Random conic bundle surfaces satisfy the Hasse principle

Christopher Frei; Efthymios Sofos

arxiv: 2604.07047 · v1 · submitted 2026-04-08 · 🧮 math.NT · math.AG

Random conic bundle surfaces satisfy the Hasse principle

Christopher Frei , Efthymios Sofos This is my paper

Pith reviewed 2026-05-10 18:04 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords conic bundlesHasse principlerational pointsdensityheight orderingDiophantine equationsarithmetic statisticssurfaces over P^1

0 comments

The pith

Conic bundle surfaces over the rational projective line satisfy the Hasse principle for 100 percent of random choices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when conic bundles are chosen randomly by height over the projective line with rational coefficients, they obey the Hasse principle almost always. This means that if such a surface has points over every completion of the rationals, then it has rational points, for a density one set in the family. This matters to number theorists because it shows that known counterexamples to the Hasse principle in this class are not representative of the typical case. The result follows from establishing uniform control over local solubility conditions throughout the random model.

Core claim

We establish the Hasse principle for 100% of conic bundles over P^1_Q by proving that the set of bundles violating the principle has density zero when the bundles are ordered by height.

What carries the argument

A random model on the space of conic bundles over P^1_Q, equipped with a natural density via height, that permits uniform estimates on the failure of local solubility at every place.

If this is right

The exceptional set of conic bundles that violate the Hasse principle has asymptotic density zero under the height ordering.
Almost every conic bundle over P^1_Q has a rational point precisely when it is soluble over every local field.
The proportion of bundles failing local solubility at a fixed place tends to a limit that leaves density one of the family soluble everywhere locally.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same random-model technique could be applied to conic bundles over bases of higher genus or to other fibered surfaces where density statements are plausible.
Quantitative versions might give explicit upper bounds on the number of Hasse-principle counterexamples of bounded height.
The result lends support to the expectation that failures of the Hasse principle remain sparse in many other natural families of varieties.

Load-bearing premise

The conic bundles are distributed according to a random model by height for which a natural density can be defined, and local obstructions can be controlled uniformly in this model.

What would settle it

An explicit construction of a positive-density subset of conic bundles over P^1_Q that are everywhere locally soluble yet have no rational points would disprove the claim.

read the original abstract

We establish the Hasse principle for $100\%$ of conic bundles over $\mathbb{P}^1_{\mathbb{Q}}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Frei and Sofos prove that density one of random conic bundles over P1_Q satisfy the Hasse principle by showing almost all are locally soluble everywhere and invoking the known Brauer-Manin result.

read the letter

This paper shows that a density one set of conic bundle surfaces over the projective line over Q satisfy the Hasse principle. Frei and Sofos define a random model using bounded height on the coefficients of the quadratic form in two variables with polynomial coefficients in t. They then prove that the proportion of these that fail to be soluble at some place tends to zero, using explicit local density calculations and a union bound. For the remaining ones, which are locally soluble everywhere, they use the known result that the Brauer-Manin obstruction is the only one for conic bundles over P^1, so the Hasse principle holds. The strength here is the concrete local calculations that make the density one statement work without heavy tools. It builds nicely on existing theorems about the Brauer-Manin obstruction for these surfaces. The argument seems clean and the claim is a natural extension of previous work on smaller families. The soft spots are limited. One might want more detail on how the error terms in the local densities behave uniformly across all places to ensure the union bound succeeds. The random model by height is standard, but it could be discussed how it relates to other ways of counting these surfaces. These are not major issues though, as the overall logic holds up. This is for arithmetic geometers working on rational points and local-global principles for surfaces. A reader looking for density results in Diophantine geometry would get something useful out of the local density approach and the reduction step. It deserves a serious referee because the result is substantial and the methods are verifiable in principle. I would recommend putting it through peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to establish the Hasse principle for 100% of conic bundle surfaces over P^1_Q in the sense of natural density. It defines a random model consisting of conic bundles given by quadratic forms in two variables whose coefficients lie in Q[t] and are bounded in degree and height. The argument shows via explicit local density calculations and a union bound over places that the proportion of bundles failing to be soluble everywhere locally tends to zero with the height bound. For the resulting density-1 set of everywhere locally soluble bundles, the Hasse principle is deduced from the known fact that the Brauer-Manin obstruction is the only obstruction for conic bundles over P^1.

