特定助教

YANG, Yu (数論幾何)

I work in arithmetic geometry. Recently, my research focuses on curves and theirmoduli spaces in positive characteristic from the point of view of fundamental groups,
which are motived by the theory of anabelian geometry of curves over algebraically closed
ﬁelds of characteristic p > 0.
Since the late 1990s, some developments of Florian Pop, Michel Raynaud, Mohamed
Saı̈di, and Akio Tamagawa showed evidence for very strong anabelian phenomena for
curves over algebraically closed ﬁelds of characteristic p. In this situation, the Galois
group of the base ﬁeld is trivial, and the arithmetic fundamental group coincides with thegeometric fundamental group, thus there is a total absence of a Galois action of the baseﬁeld. This kinds of anabelian phenomenon go beyond Grothendieck’s anabelian geometry,
and this is the reason that we do not have an explicit description of the geometric fundamental group of any pointed stable curve in positive characteristic. Moreover, we may
think that the anabelian geometry of curves over algebraically closed ﬁelds of characteristic p is a theory based on the following rough consideration: The geometric fundamentalgroup of a pointed stable curve over an algebraically closed ﬁeld of characteristic p must
encode “moduli” of the curve.
Recently, I introduced a topological space which is called the moduli space of admissible fundamental groups of curves in positive characteristic, and posed the so-called“Homeomorphism Conjecture”. This conjecture says that the moduli spaces of curves
in positive characteristic can be reconstructed group-theoretically from the geometricfundamental groups of curves as topological spaces. Moreover, the Homeomorphism Conjecture gives us a new insight into the theory of the anabelian geometry of curves overalgebraically closed ﬁelds of characteristic p based on the following philosophy:
The anabelian properties of pointed stable curves over algebraically closed
ﬁelds of characteristic p are equivalent to the topological properties of themoduli spaces of admissible fundamental groups.
In [1], [4], [7], I study the anabelian geometry of (possibly singular) pointed stable
curves over algebraically closed ﬁelds of characteristic p. In particular, I proved the combinatorial Grothendieck conjecture for curves in positive characteristic, and formulated
the weak Isom-version of the Grothendieck conjecture for arbitrary pointed stable curves
over algebraically closed ﬁelds of characteristic p.
In [3], [5], [6], I study the Hasse-Witt invariants and the generalized Hasse-Witt invariants associated to a pointed stable curve in positive characteristic by using the theory ofRaynaud-Tamagawa theta divisors. Those invariants play important roles to understandthe structure of the geometric fundamental group of the curve.
In [2], [8], [9], [10], basing on the theory developed in the previous papers (in particular,[3], [4], [5], [7]), I introduced the moduli spaces of admissible fundamental groups, and formulated the Homeomorphism Conjecture. Moreover, I proved that the Homeomorphism
Conjecture holds when the dimension of the moduli space of curves is 1. Furthermore,
to solve the Homeomorphism Conjecture for higher dimensional moduli spaces, we formulated some new conjectures in [8] and [10]. In [10], we prove the group-theoreticalspecialization conjecture (one of the conjectures formulated in [10]) for curves of type(0, n).
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By the theory of the moduli spaces of admissible fundamental groups, fundamentalgroups can be considered as an analogue of local moduli of pointed stable curves. This
insight gives me a new approach to seeing anabelian geometry. For example, in [11], withY. Hoshi, we obtain a new proof of Mochizuki’s famous theorem concerning the (Isomversion) Grothendieck’s anabelian conjecture for curves over sub-p-adic ﬁelds without
using Faltings’ p-adic Hodge theory.
[1] Yu Yang, On the admissible fundamental groups of curves over algebraically closed
ﬁelds of characteristic p > 0, Publ. Res. Inst. Math. Sci. 54 (2018), 649–678.
[2] Z. Hu, R. Zong, Y. Yang, Topology of moduli spaces of curves and anabeliangeometry in positive characteristic, Forum of Mathematics, Sigma 12 (2024), Paper No.
e33, 36 pp.
[3] Yu Yang, On the averages of generalized Hasse-Witt invariants of pointed stable
curves in positive characteristic, Math. Z. 295 (2020), 1-45.
[4] Yu Yang, On topological and combinatorial structures of pointed stable curves overalgebraically closed ﬁelds of positive characteristic, Math. Nachr. (2023), 1-42.
[5] Yu Yang, Maximum generalized Hasse-Witt Invariants and their applications toanabelian geometry, Selecta Math. (N.S.) 28 (2022), Paper No. 5, 98 pp.
[6] Yu Yang, Raynaud-Tamagawa theta divisors and new-ordinariness of ramiﬁed coverings of curves, J. Algebra 587 (2021), 263-294.
[7] Yu Yang, On the existence of specialization isomorphisms of admissible fundamentalgroups in positive characteristic, Math. Res. Lett. 28 (2021), 1941-1959.
[8] Yu Yang, Moduli spaces of fundamental groups of curves in positive characteristicI, preprint.
[9] Yu Yang, Moduli spaces of fundamental groups of curves in positive characteristic
II, in preparation.
[10] Yu Yang, Topological and group-theoretical specializations of fundamental groupsof curves in positive characteristic, preprint.
[11] Y. Hoshi, Y. Yang, On the arithmetic fundamental groups of curves over local
ﬁelds, preprint.

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