Results 11 to 20 of about 339 (22)
Rational points and non-anticanonical height functions [PDF]
, 2019 A conjecture of Batyrev and Manin predicts the asymptotic behaviour of rational points of bounded height on smooth projective varieties over number fields.
Frei, Christopher, Loughran, Daniel
core +2 more sources
On Selmer groups of abelian varieties over $\ell$-adic Lie extensions of global function fields [PDF]
, 2013 Let $F$ be a global function field of characteristic $p>0$ and $A/F$ an abelian variety. Let $K/F$ be an $\l$-adic Lie extension ($\l\neq p$) unramified outside a finite set of primes $S$ and such that $\Gal(K/F)$ has no elements of order $\l$.
Bandini, Andrea, Valentino, Maria
core +2 more sources
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, SigmaThe 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 +3 more
doaj +1 more source
Average Analytic Ranks of Elliptic Curves over Number Fields
Forum of Mathematics, SigmaWe 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
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 +4 more
core +2 more sources
, 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
Improvements on dimension growth results and effective Hilbert’s irreducibility theorem
Forum of Mathematics, SigmaWe 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 +4 more
doaj +1 more source
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
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