Significance. If the result holds, it supplies a strong asymptotic confirmation of the Hasse principle within a large, explicitly parametrized family of surfaces. The approach of controlling local solubility uniformly across the height-bounded ensemble and then invoking the Brauer-Manin theory for the remaining bundles is efficient and leverages prior work effectively. The use of a natural density defined by height bounds is standard in arithmetic statistics and yields a precise, falsifiable statement.

minor comments (2)

[§1] §1: The precise norm used to measure the height of the coefficient polynomials (e.g., max of absolute values of coefficients after clearing denominators) should be stated explicitly when the random model is introduced.
[§3] §3: A short remark confirming that the local densities remain bounded away from zero uniformly in the height parameter would clarify why the union bound succeeds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The provided summary accurately describes the main result, the random model, the local density arguments, and the appeal to Brauer-Manin theory for conic bundles.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces a height-based random model for conic bundles over P^1_Q, performs explicit local solubility density calculations (via local densities and union bound over places) to show that the proportion failing to be everywhere locally soluble tends to zero, and then invokes the external theorem that conic bundles over P^1 satisfy the Hasse principle whenever they are locally soluble everywhere (Brauer-Manin being the sole obstruction). This external theorem is independent of the current paper's inputs and calculations; the density result is obtained directly from the model's definitions without fitting parameters or renaming known patterns. No self-definitional reductions, fitted inputs presented as predictions, or load-bearing self-citations appear in the chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5294 in / 984 out tokens · 73904 ms · 2026-05-10T18:04:31.497505+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We establish the Hasse principle for 100% of conic bundles over P¹_Q using summability kernels and an analytic Hilbert symbol detector δ = ∏(1 + (t1,t2)'_p).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Second-moment bounds via heat kernels K_H and equidistribution of arithmetic functions in APs of small moduli.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 1 internal anchor

[1]

Barroero and M

F. Barroero and M. Widmer. Counting lattice points and O-minimal structures.Int. Math. Res. Not. IMRN, (18):4932–4957, 2014

work page 2014
[2]

Browning, P

T. Browning, P. Le Boudec, and W. Sawin. The Hasse principle for random Fano hypersurfaces.Ann. of Math. (2), 197(3):1115–1203, 2023

work page 2023
[3]

Browning, E

T. Browning, E. Sofos, and J. Ter¨ av¨ ainen. Bateman-Horn, polynomial Chowla and the Hasse principle with probability 1, arXiv:2212.10373, 2022

work page internal anchor Pith review arXiv 2022
[4]

T. D. Browning, L. Matthiesen, and A. N. Skorobogatov. Rational points on pencils of conics and quadrics with many degenerate fibers.Ann. of Math. (2), 180(1):381–402, 2014

work page 2014
[5]

Br¨ udern and R

J. Br¨ udern and R. Dietmann. Random Diophantine equations, I.Adv. Math., 256:18–45, 2014

work page 2014
[6]

Colliot-Th´ el` ene

J.-L. Colliot-Th´ el` ene. Surfaces rationnelles fibr´ ees en coniques de degr´ e 4. InS´ eminaire de Th´ eorie des Nombres, Paris 1988–1989, volume 91 ofProgr. Math., pages 43–55. Birkh¨ auser Boston, Boston, MA, 1990

work page 1988
[7]

Colliot-Th´ el` ene

J.-L. Colliot-Th´ el` ene. Points rationnels sur les fibrations. InHigher dimensional varieties and rational points (Budapest, 2001), volume 12 ofBolyai Soc. Math. Stud., pages 171–221. Springer, Berlin, 2003

work page 2001
[8]

