Results 11 to 20 of about 339 (22)

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

The Manin–Peyre conjecture for smooth spherical Fano varieties of semisimple rank one

Forum of Mathematics, Sigma
The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.
Valentin Blomer, Jörg Brüdern, Ulrich Derenthal, Giuliano Gagliardi   +3 more
doaj   +1 more source

Average Analytic Ranks of Elliptic Curves over Number Fields

Forum of Mathematics, Sigma
We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are modular and have L-functions which satisfy the ...
Tristan Phillips
doaj   +1 more source

Probabilistic Galois Theory

, 2011
We show that there are at most $O_{n,\epsilon}(H^{n-2+\sqrt{2}+\epsilon})$ monic integer polynomials of degree $n$ having height at most $H$ and Galois group different from the full symmetric group $S_n$, improving on the previous 1973 world record $O_{n}
Dietmann, Rainer
core   +1 more source

On the computation of the Picard group for $K3$ surfaces

, 2010
We construct examples of $K3$ surfaces of geometric Picard rank $1$. Our method is a refinement of that of R. van Luijk. It is based on an analysis of the Galois module structure on \'etale cohomology.
ANDREAS-STEPHAN ELSENHANS, Beauville, Elsenhans, JÖRG JAHNEL, Milne   +4 more
core   +2 more sources

Improvements on dimension growth results and effective Hilbert’s irreducibility theorem

Forum of Mathematics, Sigma
We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree d, over any global field.
Raf Cluckers, Pierre Dèbes, Yotam I. Hendel, Kien Huu Nguyen, Floris Vermeulen   +4 more
doaj   +1 more source

Constructions of diagonal quartic and sextic surfaces with infinitely many rational points

, 2014
In this note we construct several infinite families of diagonal quartic surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where $a,b,c,d\in\Z\setminus\{0\}$ with infinitely many rational points and satisfying the condition $abcd\neq ...
Ajai Choudhry, Andrew Bremner, Dem'janenko V. A., Elkies N. D., Elkies N. D., Maciej Ulas, Mordell L. J.   +6 more
core   +1 more source

Brauer Groups and Tate-Shafarevich Groups [PDF]

, 2003
Let XK be a proper, smooth and geometrically connected curve over a global field K. In this paper we generalize a formula of Milne relating the order of the Tate-Shafarevich group of the Jacobian of XK to the order of the Brauer group of a proper regular
Gonzalez-Aviles, Cristian D
core   +4 more sources

Rational points on varieties and Morita equivalences of $C^*$-algebras

, 2020
Let $V(k)$ be a projective variety over a number field $k\subset\mathbf{C}$ and let $\mathscr{A}_V$ be the Serre $C^*$-algebra of $V(k)$. We construct a functor $F: V(k)\mapsto \mathscr{A}_V$, such that the $\mathbf{C}$-isomorphic ($k$-isomorphic, resp.)
Nikolaev, Igor
core  

Integral points of bounded height on a log Fano threefold

, 2019
We determine an asymptotic formula for the number of integral points of bounded height on a blow-up of $\mathbb{P}^3$ outside certain planes using universal torsors.Comment: 18 ...
Wilsch, Florian
core