo surfaces and K3 surfaces

Dong Quan Ngoc Nguyen (2012)

Journal de Théorie des Nombres de Bordeaux

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 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 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 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 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 ∈ 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.

Currently displaying 341 – 360 of 397