The Varieties over global fields session is one of the afternoon sessions during week 3 of the 2025 Summer Research Institute in Algebraic Geometry, at Colorado State University in Fort Collins, Colorado. This page contains the information for this session only; please refer to the main SRI page for general information, including registration.
Sarah Frei, Dartmouth College
Ziyang Gao, University of California, Los Angeles
Brendan Hassett, Brown University
Ariyan Javanpeykar, Radboud University Nijmegen
Zhuchao Ji, Westlake University
Brian Lehmann, Boston College
Nicole Looper, University of Illinois, Chicago
Yohsuke Matsuzawa, Osaka Metropolitan University
Rachel Pries, Colorado State University
Sho Tanimoto, Nagoya University
Zhiyu Tian, Peking University
Junyi Xie, Peking University
1:30 - 2:20 pm
Ariyan Javanpeykar
The Weakly Special Conjecture contradicts Orbifold Mordell (and hence abc)
Abstract: Which varieties over a number field have a potentially dense set of rational points? Lang conjectured that varieties of general type over a number field have very few rational points. In 2000, guided by Lang’s conjecture and in search of classifying all varieties with a potentially dense set of rational points, Abramovich, Colliot-Thélène, Harris, and Tschinkel formulated the ”Weakly Special Conjecture”: every weakly special variety over a number field has a potentially dense set of rational points. In this talk I will explain how this conjecture contradicts the abc conjecture, and more precisely Campana’s ”Orbifold Mordell” conjecture, as well as explain that the function field analogue of the Weakly Special Conjecture fails (unconditionally). This is joint work with Finn Bartsch, Frederic Campana, and Olivier Wittenberg.
2:50 - 3:40 pm
Sho Tanimoto
Homological stability and Manin's conjecture
Abstract: I present our proofs for a version of Manin's conjecture over F_q for q large and Cohen—Jones—Segal conjecture over C for rational curves on split quartic del Pezzo surfaces. The proofs share a common method which builds upon prior work of Das—Tosteson. The main ingredients of this method are (i) the construction of bar complexes formalizing the inclusion-exclusion principle and its point counting estimates, (ii) dimension estimates for spaces of rational curves using conic bundle structures, (iii) estimates of error terms using arguments of Sawin—Shusterman based on Katz's results, and (iv) a certain virtual height zeta function revealing the compatibility of bar complexes and Peyre's constant. Our argument verifies the heuristic approach to Manin's conjecture over global function fields given by Batyrev and Ellenberg-Venkatesh, and it is a nice combination of various tools from algebraic geometry (birational geometry of moduli spaces of rational curves), arithmetic geometry (simplicial schemes, their homotopy theory, and Grothendieck—Lefschetz trace formula), algebraic topology (the inclusion-exclusion principle and Vassiliev type method of the bar complexes) and some elementary analytic number theory. This is joint work with Ronno Das, Brian Lehmann, and Phil Tosteson with a help by Will Sawin and Mark Shusterman.
4:10 - 5:00 pm
Junyi Xie
Geometric Bombieri-Lang conjecture
The geometric Bombieri Lang conjecture is an analogue of the Bombieri-Lang conjecture over function fields. With Yuan, we developed a mechanism to realize Vojtas dictionary in a reasonably concrete way. Applying this mechanism, Xie, Yuan and Gao proved the geometric Bombieri Lang conjecture for varieties having a finite map to an abelian variety.
1:30 - 2:20 pm
Nicole Looper
The Tate-Voloch conjecture and applications to the arithmetic of abelian varieties
Abstract: The Tate-Voloch conjecture says that given a closed subvariety X of a semiabelian variety G over C_p, the distance between X and any torsion point in the complement of X is uniformly bounded away from 0. This conjecture was proved by Scanlon in the case where G is defined over the algebraic closure of Q_p. In this talk I will present how this conjecture may be approached via difference equations, as well as applications to integral torsion points on abelian varieties.
