@article{19054,
  abstract     = {This work concerns asymptotical stabilisation phenomena occurring in the moduli space of sections of certain algebraic families over a smooth projective curve, whenever the generic fibre of the family is a smooth projective Fano variety, or not far from being Fano.
 We describe the expected behaviour of the class, in a ring of motivic integration, of the moduli space of sections of given numerical class. Up to an adequate normalisation, it should converge, when the class of the sections goes arbitrarily far from the boundary of the dual of the effective cone, to an effective element given by a motivic Euler product. Such a principle can be seen as an analogue for rational curves of the Batyrev-Manin-Peyre principle for rational points.
 The central tool of this article is the property of equidistribution of curves. We show that this notion does not depend on the choice of a model of the generic fibre, and that equidistribution of curves holds for smooth projective split toric varieties. As an application, we study the Batyrev-Manin-Peyre principle for curves on a certain kind of twisted products.},
  author       = {Faisant, Loïs},
  issn         = {1944-7833},
  journal      = {Algebra & Number Theory},
  pages        = {883--965},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Motivic distribution of rational curves and twisted products of toric varieties}},
  doi          = {10.2140/ant.2025.19.883},
  volume       = {19},
  year         = {2025},
}

@article{21244,
  abstract     = {Given a family of varieties over the projective line, we study the density of fibres that are everywhere locally soluble in the case that components of higher multiplicity are allowed. We use log geometry to formulate a new sparsity criterion for the existence of everywhere locally soluble fibres and formulate new conjectures that generalise previous work of Loughran and Smeets. These conjectures involve geometric invariants of the associated multiplicity orbifolds on the base of the fibration in the spirit of Campana. We give evidence for the conjectures by providing an assortment of bounds using Chebotarev’s theorem and sieve methods, with most of the evidence involving upper bounds. },
  author       = {Browning, Timothy D and Lyczak, Julian and Smeets, Arne},
  issn         = {1944-7833},
  journal      = {Algebra & Number Theory},
  number       = {10},
  pages        = {2049--2090},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Paucity of rational points on fibrations with multiple fibres}},
  doi          = {10.2140/ant.2025.19.2049},
  volume       = {19},
  year         = {2025},
}

@article{17449,
  abstract     = {We prove that the $k$-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval $(x, x+H]$ matches the corresponding Gaussian moment, as long as $H\ll x/(\log x)^{2k^2+2+o(1)}$ and $H$ tends to infinity with $x$. We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian limiting distribution in short moving intervals $(x, x+H]$ with $H\ll X/(\log X)^{W(X)}$ tending to infinity with $X$, where $x$ is uniformly chosen from $\{1,2,\dots, X\}$, and $W(X)$ tends to infinity with $X$ arbitrarily slowly. This makes some initial progress on a recent question of Harper.},
  author       = {Pandey, Mayank and Wang, Victor and Xu, Max Wenqiang},
  issn         = {1944-7833},
  journal      = {Algebra & Number Theory},
  number       = {2},
  pages        = {389--408},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Partial sums of typical multiplicative functions over short moving intervals}},
  doi          = {10.2140/ant.2024.18.389},
  volume       = {18},
  year         = {2024},
}

@article{13091,
  abstract     = {We use a function field version of the Hardy–Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre’s reformulation of the Batyrev–Manin conjecture in terms of slopes with respect to the tangent bundle.},
  author       = {Browning, Timothy D and Sawin, Will},
  issn         = {1944-7833},
  journal      = {Algebra and Number Theory},
  number       = {3},
  pages        = {719--748},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Free rational curves on low degree hypersurfaces and the circle method}},
  doi          = {10.2140/ant.2023.17.719},
  volume       = {17},
  year         = {2023},
}

@article{9199,
  abstract     = {We associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre's programme on "freeness" for rational points of bounded height on Fano
varieties.},
  author       = {Browning, Timothy D and Horesh, Tal and Wilsch, Florian Alexander},
  issn         = {1944-7833},
  journal      = {Algebra & Number Theory},
  number       = {10},
  pages        = {2385--2407},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Equidistribution and freeness on Grassmannians}},
  doi          = {10.2140/ant.2022.16.2385},
  volume       = {16},
  year         = {2022},
}

@article{15279,
  abstract     = {We formulate and prove an analog of Poonen’s finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil and Wood, who treated the case of smooth hypersurface sections, and is made possible by the use of motivic Euler products to write down candidate motivic probabilities. As applications, we give motivic analogs of many results in arithmetic statistics that have been proven using Poonen’s sieve, including work of Bucur and Kedlaya on complete intersections and Erman and Wood on semiample Bertini theorems.},
  author       = {Bilu, Margaret and Howe, Sean},
  issn         = {1944-7833},
  journal      = {Algebra & Number Theory},
  keywords     = {Algebra and Number Theory},
  number       = {9},
  pages        = {2195--2259},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Motivic Euler products in motivic statistics}},
  doi          = {10.2140/ant.2021.15.2195},
  volume       = {15},
  year         = {2021},
}

@article{265,
  abstract     = {We establish the dimension and irreducibility of the moduli space of rational curves (of fixed degree) on arbitrary smooth hypersurfaces of sufficiently low degree. A spreading out argument reduces the problem to hypersurfaces defined over finite fields of large cardinality, which can then be tackled using a function field version of the Hardy-Littlewood circle method, in which particular care is taken to ensure uniformity in the size of the underlying finite field.},
  author       = {Browning, Timothy D and Vishe, Pankaj},
  issn         = {1944-7833},
  journal      = {Geometric Methods in Algebra and Number Theory},
  number       = {7},
  pages        = {1657 -- 1675},
  publisher    = { Mathematical Sciences Publishers},
  title        = {{Rational curves on smooth hypersurfaces of low degree}},
  doi          = {10.2140/ant.2017.11.1657},
  volume       = {11},
  year         = {2017},
}