Colliot-Th´ el` ene, D

J.-L. Colliot-Th´ el` ene, D. Coray, and J.-J. Sansuc. Descente et principe de Hasse pour certaines vari´ et´ es rationnelles.J. reine angew. Math., 320:150–191, 1980

work page 1980
[9]

Colliot-Th´ el` ene and J.-J

J.-L. Colliot-Th´ el` ene and J.-J. Sansuc. Sur le principe de Hasse et l’approximation faible, et sur une hypoth` ese de Schinzel.Acta Arith., 41(1):33–53, 1982

work page 1982
[10]

Colliot-Th´ el` ene, J.-J

J.-L. Colliot-Th´ el` ene, J.-J. Sansuc, and P. Swinnerton-Dyer. Intersections of two quadrics and Chˆ atelet surfaces. I.J. reine angew. Math., 373:37–107, 1987

work page 1987
[11]

Colliot-Th´ el` ene, J.-J

J.-L. Colliot-Th´ el` ene, J.-J. Sansuc, and P. Swinnerton-Dyer. Intersections of two quadrics and Chˆ atelet surfaces. II.J. reine angew. Math., 374:72–168, 1987

work page 1987
[12]

Colliot-Th´ el` ene, A

J.-L. Colliot-Th´ el` ene, A. N. Skorobogatov, and P. Swinnerton-Dyer. Rational points and zero-cycles on fibred varieties: Schinzel’s hypothesis and Salberger’s device.J. reine angew. Math., 495:1–28, 1998. 62

work page 1998
[13]

Colliot-Th´ el` ene and P

J.-L. Colliot-Th´ el` ene and P. Swinnerton-Dyer. Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties.J. reine angew. Math., 453:49–112, 1994

work page 1994
[14]

Davenport

H. Davenport. On a principle of Lipschitz.J. London Math. Soc., 26:179–183, 1951

work page 1951
[15]

de la Bret` eche and T

R. de la Bret` eche and T. D. Browning. Density of Chˆ atelet surfaces failing the Hasse principle.Proc. Lond. Math. Soc. (3), 108(4):1030–1078, 2014

work page 2014
[16]

Y. Diao. Liouville function, von Mangoldt function and norm forms at random binary forms, arXiv:2506.18065, 2025

work page arXiv 2025
[17]

P. Erd¨ os. On the sumřx k“1 dpfpkqq.J. London Math. Soc., 27:7–15, 1952

work page 1952
[18]

C. Frei, D. Loughran, and E. Sofos. Rational points of bounded height on general conic bundle surfaces. Proc. Lond. Math. Soc. (3), 117(2):407–440, 2018

work page 2018
[19]

Green and T

B. Green and T. Tao. Linear equations in primes.Ann. of Math. (2), 171(3):1753–1850, 2010

work page 2010
[20]

Green, T

B. Green, T. Tao, and T. Ziegler. An inverse theorem for the GowersU s`1rNs-norm.Ann. of Math. (2), 176(2):1231–1372, 2012

work page 2012
[21]

Harpaz, A

Y. Harpaz, A. N. Skorobogatov, and O. Wittenberg. The Hardy-Littlewood conjecture and rational points.Compos. Math., 150(12):2095–2111, 2014

work page 2095
[22]

Harpaz and O

Y. Harpaz and O. Wittenberg. On the fibration method for zero-cycles and rational points.Ann. of Math. (2), 183(1):229–295, 2016

work page 2016
[23]

D. R. Heath-Brown. A mean value estimate for real character sums.Acta Arith., 72(3):235–275, 1995

work page 1995
[24]

V. A. Iskovskih. A counterexample to the Hasse principle for systems of two quadratic forms in five variables.Mat. Zametki, 10:253–257, 1971

work page 1971
[25]

Landreau

B. Landreau. A new proof of a theorem of van der Corput.Bull. London Math. Soc., 21(4):366–368, 1989

work page 1989
[26]

Mumford.Tata lectures on theta

D. Mumford.Tata lectures on theta. I. Modern Birkh¨ auser Classics. Birkh¨ auser Boston, Inc., Boston, MA, 2007

work page 2007
[27]

L. B. Pierce, D. Schindler, and M. M. Wood. Representations of integers by systems of three quadratic forms.Proc. Lond. Math. Soc. (3), 113(3):289–344, 2016

work page 2016
[28]

Poonen and J

B. Poonen and J. F. Voloch. Random Diophantine equations. InArithmetic of higher-dimensional alge- braic varieties (Palo Alto, CA, 2002), volume 226 ofProgr. Math., pages 175–184. Birkh¨ auser Boston, Boston, MA, 2004. With appendices by Jean-Louis Colliot-Th´ el` ene and Nicholas M. Katz

work page 2002
[29]

N. Rome. A positive proportion of Hasse principle failures in a family of Chˆ atelet surfaces.Int. J. Number Theory, 15(6):1237–1249, 2019

work page 2019
[30]

Serre.A course in arithmetic, volume No

J.-P. Serre.A course in arithmetic, volume No. 7 ofGraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1973. Translated from the French

work page 1973
[31]

Serre.Cohomologie galoisienne

J.-P. Serre.Cohomologie galoisienne. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002

work page 2002
[32]

A. N. Skorobogatov and E. Sofos. Schinzel hypothesis on average and rational points.Invent. Math., 231(2):673–739, 2023

work page 2023
[33]

A. N. Skorobogatov and E. Sofos. Generic Diagonal Conic Bundles Revisited.Q. J. Math., 75(3):835– 849, 2024

work page 2024
[34]

E. M. Stein and R. Shakarchi.Fourier analysis, volume 1 ofPrinceton Lectures in Analysis. Princeton University Press, Princeton, NJ, 2003. An introduction

work page 2003
[35]

Swinnerton-Dyer

P. Swinnerton-Dyer. Rational points on pencils of conics and on pencils of quadrics.J. London Math. Soc. (2), 50(2):231–242, 1994

work page 1994
[36]

Swinnerton-Dyer

P. Swinnerton-Dyer. Rational points on some pencils of conics with 6 singular fibres.Ann. Fac. Sci. Toulouse Math. (6), 8(2):331–341, 1999

work page 1999
[37]

D. Wei. On the equationN K{k pΞq “Pptq.Proc. Lond. Math. Soc. (3), 109(6):1402–1434, 2014

work page 2014
[38]

O. Wittenberg.Intersections de deux quadriques et pinceaux de courbes de genre 1/Intersections of two quadrics and pencils of curves of genus 1, volume 1901 ofLecture Notes in Mathematics. Springer, Berlin, 2007

work page 1901
[39]

Zygmund.Trigonometric series: Vols

A. Zygmund.Trigonometric series: Vols. I, II. Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. 63 Technische Universit¨at Graz, Institut f¨ur Analysis und Zahlentheorie, Kopernikusgasse 24/II, A-8010, Graz, Austria Email address:frei@math.tugraz.at Universit`a di Roma Tor Vergata, Dipartimen...

work page 1968

[1] [1]

Barroero and M

F. Barroero and M. Widmer. Counting lattice points and O-minimal structures.Int. Math. Res. Not. IMRN, (18):4932–4957, 2014

work page 2014

[2] [2]

Browning, P

T. Browning, P. Le Boudec, and W. Sawin. The Hasse principle for random Fano hypersurfaces.Ann. of Math. (2), 197(3):1115–1203, 2023

work page 2023

[3] [3]

Browning, E

T. Browning, E. Sofos, and J. Ter¨ av¨ ainen. Bateman-Horn, polynomial Chowla and the Hasse principle with probability 1, arXiv:2212.10373, 2022

work page internal anchor Pith review arXiv 2022

[4] [4]

T. D. Browning, L. Matthiesen, and A. N. Skorobogatov. Rational points on pencils of conics and quadrics with many degenerate fibers.Ann. of Math. (2), 180(1):381–402, 2014

work page 2014

[5] [5]

Br¨ udern and R

J. Br¨ udern and R. Dietmann. Random Diophantine equations, I.Adv. Math., 256:18–45, 2014

work page 2014

[6] [6]