2:50 - 3:40 pm
Zhuchao Ji
4:10 - 5:00 pm
Yohsuke Matsuzawa
Generalized greatest common divisor along orbits of rational self-maps
Abstract: The global height function associated with a closed subscheme of codimension at least two is called generalized greatest common divisor. It is a generalization of (log of) greatest common divisor among coordinates.Geometrically, it is equal to the usual Weil height function associated to the exceptional divisor of the blow up along the subscheme. Given a dominant rational self-map f on a projective variety over a number field and a rational point x, it is expected that the generalized greatest common divisor of f^n(x) is small compared with height of f^n(x), namely the ratio goes to zero. Such a statement has been known only very special cases or under Vojta's conjecture. We prove an unconditional sufficient condition in terms of dynamical degrees of the map and arithmetic degree of the orbit (, which is exponential growth rate of height along the orbit).
1:30 - 2:20 pm
Zhiyu Tian
2:50 - 3:40 pm
Brian Lehmann
Integral points on log Fano varieties over complex function fields
Abstract:Suppose U is a smooth quasiprojective variety over a global field. When U is "positively curved" -- e.g. when U is compactified by a log Fano pair -- it is conjectured that integral points on U are potentially dense. I will report on new progress toward establishing potential density over a complex function field. I will also discuss certain geometric properties of log Fano varieties which obstruct stronger conjectures about integral points. This is joint work with Eric Jovinelly and Eric Riedl.
4:10 - 5:00 pm
Rachel Pries
Four perspectives on rational points on one Shimura curve
Abstract: In this talk, I describe the points on a special Shimura curve from four perspectives: a family of cyclic covers of the projective line, a hyperbolic triangle, quadratic forms, and a unitary similitude group. By leveraging these four perspectives, we generalize a result of Elkies about supersingular reduction of elliptic curves to the case of genus four curves having an automorphism of order five. This is joint work with Wanlin Li, Elena Mantovan, and Yunqing Tang.
1:30 - 2:20 pm
Ziyang Gao
Generic positivity of the Beilinson-Bloch height
Abstract: The Beilinson-Bloch height is conjecturally defined for homologically trivial cycles in algebraic varieties defined over number fields. It is conjectured to be non-negative and to vanish if and only if the cycle is trivial in the Chow group. In the case of the Gross-Schoen and Ceresa cycles, the Beilinson-Bloch height is known to be unconditionally defined, and thus can be seen as a height function on M_g (the moduli space of curves of genus g) which we call the BB height for short.
I will explain a program recently initiated with Shouwu Zhang to study the Beilinson-Bloch height. More precisely, we construct a rational Zariski open subset U of M_g on which the BB height is bounded below linearly in terms of the Falting height, and show that the subset is proper if g is at least 3. This proves a uniform lower bound and the Northcott property of the BB height on U.
2:50 - 3:40 pm
Sarah Frei
Rationality in arithmetic families
Abstract: The rationality problem in algebraic geometry asks whether a given algebraic variety can be parameterized by algebraic functions. A natural question in the study of rationality is the following: is rationality a deformation invariant in smooth families? In arithmetic families, this question asks how the rationality of a variety defined over Q interacts with the rationality of its modulo p reductions for various primes p. In this talk, I'll discuss work in progress with Asher Auel and Alena Pirutka on algebraic varieties over Q that are not rational over the complex numbers but are rational modulo p for infinitely many primes p. Our construction involves the “famous 95” K3 surfaces in weighted projective space and arithmetic work on the reduction of Brauer classes.
4:10 - 5:00 pm
Brendan Hassett
Birational and equivariant geometry
Abstract: Galois theory offers a strong link between equivariant geometry under finite groups and geometry over non-closed fields. We consider birational questions for combinatorially-rich varieties including moduli spaces of pointed rational curves and quadric hypersurfaces. (joint with Bethea, Tschinkel, and Zhang)