TY - GEN
AB - It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least 3 over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thélène that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least 3. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.
AU - Browning, Timothy D
AU - Boudec, Pierre Le
AU - Sawin, Will
ID - 8682
T2 - arXiv
TI - The Hasse principle for random Fano hypersurfaces
ER -
TY - JOUR
AB - We develop a version of Ekedahl’s geometric sieve for integral quadratic forms of rank at least five. As one ranges over the zeros of such quadratic forms, we use the sieve to compute the density of coprime values of polynomials, and furthermore, to address a question about local solubility in families of varieties parameterised by the zeros.
AU - Browning, Timothy D
AU - Heath-Brown, Roger
ID - 8742
JF - Forum Mathematicum
SN - 09337741
TI - The geometric sieve for quadrics
ER -
TY - JOUR
AB - An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface x1y21+⋯+x4y24=0 in ℙ3×ℙ3. This confirms the modified Manin conjecture for this variety, in which the removal of a thin set of rational points is allowed.
AU - Browning, Timothy D
AU - Heath Brown, Roger
ID - 179
IS - 16
JF - Duke Mathematical Journal
TI - Density of rational points on a quadric bundle in ℙ3×ℙ3
VL - 169
ER -
TY - JOUR
AB - We develop a geometric version of the circle method and use it to compute the compactly supported cohomology of the space of rational curves through a point on a smooth affine hypersurface of sufficiently low degree.
AU - Browning, Timothy D
AU - Sawin, Will
ID - 177
IS - 3
JF - Annals of Mathematics
TI - A geometric version of the circle method
VL - 191
ER -
TY - JOUR
AB - An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariskiopen subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses the Hardy–Littlewood circle method.
AU - Browning, Timothy D
AU - Hu, L.Q.
ID - 6310
JF - Advances in Mathematics
SN - 00018708
TI - Counting rational points on biquadratic hypersurfaces
VL - 349
ER -
TY - JOUR
AB - This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of ℙ3ℚ given by the following equation 𝑥0(𝑥21+𝑥22)−𝑥33=0 in agreement with the Manin-Peyre conjectures.
AU - De La Bretèche, Régis
AU - Destagnol, Kevin N
AU - Liu, Jianya
AU - Wu, Jie
AU - Zhao, Yongqiang
ID - 6620
IS - 12
JF - Science China Mathematics
SN - 16747283
TI - On a certain non-split cubic surface
VL - 62
ER -
TY - JOUR
AB - An upper bound sieve for rational points on suitable varieties isdeveloped, together with applications tocounting rational points in thin sets,to local solubility in families, and to the notion of “friable” rational pointswith respect to divisors. In the special case of quadrics, sharper estimates areobtained by developing a version of the Selberg sieve for rational points.
AU - Browning, Timothy D
AU - Loughran, Daniel
ID - 175
IS - 8
JF - Transactions of the American Mathematical Society
SN - 00029947
TI - Sieving rational points on varieties
VL - 371
ER -
TY - JOUR
AB - We derive the Hasse principle and weak approximation for fibrations of certain varieties in the spirit of work by Colliot-Thélène–Sansuc and Harpaz–Skorobogatov–Wittenberg. Our varieties are defined through polynomials in many variables and part of our work is devoted to establishing Schinzel's hypothesis for polynomials of this kind. This last part is achieved by using arguments behind Birch's well-known result regarding the Hasse principle for complete intersections with the notable difference that we prove our result in 50% fewer variables than in the classical Birch setting. We also study the problem of square-free values of an integer polynomial with 66.6% fewer variables than in the Birch setting.
AU - Destagnol, Kevin N
AU - Sofos, Efthymios
ID - 6835
IS - 11
JF - Bulletin des Sciences Mathematiques
SN - 0007-4497
TI - Rational points and prime values of polynomials in moderately many variables
VL - 156
ER -