Nagoya Mathematical Journal Articles (Project Euclid)

Hilbert-Samuel polynomials for the contravariant extension functor

Thu, 05 Aug 2010 15:41 EDT

Andrew Crabbe, Daniel Katz, Janet Striuli, Emanoil Theodorescu

Source: Nagoya Math. J., Volume 198, 1--22.

Abstract:
Let $(R,\mathfrak {m})$ be a local ring, and let $M$ and $N$ be finite $R$ -modules. In this paper we give a formula for the degree of the polynomial giving the lengths of the modules $\operatorname{Ext}^{i}_{R}(M,N/\mathfrak{m}^{n}N)$ . A number of corollaries are given, and more general filtrations are also considered.

On Eells-Sampson’s existence theorem for harmonic maps via exponentially harmonic maps

Fri, 11 Feb 2011 09:15 EST

Toshiaki Omori

Source: Nagoya Math. J., Volume 201, 133--146.

Abstract:
In this note, we introduce an approximation of harmonic maps via a sequence of exponentially harmonic maps. We then reestablish the existence theorem of harmonic maps due to Eells and Sampson.

References:
[1] D. M. Duc, Variational problems of certain functionals, Internat. J. Math. 6 (1995), 503–535.

[2] D. M. Duc and J. Eells, Regularity of exponentially harmonic functions, Internat. J. Math. 2 (1991), 395–408.

[3] J. Eells and L. Lemaire, “Some properties of exponentially harmonic maps” in Partial Differential Equations, Part 1, 2 (Warsaw, 1990), Banach Center Publ. 27, Part 1, Vol. 2, Polish Acad. Sci., Warsaw, 1992, 129–136.

[4] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160.

[5] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, reprint of the 1998 original, Classics Math., Springer, Berlin, 2001.

[6] J. Q. Hong and Y. H. Yang, Some results on exponentially harmonic maps (in Chinese), