HIRANOUCHI, Toshiro

(pre)Publications

1.  Zero-cycles on varieties over local fields and the Kato homology groups. (with R. Sugiyama)
   Abstract: For a d-dimensional variety X over a local field F, we study the structure of the higher Chow groups CH^{d+s}(X,s) for s > 1.  In particular, we prove that CH^{d+s}(X,s) is divisible for s > 2, and CH^{d+2}(X,2) is isomorphic to the direct sum of a finite group and a divisible group. As an application, we study the Kato homology groups KH0(X,Z/lr) when F is finite or local. 

2. A Hasse principle of the higher Chow groups for an elliptic curve over a global function field. arXiv:2507.22319
   Abstract: We investigate the structure of the higher Chow groups $CH^2(E,1)$ for an elliptic curve $E$ over a global function field $F$. Focusing on the kernel $V(E)$ of the push-forward map $CH^2(E,1)\to CH^1(\Spec(F),1) = K_1(F)$ associated to the structure map $E\to \Spec(F)$, we analyze the torsion part $V(E)$ based on the mod $l$ Galois representations associated to the $l$-torsion points $E[l]$.

3.  A Hasse principle for GL2(𝔽p) and Bloch's exact sequence for elliptic curves over number fields. arXiv:2504.05595
   Abstract: We investigate the higher Chow groups, specifically SK1(E) for elliptic curves E over number fields F. Focusing on the kernel V(E) of the norm map SK1(E)->F*, we analyze its mod p structure. We provide conditions, based on the mod p Galois representations associated to E[p], under which the torsion subgroup of V(E) is infinite. 

4.  Extended differential symbol and the Kato homology groups, Eur. J. Math. 11 (2025). doi: 10.1007/s40879-025-00843-8 (with R. Sugiyama) arXiv:2501.11224
   Abstract: For a product of curves X over a local field or a global field F of positive characteristic p, we provide a group-theoretic presentation of the degree 0-part of the Kato homology group KH0(X,Z/pr) for any r>0. This generalizes the classical theorem that the p-torsion part of the Brauer group of the field F can be described using logarithmic differential forms.  

5.  An additive variant of the differential symbol maps, Ann. K-Theory 9, (2024), no. 3, 499--518. doi: 10.2140/akt.2024.9.499. arXiv:2403.19974 [math.KT]
   Abstract: We investigate an additive analogue of the differential symbol map which establishes a relation between the Milnor K-group of a field of the positive characteristic and the Galois cohomology group of the field. Just as the Bloch-Gabber-Kato theorem, our differential symbol map gives an isomorphism between an additive variant of the Milnor K-group of the field and a Galois cohomology group. 

6.  Asymptotic behavior of class groups and Iwasawa theory of elliptic curves, J. Théor. Nombres Bordeaux 35 (2023), no. 2, 591--657. doi: 10.5802/jtnb.1258. arXiv:2203.16039 [math.NT] (with T. Ohshita)
   Abstaract: In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module X∞ of the Pontrjagin dual of the fine Selmer group of an elliptic curve E defined over Q. We consider the Galois extension field KnE of Q generated by coordinates of all pn-torsion points of E, and introduce a quotient AnE of the p-sylow subgroup of the ideal class group of KnE cut out by the modulo pn Galois representation E[pn]. We describe the asymptotic behavior of AnE by using the Iwasawa module X∞. In particular, under certain conditions, we obtain an asymptotic formula as Iwasawa's class number formula on the order of AnE by using Iwasawa's invariants of X∞.

7.  Abelian geometric fundamental groups for curves over a p-adic field, J. Théor. Nombres Bordeaux 35 (2023), no. 3, 905--946. doi: 10.5802/jtnb.1269 arXiv:2201.05982 (with E. Gazaki) 
   Abstract: For a curve X over a p-adic field k, using the class field theory of X due to S. Bloch and S. Saito we investigate the ``ramified part'' of the geometric (abelian) fundamental group of X. This classifies the geometric and abelian coverings of X which allow possible ramification over the special fiber of the model of X. Under the assumptions that X has a k-rational point, X has good reduction and its Jacobian variety has good ordinary reduction, we give some upper and lower bounds of this ramified part of the geometric fundamental group of X. Under some additional assumptions, we also construct the maximal covering of X which produces all the ramified part.

8.  Divisibility results for zero-cycles, Eur. J. Math. 7 (2021), no. 4, 1458--1501. arXiv:2004.05255 [math.AG] doi: 10.1007/s40879-021-00471-y (with E. Gazaki)
   Abstract: Let X be an abelian variety or a product of curves over a finite unramified extension k of Qp. Suppose that the Albanese variety of X has good reduction. We propose the following conjecture. The kernel of the Albanese map CH0(X)0 -> AlbX(k) is p-divisible. We prove this conjecture for a large family of products of elliptic curves and certain principal homogeneous spaces of abelian varieties. Using this, we provide some evidences for a local-to-global conjecture for zero-cycles of Colliot-Thélène and Sansuc, and Kato and Saito.

9.  Galois symbol maps for abelian varieties over a p-adic field, Acta Arith. 197 (2021), no. 2, 137-157. doi: 10.4064/aa191129-11-4 arXiv:1911.10669
   Abstract: We study the Galois symbol map associated to the multiplicative group and an abelian variety which has good ordinary reduction over a p-adic field. As a byproduct, one can calculate the "class group" in the view of the class field theory for curves over a p-adic field.

10.  A vanishing theorem of the additive higher Chow groups, Sci. Math. Jpn., 81 (2019), no. 3, 247-256 (Special Issue on FIM 2017) doi:10.32219/isms.81.3_247.
    Abstract: We show that the additive higher Chow group of the form TCHdim(X)+q(X,q;m) becomes 0 for some scheme X over a perfect field of positive characteristic and for q>1. This is an analogy of Akhtar's theorem on the higher Chow groups: CHdim(X)+q(X,q) = 0 for q>1. This short note is taken from Sect. 5 in the old version (arXiv:1208.6455v2) of the article titled ``An additive variant of Somekawa's K-groups...'' below which has been deleted before publication.

11.  Local torsion primes and the class numbers associated to an elliptic curve over Q, Hiroshima Math. J. 49 (Mar. 2019), no. 1, 117-128. arXiv:1703.08275 [math.NT]
    Abstract: Using the rank of the Mordell-Weil group E(Q) of an elliptic curve E over Q, we give a lower bound of the class number of the number field Q(E[p^n]) generated by pn-division points of E when the curve E does not possess a p-adic point of order p: E(Qp)[p]=0.

12.  A Hermite-Minkowski type theorem of varieties over finite fields, J. Number Theory, 176 (2017) 473-499. arXiv:1512.02348 [math.NT]
    Abstract: As an application of P. Delgine's theorem (Esnault and Kerz in Acta Math. Vietnam. 37:531-562, 2012) on a finiteness of l-adic sheaves on a variety over a finite field, we show the finiteness of étale coverings of such a variety with given degree whose ramification bounded along an effective Cartier divisor. This can be thought of a higher dimensional analogue of the classical Hermite-Minkowski theorem.

13.  Class field theory for open curves over local fields, J. Théor. Nombres Bordeaux 30 (2018), no. 2, 501-524. doi: 10.5802/jtnb.1036
    Abstract: We investigate the class field theory for an open curve over a local field. In particular, we determine the kernel and the cokernel of the reciprocity homomorphism and obtain the one to one correspondence between the set of finite abelian étale coverings which are not completly split and the set of finite index open subgroups of the idèle class group as in the classical class field theory.

14.  Milnor K-groups attached to elliptic curves over a p-adic field, Funct. Approx. Comment. Math. 54 (2016), no. 1, 39-55
    Abstract: We study the Galois symbol map of the Milnor K-group attached to elliptic curves over a p-adic field. As by-products, we determine the structure of the Chow group for the product of elliptic curves over a p-adic field under some assumptions.

15.  Finiteness of certain products of algebraic groups over a finite field, Algebraic number theory and related topics 2012, 3-14, RIMS Kôkyûroku Bessatsu, B51, Res. Inst. Math. Sci. (RIMS), Kyoto, 2014. arXiv:1209.4457
    Abstract: Let G1,...,Gq be algebraic varieties over a finite field k. We show that, if q>1, the finiteness of the tensor product (= the Mackey product) of the Mackey functors G1,...,Gq. We apply this to prove the finiteness of an abelian fundamental group which classifies abelian coverings with bounded ramification along the boundary.

