13 citations · 78 across the 26 of their papers we have counts for
32 papers
Explicit Brauer-Manin obstructions on plane quartics
Nils Bruin, Brendan Creutz
We describe a method to show a plane quartic over a number field has no rational points. The method can be adapted to show that a curve does not have divisors of degree 1 or 2 and…
Adelic Mordell-Lang and the Brauer-Manin obstruction
Brendan Creutz
Let be a closed subvariety of an abelian variety over a global function field such that the base change of to an algebraic closure does not have any positive dimens…
Quartic del Pezzo surfaces without quadratic points
Brendan Creutz, Bianca Viray
Previous work of the authors showed that every quartic del Pezzo surface over a number field has index dividing (i.e., has a closed point of degree modulo ),, and asked…
Galois invariants of finite abelian descent and Brauer sets
Brendan Creutz, Jesse Pajwani, Jose Felipe Voloch
For a variety over a global field, one can consider subsets of the set of adelic points of the variety cut out by finite abelian descent or Brauer-Manin obstructions. Given a Galoi…
Weak approximation versus the Hasse principle for subvarieties of abelian varieties
Brendan Creutz
For varieties over global fields, weak approximation in the space of adelic points can fail. For a subvariety of an abelian variety one expects this failure is always explained by…
The Brauer-Manin obstruction for nonisotrivial curves over global function fields
Brendan Creutz, José Felipe Voloch
We prove that the set of rational points on a nonisotrivial curves of genus at least 2 over a global function field is equal to the set of adelic points cut out by the Brauer-Manin…
Degrees of points on varieties over Henselian fields
Brendan Creutz, Bianca Viray
Let be a nonempty scheme over the field of fractions of a Henselian local ring . A result of Gabber, Liu and Lorenzini shows that the GCD of the set of degrees of closed p…
Etale descent obstruction and anabelian geometry of curves over finite fields
Brendan Creutz, Jose Felipe Voloch
Let and be smooth, proper and geometrically integral curves over a finite field . Any morphism from to induces a morphism of their étale fundamental groups. The…
The local-global principle for divisibility in CM elliptic curves
Brendan Creutz, Sheng Lu
We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime we give sharp lower bounds…
Brauer-Manin obstructions on hyperelliptic curves
Brendan Creutz, Duttatrey Nath Srivastava
We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advant…
Quadratic points on intersections of two quadrics
Brendan Creutz, Bianca Viray
We prove that a smooth complete intersection of two quadrics of dimension at least over a number field has index dividing , i.e., that it possesses a rational -cycle of d…
The Brauer-Manin obstruction for constant curves over global function fields
Brendan Creutz, José Felipe Voloch
Let be a finite field and smooth, geometrically irreducible proper curves over and set . We consider Brauer-Manin and abelian des…
Tate-Shafarevich groups of constant elliptic curves and isogeny volcanos
Brendan Creutz, Jose Felipe Voloch
We describe the structure of Tate-Shafarevich groups of a constant elliptic curves over function fields by exploiting the volcano structure of isogeny graphs of elliptic curves ove…
Three-dimensional connected groups of automorphisms of toroidal circle planes
Brendan Creutz, Duy Ho, Günter F. Steinke
We contribute to the classification of toroidal circle planes and flat Minkowski planes possessing three-dimensional connected groups of automorphisms. When such a group is an almo…
The d-primary Brauer-Manin obstruction for Curves
Brendan Creutz, Bianca Viray, José Felipe Voloch
For a curve over a global field we consider for which integers d the d-primary part of the Brauer group can obstruct the existence of rational points. We give examples showing it i…
No transcendental Brauer-Manin obstructions on abelian varieties
Brendan Creutz
Suppose is a torsor under an abelian variety over a number field. We show that any adelic point of that is orthogonal to the algebraic Brauer group of is orthogonal…
On automorphism groups of toroidal circle planes
Brendan Creutz, Duy Ho, Günter F. Steinke
Schenkel proved that the automorphism group of a flat Minkowski plane is a Lie group of dimension at most 6 and described planes whose automorphism group has dimension at least 4 o…
Improved rank bounds from 2-descent on hyperelliptic Jacobians
Brendan Creutz
We describe a qualitative improvement to the algorithms for performing 2-descents to obtain information regarding the Mordell-Weil rank of a hyperelliptic Jacobian. The improvement…
Degree and the Brauer-Manin obstruction
Brendan Creutz, Bianca Viray
Let X be a smooth variety over a number field k embedded as a degree d subvariety of and suppose that X is a counterexample to the Hasse principle explained by the B…
Zero-cycles of degree one on Skorobogatov's bielliptic surface
Brendan Creutz
Skorobogatov constructed a bielliptic surface which is a counterexample to the Hasse principle not explained by the Brauer-Manin obstruction. We show that this surface has a -cy…
Local-global principles for Weil-Châtelet divisibility in positive characteristic
Brendan Creutz, José Felipe Voloch
We extend existing results characterizing Weil-Châtelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of…
Generalized Jacobians and explicit descents
Brendan Creutz
We develop a cohomological description of various explicit descents in terms of generalized Jacobians, generalizing the known description for hyperelliptic curves. Specifically, gi…
Most binary forms come from a pencil of quadrics
Brendan Creutz
A pair of symmetric bilinear forms A and B determine a binary form . We prove that the question of whether a given binary form can be written in this way as a…
Relative Brauer groups of torsors of period two
Brendan Creutz
We consider the problem of computing the relative Brauer group of a torsor of period 2 under an elliptic curve E. We show how this problem can be reduced to finding a set of genera…