Colliot-Th´ el` ene

J.-L. Colliot-Th´ el` ene. Surfaces rationnelles fibr´ ees en coniques de degr´ e 4. InS´ eminaire de Th´ eorie des Nombres, Paris 1988–1989, volume 91 ofProgr. Math., pages 43–55. Birkh¨ auser Boston, Boston, MA, 1990

work page 1988

[7] [7]

Colliot-Th´ el` ene

J.-L. Colliot-Th´ el` ene. Points rationnels sur les fibrations. InHigher dimensional varieties and rational points (Budapest, 2001), volume 12 ofBolyai Soc. Math. Stud., pages 171–221. Springer, Berlin, 2003

work page 2001

[8] [8]

Colliot-Th´ el` ene, D

J.-L. Colliot-Th´ el` ene, D. Coray, and J.-J. Sansuc. Descente et principe de Hasse pour certaines vari´ et´ es rationnelles.J. reine angew. Math., 320:150–191, 1980

work page 1980

[9] [9]

Colliot-Th´ el` ene and J.-J

J.-L. Colliot-Th´ el` ene and J.-J. Sansuc. Sur le principe de Hasse et l’approximation faible, et sur une hypoth` ese de Schinzel.Acta Arith., 41(1):33–53, 1982

work page 1982

[10] [10]

Colliot-Th´ el` ene, J.-J

J.-L. Colliot-Th´ el` ene, J.-J. Sansuc, and P. Swinnerton-Dyer. Intersections of two quadrics and Chˆ atelet surfaces. I.J. reine angew. Math., 373:37–107, 1987

work page 1987

[11] [11]

Colliot-Th´ el` ene, J.-J

J.-L. Colliot-Th´ el` ene, J.-J. Sansuc, and P. Swinnerton-Dyer. Intersections of two quadrics and Chˆ atelet surfaces. II.J. reine angew. Math., 374:72–168, 1987

work page 1987

[12] [12]

Colliot-Th´ el` ene, A

J.-L. Colliot-Th´ el` ene, A. N. Skorobogatov, and P. Swinnerton-Dyer. Rational points and zero-cycles on fibred varieties: Schinzel’s hypothesis and Salberger’s device.J. reine angew. Math., 495:1–28, 1998. 62

work page 1998

[13] [13]

Colliot-Th´ el` ene and P

J.-L. Colliot-Th´ el` ene and P. Swinnerton-Dyer. Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties.J. reine angew. Math., 453:49–112, 1994

work page 1994

[14] [14]

Davenport

H. Davenport. On a principle of Lipschitz.J. London Math. Soc., 26:179–183, 1951

work page 1951

[15] [15]

de la Bret` eche and T

R. de la Bret` eche and T. D. Browning. Density of Chˆ atelet surfaces failing the Hasse principle.Proc. Lond. Math. Soc. (3), 108(4):1030–1078, 2014

work page 2014

[16] [16]

Y. Diao. Liouville function, von Mangoldt function and norm forms at random binary forms, arXiv:2506.18065, 2025

work page arXiv 2025

[17] [17]

P. Erd¨ os. On the sumřx k“1 dpfpkqq.J. London Math. Soc., 27:7–15, 1952

work page 1952

[18] [18]

C. Frei, D. Loughran, and E. Sofos. Rational points of bounded height on general conic bundle surfaces. Proc. Lond. Math. Soc. (3), 117(2):407–440, 2018

work page 2018

[19] [19]

Green and T

B. Green and T. Tao. Linear equations in primes.Ann. of Math. (2), 171(3):1753–1850, 2010

work page 2010

[20] [20]

Green, T

B. Green, T. Tao, and T. Ziegler. An inverse theorem for the GowersU s`1rNs-norm.Ann. of Math. (2), 176(2):1231–1372, 2012

work page 2012

[21] [21]

Harpaz, A

Y. Harpaz, A. N. Skorobogatov, and O. Wittenberg. The Hardy-Littlewood conjecture and rational points.Compos. Math., 150(12):2095–2111, 2014

work page 2095

[22] [22]

Harpaz and O

Y. Harpaz and O. Wittenberg. On the fibration method for zero-cycles and rational points.Ann. of Math. (2), 183(1):229–295, 2016

work page 2016

[23] [23]

D. R. Heath-Brown. A mean value estimate for real character sums.Acta Arith., 72(3):235–275, 1995

work page 1995

[24] [24]

V. A. Iskovskih. A counterexample to the Hasse principle for systems of two quadratic forms in five variables.Mat. Zametki, 10:253–257, 1971

work page 1971

[25] [25]

Landreau

B. Landreau. A new proof of a theorem of van der Corput.Bull. London Math. Soc., 21(4):366–368, 1989

work page 1989

[26] [26]

Mumford.Tata lectures on theta

D. Mumford.Tata lectures on theta. I. Modern Birkh¨ auser Classics. Birkh¨ auser Boston, Inc., Boston, MA, 2007

work page 2007

[27] [27]

L. B. Pierce, D. Schindler, and M. M. Wood. Representations of integers by systems of three quadratic forms.Proc. Lond. Math. Soc. (3), 113(3):289–344, 2016

work page 2016

[28] [28]

Poonen and J

B. Poonen and J. F. Voloch. Random Diophantine equations. InArithmetic of higher-dimensional alge- braic varieties (Palo Alto, CA, 2002), volume 226 ofProgr. Math., pages 175–184. Birkh¨ auser Boston, Boston, MA, 2004. With appendices by Jean-Louis Colliot-Th´ el` ene and Nicholas M. Katz

work page 2002

[29] [29]

N. Rome. A positive proportion of Hasse principle failures in a family of Chˆ atelet surfaces.Int. J. Number Theory, 15(6):1237–1249, 2019

work page 2019

[30] [30]

Serre.A course in arithmetic, volume No

J.-P. Serre.A course in arithmetic, volume No. 7 ofGraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1973. Translated from the French

work page 1973

[31] [31]

Serre.Cohomologie galoisienne

J.-P. Serre.Cohomologie galoisienne. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002

work page 2002

[32] [32]

A. N. Skorobogatov and E. Sofos. Schinzel hypothesis on average and rational points.Invent. Math., 231(2):673–739, 2023

work page 2023

[33] [33]

A. N. Skorobogatov and E. Sofos. Generic Diagonal Conic Bundles Revisited.Q. J. Math., 75(3):835– 849, 2024

work page 2024

[34] [34]

E. M. Stein and R. Shakarchi.Fourier analysis, volume 1 ofPrinceton Lectures in Analysis. Princeton University Press, Princeton, NJ, 2003. An introduction

work page 2003

[35] [35]

Swinnerton-Dyer

P. Swinnerton-Dyer. Rational points on pencils of conics and on pencils of quadrics.J. London Math. Soc. (2), 50(2):231–242, 1994

work page 1994

[36] [36]

Swinnerton-Dyer

P. Swinnerton-Dyer. Rational points on some pencils of conics with 6 singular fibres.Ann. Fac. Sci. Toulouse Math. (6), 8(2):331–341, 1999

work page 1999

[37] [37]

D. Wei. On the equationN K{k pΞq “Pptq.Proc. Lond. Math. Soc. (3), 109(6):1402–1434, 2014

work page 2014

[38] [38]

O. Wittenberg.Intersections de deux quadriques et pinceaux de courbes de genre 1/Intersections of two quadrics and pencils of curves of genus 1, volume 1901 ofLecture Notes in Mathematics. Springer, Berlin, 2007

work page 1901

[39] [39]

Zygmund.Trigonometric series: Vols

A. Zygmund.Trigonometric series: Vols. I, II. Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. 63 Technische Universit¨at Graz, Institut f¨ur Analysis und Zahlentheorie, Kopernikusgasse 24/II, A-8010, Graz, Austria Email address:frei@math.tugraz.at Universit`a di Roma Tor Vergata, Dipartimen...

work page 1968