Organizers: Takehiko Yasuda (main), Kohei Aoyama, Yutaro Kaijima
Date: June 27, 2025
Venus: Room E404, Science Building (the 4th floor of Building E), Toyonaka Campus, the University of Osaka
Linghu Fan (Kavli IPMU, the University of Tokyo)
Runxuan Gao (Nagoya University)
Changho Han (Korea University)
Takashi Taniguchi (Kobe University)
Julie Tavernier (University of Bath)
10:00 - 11:00 Fan
11:20 - 12:20 Gao
Lunch Time
14:00 - 15:00 Han
15:15 - 16:15 Tavernier
16:30 - 17:30 Taniguchi
Free Discussion
Linghu Fan (Kavli IPMU, the University of Tokyo)
Title: Euler characteristic of crepant resolutions of specific modular quotient singularities
Abstract: From the perspective of McKay correspondence, the Euler characteristic of crepant resolutions of quotient singularities, associated with given finite groups (and their special linear actions), is an important geometric invariant, which is usually conjectured to be related with certain algebraic invariants of the groups. Over the complex numbers, Batyrev's theorem provides such a relation, of which the direct analogue fails in positive characteristic. In this talk, we study the Euler characteristic of crepant resolutions for modular groups with specific semidirect product structure, by considering a counting problem of coverings. Based on this result, we propose a conjectural version of Batyrev's theorem in positive characteristic.
Runxuan Gao (Nagoya University)
Title: Quasi-étale covers of Du Val del Pezzo surfaces and Zariski dense exceptional sets in Manin’s conjecture
Abstract: On rationally connected varieties over number fields, an exceptional set is a subset where the number of rational points of bounded height grows faster than expected. While Zariski dense exceptional sets have long been known in dimension ≥3, we provide the first such examples in dimension 2. To construct and classify these examples, we classify all quasi-étale covers of Du Val del Pezzo surfaces, analyze the descent problem for these covers, and study the relationship between the pseudo-effective cones of the surface and its cover. If time permits, I will also explain how these provide interesting examples for Manin’s conjecture for Deligne–Mumford stacks.
Changho Han (Korea University)
Title: Counting 5-isogenies of Elliptic Curves over Q
Abstract: In Number Theory, an important class of questions is to asymptotically count rational points of a variety/stack of arithmetic interest with respect to a height function. As a consequence of this viewpoint, it is well-known that the number of Q-elliptic curves with naive discriminant bounded by B is asymptotic to a scalar multiple of H^{5/6}; this is obtained from counting Q-points of the moduli stack X(1) of stable elliptic curves. A related question is then to asymptotically count Q-points of modular curves X_0(m), which is to count elliptic curves with m-isogenies defined over Q. As a joint work with Santiago Arango-Piñeros, Oana Padurariu, and Sun Woo Park, I will explain how to count Q-points of X_0(5), which is about counting elliptic curves with 5-isogenies defined over Q.
Julie Tavernier (University of Bath)
Title: Counting abelian number fields with restricted ramification via rational points on stacks.
Abstract: In this talk we will explain how counting number fields with restricted ramification can be interpreted via rational points on the stack BG. We give an asymptotic formula for the number of these number fields with restricted ramification and provide an explicit formula for the leading constant. We use this stacky viewpoint to show that the existence of number fields with certain prescribed ramification is controlled by a Brauer-Manin obstruction on the stack BG. Finally we show using rational points on BG that these number fields are equidistributed with respect to suitable collections of infinitely many local conditions.
Takashi Taniguchi (Kobe University)
Title: Exponential sums over singular binary quartic forms and applications
Abstract: We study the exponential sum over singular binary quartic forms, proving an explicit formula for the finite field Fourier transform of this set. At most singular points, the exponential sum becomes a polynomial in the prime p, but interestingly, at non-singular points, the point counting function of an elliptic curve appears in the formula. As an application, we show the existence of "many" 2-Selmer elements for elliptic curves with discriminants that are squarefree and have at most four prime factors.
This workshop is supported by JSPS KAKENHI Grant Number JP23H01070.