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Damaris Schindler

Publications and source records attributed to Damaris Schindler.

25 records · Page 2Linked to original sources
arXiv2015

On the number of certain Del Pezzo surfaces of degree four violating the Hasse principle

We give an asymptotic expansion for the density of del Pezzo surfaces of degree four in a certain Birch Swinnerton-Dyer family violating the Hasse principle due to a Brauer-Manin obstruction. Under the assumption of Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups for elliptic curves, we obtain an asymptotic formula for the number of all del Pezzo surfaces in the family, which violate the Hasse principle.

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arXiv2015

On the Brauer-Manin obstruction for degree four del Pezzo surfaces

We show that, for every integer $1 \leq d \leq 4$ and every finite set $S$ of places, there exists a degree $d$ del Pezzo surface $X$ over ${\mathbb Q}$ such that ${\rm Br}(X)/{\rm Br}({\mathbb Q}) \cong {\mathbb Z}/2{\mathbb Z}$ and the Brauer-Manin obstruction works exactly at the places in $S$. For $d = 4$, we prove that in all cases, with the exception of $S = \{\infty\}$, this surface may be chosen diagonalizably over ${\mathbb Q}$.

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arXiv2014

Counting rational points on hypersurfaces and higher order expansions

We study the number of representations of an integer n=F(x_1,...,x_s) by a homogeneous form in sufficiently many variables. This is a classical problem in number theory to which the circle method has been succesfully applied to give an asymptotic for the number of such representations where the integer vector (x_1,...,x_s) is restricted to a box of side length P for P sufficiently large. In the special case of Waring's problem, Vaughan and Wooley have recently established for the first time a higher order expansion for the corresponding asymptotic formula. Via a different and much more general approach we derive a multi-term asymptotic for this problem for general forms F(x_1,...,x_s) and give an interpretation for the occurring lower order terms. As an application we derive higher order expansions for the number of rational points of bounded anticanonical height on the projective hypersurface F(x_1,...,x_s)=0 for forms F in sufficiently many variables.

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arXiv2014

A variant of Weyl's inequality for systems of forms and applications

We give a variant of Weyl's inequality for systems of forms together with applications. First we use this to give a different formulation of a theorem of B. J. Birch on forms in many variables. More precisely, we show that the dimension of the locus V^* introduced in this work can be replaced by the maximal dimension of the singular loci of forms in the linear system of the given forms. In some cases this improves on the aforementioned theorem of Birch. We say that a system of forms is a Hardy-Littlewood system if the number of integer points on the corresponding variety restricted to a box satisfies the asymptotic behaviour predicted by the classical circle method. As a second application, we improve on a theorem of W. M. Schmidt which states that a system of homogeneous forms of same degree is a Hardy-Littlewood system as soon as the so called $h$-invariant of the system is sufficiently large. In this direction we generalise previous improvements of R. Dietmann on systems of quadratic and cubic forms to systems of forms of general degree.

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arXiv2013

Norms as products of linear polynomials

Let F be a number field, and let F\subset K be a field extension of degree n. Suppose that we are given 2r sufficiently general linear polynomials in r variables over F. Let X be the variety over F such that the F-points of X bijectively correspond to the representations of the product of these polynomials by a norm from K to F. Combining the circle method with descent we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X.

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arXiv2013

Bihomogeneous forms in many variables

We count integer points on bihomogeneous varieties using the Hardy-Littlewood method. The main novelty lies in using the structure of bihomogeneous equations to obtain asymptotics in generically fewer variables than would be necessary in using the standard approach for homogeneous varieties. Also, we consider counting functions where not all the variables have to lie in intervals of the same size, which arises as a natural question in the setting of bihomogeneous varieties.

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