16.  An additive variant of Somekawa's K-groups and Kähler differentials, J. of K-theory 13 (2014), no. 3, 481-516. doi:10.1017/is014003007jkt257 arXiv:1208.6455
    Abstract: We introduce a Milnor type K-group associated to commutative algebraic groups over a perfect field. It is an additive variant of Somekawa's K-group. We show that the K-group associated to the additive group and q multiplicative groups of a field is isomorphic to the space of absolute Kaehler differentials of degree q of the field, thus giving us a geometric interpretation of the space of absolute Kaehler differentials. We also show that the K-group associated to the additive group and Jacobian variety of a curve is isomorphic to the homology group of a certain complex.

17.  On the cycle map for products of elliptic curves over a p-adic field, Acta Arith. 157 (2013) no. 2, 101-118. doi:10.4064/aa157-2-1 (with S. Hirayama). arXiv:1010.2600
    Abstract: We study the Chow group of 0-cycles on the product of elliptic curves over a p-adic field. In particular, we obtain the structure of the image of the Albanese kernel for such abelian variety by the cycle class map. The key ingredient is the structure theorem of the graded quotients associated with a certain isogeny of formal groups. In Appendix, we determine the graded quotients of the induced filtration on the Milnor K-group modulo pn for a mixed characteristic Henselian discrete valuation field which contains a pn-th root of unity, in terms of differential forms of the residue field.

18.  Milnor K-groups modulo pn of a complete discrete valuation field, Proc. Japan Acad., Ser. A. 88 (2012), no. 4, 59-61. doi:10.3792/pjaa.88.59
    Abstract: For a mixed characteristic complete discrete valuation field K which contains a pn-th root of unity, we determine the graded quotients of the filtration on the Milnor K-groups KqM(K) modulo pn in terms of differential forms of the residue field of K.

19.  Ramification of truncated discrete valuation rings: a survey, Algebraic number theory and related topics 2008, 35-43, RIMS Kôkyûroku Bessatsu, B19, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010. DOI 10.4036/iis.2010.33
    Abstract: This note is a short survey of the author's results which are concerned with the ramification of extensions of truncated discrete valuation rings joint work with Y. Taguchi.

20.  Flat modules and Gröbner bases over truncated discrete valuation rings, Interdiscip. Inform. Sci. 16 (2010), no. 1, 33-37. doi:10.4036/iis.2010.33 (with Y. Taguchi).
    Abstract: We present basic properties of Gröbner bases of submodules of a free module of finite rank over a polynomial ring R with coefficients in a graded truncated discrete valuations ring A. As an application, we give a criterion for a finitely generated R-module to be flat over A. Its non-graded version is also given.

21.  Class field theory for open curves over p-adic fields, Math. Z., 266 (2010), no. 1, 107-113. DOI 10.1007/s00209-009-0556-1
    Abstract: We introduce the idèle class group for quasi-projective curves over p-adic fields and show that the kernel of the reciprocity map is divisible. This extends S. Saito's class field theory for projective curves.

22.  Pure weight perfect Modules on divisorial schemes, Deformation spaces, 75-89, Aspects Math., E40, Vieweg + Teubner, Wiesbaden, 2010. doi:10.1007/978-3-8348-9680-3_3 (with S. Mochizuki)
    Abstract: We introduce the notion of weight for pseudo-coherent Modules on a scheme. For a divisorial scheme X and a regular closed immersion i:Y -> X of codimension r, We show that there is a canonical Morita equivalence between the DG-category of perfect complexes on X whose cohomological supports are in Y and the DG-category of bounded complexes of weight r pseudo-coherent OX-Modules supported on Y. This implies that there is a canonical isomorphism between the Bass-Thomason-Trobaugh non-connected K-theory (resp. the Keller-Weibel cyclic homology) for the immersion i and the Schlichting non-connected K-theory associated with (resp. that of) the exact category of weight r pseudo-coherent Modules supported on Y. As an application, we decide a generator of the topological filtration on the non-connected K-theory (resp. cyclic homology theory) for affine Cohen-Macaulay schemes.

