scholarly journals Counting imaginary quadratic points via universal torsors

2014 ◽  
Vol 150 (10) ◽  
pp. 1631-1678 ◽  
Author(s):  
Ulrich Derenthal ◽  
Christopher Frei
Keyword(s):  
Arbitrary Number ◽  
Number Fields ◽  
Quadratic Fields ◽  
Del Pezzo Surfaces ◽  
Fano Varieties ◽  
Imaginary Quadratic Fields ◽  
Del Pezzo Surface ◽  
Singularity Type ◽  
Del Pezzo ◽  
Manin’S Conjecture

AbstractA conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields$K$for the quartic del Pezzo surface$S$of singularity type${\boldsymbol{A}}_{3}$with five lines given in${\mathbb{P}}_{K}^{4}$by the equations${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$.

scholarly journals On Manin's conjecture for singular del Pezzo surfaces of degree four, II

2007 ◽  
Vol 143 (3) ◽  
pp. 579-605 ◽  
Author(s):  
R. DE LA BRETÈCHE ◽  
T. D. BROWNING
Keyword(s):  
Zeta Function ◽  
Half Plane ◽  
Open Subset ◽  
Height Function ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Del Pezzo ◽  
Manin’S Conjecture ◽  
Height Zeta Function

AbstractThis paper establishes the Manin conjecture for a certain non-split singular del Pezzo surface of degree four$X \subset \bfP^4$. In fact, ifU⊂Xis the open subset formed by deleting the lines fromX, andHis the usual projective height function on$\bfP^4(\Q)$, then the height zeta function$ \sum_{x \in U(\Q)}{H(x)^{-s}} $is analytically continued to the half-plane ℜe(s) > 17/20.

scholarly journals GENERALISED DIVISOR SUMS OF BINARY FORMS OVER NUMBER FIELDS

2017 ◽  
Vol 19 (1) ◽  
pp. 137-173 ◽  
Author(s):  
Christopher Frei ◽  
Efthymios Sofos
Keyword(s):  
Dirichlet Character ◽  
Number Fields ◽  
Del Pezzo Surfaces ◽  
Binary Forms ◽  
Subsequent Work ◽  
Order Of Magnitude ◽  
Del Pezzo ◽  
Divisor Sums ◽  
Manin’S Conjecture ◽  
Sparse Set

Estimating averages of Dirichlet convolutions $1\ast \unicode[STIX]{x1D712}$, for some real Dirichlet character $\unicode[STIX]{x1D712}$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin’s conjecture for Châtelet surfaces. We introduce a far-reaching generalisation of this problem, in particular replacing $\unicode[STIX]{x1D712}$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin’s conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number.

scholarly journals Counting imaginary quadratic points via universal torsors, II

2014 ◽  
Vol 156 (3) ◽  
pp. 383-407 ◽  
Author(s):  
ULRICH DERENTHAL ◽  
CHRISTOPHER FREI
Keyword(s):  
Number Fields ◽  
Del Pezzo Surfaces ◽  
Quadratic Number ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number ◽  
Del Pezzo ◽  
Manin’S Conjecture

AbstractWe prove Manin's conjecture for four singular quartic del Pezzo surfaces over imaginary quadratic number fields, using the universal torsor method.

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

scholarly journals Manin’s conjecture for quartic Del Pezzo surfaces with a conic fibration

2011 ◽  
Vol 160 (1) ◽  
pp. 1-69 ◽  
Author(s):  
R. De la Bretèche ◽  
T. D. Browning
Keyword(s):  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Manin’S Conjecture

scholarly journals Quotients of del Pezzo surfaces

2019 ◽  
Vol 30 (12) ◽  
pp. 1950068
Author(s):  
Andrey Trepalin
Keyword(s):  
Complete Classification ◽  
Finite Subgroup ◽  
Del Pezzo Surfaces ◽  
Picard Number ◽  
Del Pezzo Surface ◽  
Cyclic Groups ◽  
Trivial Group ◽  
Quotient Surface ◽  
Del Pezzo

Let [Formula: see text] be any field of characteristic zero, [Formula: see text] be a del Pezzo surface and [Formula: see text] be a finite subgroup in [Formula: see text]. In this paper, we study when the quotient surface [Formula: see text] can be non-rational over [Formula: see text]. Obviously, if there are no smooth [Formula: see text]-points on [Formula: see text] then it is not [Formula: see text]-rational. Therefore, under assumption that the set of smooth [Formula: see text]-points on [Formula: see text] is not empty we show that there are few possibilities for non-[Formula: see text]-rational quotients. The quotients of del Pezzo surfaces of degree [Formula: see text] and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree [Formula: see text]. We show that they can be non-[Formula: see text]-rational only for the trivial group or cyclic groups of order [Formula: see text], [Formula: see text] and [Formula: see text]. For the trivial group and the group of order [Formula: see text], we show that both [Formula: see text] and [Formula: see text] are not [Formula: see text]-rational if the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. For the groups of order [Formula: see text] and [Formula: see text], we construct examples of both [Formula: see text]-rational and non-[Formula: see text]-rational quotients of both [Formula: see text]-rational and non-[Formula: see text]-rational del Pezzo surfaces of degree [Formula: see text] such that the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. As a result of complete classification of non-[Formula: see text]-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of [Formula: see text]-rational surfaces, and obtain some corollaries concerning fields of invariants of [Formula: see text].

scholarly journals Cylinders in singular del Pezzo surfaces

2016 ◽  
Vol 152 (6) ◽  
pp. 1198-1224 ◽  
Author(s):  
Ivan Cheltsov ◽  
Jihun Park ◽  
Joonyeong Won
Keyword(s):  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Open Set ◽  
Del Pezzo

For each del Pezzo surface $S$ with du Val singularities, we determine whether it admits a $(-K_{S})$-polar cylinder or not. If it allows one, then we present an effective $\mathbb{Q}$-divisor $D$ that is $\mathbb{Q}$-linearly equivalent to $-K_{S}$ and such that the open set $S\setminus \text{Supp}(D)$ is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit non-trivial $\mathbb{G}_{a}$-actions on their affine cones defined by their anticanonical divisors.

scholarly journals Equivariant Compactifications of Two-Dimensional Algebraic Groups

2014 ◽  
Vol 58 (1) ◽  
pp. 149-168 ◽  
Author(s):  
Ulrich Derenthal ◽  
Daniel Loughran
Keyword(s):  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Rational Points ◽  
Del Pezzo Surfaces ◽  
Direct Products ◽  
Two Dimensional ◽  
Del Pezzo ◽  
Manin’S Conjecture ◽  
Transitive Actions

AbstractWe classify generically transitive actions of semi-direct products on ℙ2. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for semi-direct products .

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