23.  Smallness of fundamental groups for arithmetic schemes, J. Number Theory 129 (2009), no. 11, 2702-2712 (with S. Harada). DOI: 10.1007/978-3-8348-9680-3_3
    Abstract: The smallness is proved of fundamental groups for arithmetic schemes. This is a generalization of the Hermite-Minkowski theorem. We also refer to the case of varieties over finite fields. As an application, we consider the finiteness of representations of the fundamental groups over algebraically closed fields.
    Errata: The first claim in Thm. 3.6 is wrong (pointed out by Dr. Yu Yang). The geometric part of the fundamental group with restricted ramification for a variety over a finite field is not small. For the correction of the proof, see the old version in the arXiv.

24.  Extensions of truncated discrete valuation rings, Pure Appl. Math. Q. (J.-P. Serre special issue) 4 (2008), no. 4, part 1, 1205-1214 (with Y. Taguchi).
    Abstract: An equivalence is established between the category of at most a-ramified finite separable extensions of a complete discrete valuation field K and the category of at most a-ramified finite extensions of the ``length-a truncation" OK/mKa of the integer ring of K. This extends the main theorem of Deligne to the imperfect residue field case.
    Errata: "Extensions of truncated discrete valuation rings'', Pure Appl. Math. Q. 11 (2015), no. 1, 171-174 (with Y. Taguchi)

25.  Finiteness of abelian fundamental groups with restricted ramification, C. R. Math. Acad. Sci. Paris 341 (2005), no. 4, 207-210.
    Abstract: We define a certain quotient of the étale fundamental group of a scheme which classifies étale coverings with bounded ramification along the boundary, and show the finiteness of the abelianization of this group for an arithmetic scheme.

Notes

1.  Projection formula and the p-torsion subgroup of the Brauer group for arithmetic fields of characteristic p>0, Bull. Kyushu Inst. Tech. Pure Appl. Math. No. 72, 2025, pp. 1–13.
   Abstract: For a field k of characteristic p>0, in our previous work (T. Hiranouchi Ann. K-Theory 9, (2024)), it has been given that a group representation of the pr-torsion subgroup of the Brauer group of k arising from the projection formula. This note provides an alternative proof of this group representation using classical class field theory for local or global fields, based on Tate's methods.

2.  Elliptic curves over a p-adic field with fake ordinary good reduction, Bull. Kyushu Inst. Technol. Pure Appl. Math. 71 (2024), 1--12. doi 10.18997/0002000379
   Abstract: We investigate the structure of the kernel of the Albanese map for a product of curves over a $p$-adic field. Specifically, we prove that the kernel becomes p-divisible when the ramification of the base field is sufficiently small.

3.  Bounds for the K-groups associated to abelian varieties over a p-adic field, Bull. Kyushu Inst. Technol. Pure Appl. Math. 70 (2023), 25--32.  doi:10.18997/00009111
   Abstract: For a  product of curves X over a p-adic field k,  in [Gazaki-Hiranouchi2021] we proposed a conjecture that the kernel of the Albanese map for X is p-divisible In this note, we report that when the Jacobian varieties of such curves all have good ordinary reduction, the Albanese kernel for the product X is still p-divisible even if the base field is not unramified but its ramification is small enough.  

4.  Galois symbol map for a Tate curve, Bull. Kyushu Inst. Technol. Pure Appl. Math. 69 (2022), 1--6. doi:10.18997/00008772
   Abstract: We calculate the ``class group'' in the view of the class field theory for Tate curves over a p-adic field.

5.  Albanese kernel of the product of curves over a p-adic field, Bull. Kyushu Inst. Technol. Pure Appl. Math. 68 (2021), 1--7.  doi:10.18997/00008063
   Abstract: In this short note, we investigate the image of the Kummer map associated to an abelian variety over a p-adic field. As a byproduct, we give the structure of the Albanese kernel of the product of curves over a p-adic field under some assumptions. The result has already known by E. Gazaki (J. Algebra 509, 2018), but the proof is completely different.

6.  Delooping of relative exact categories, arXiv:1304:0557 (with S. Mochizuki)
   Abstract: We introduce a delooping model of relative exact categories. It gives us a condition that the negative K-group of a relative exact category becomes trivial. Keywords: Negative K-theory Derived category
   (old ver. Quasi-weak equivalences in complicial exact categories, arXiv:1009.4608).

7.  On the fundamental groups with restricted ramification, Trends in Mathematics 9, 1 (2006) 35--37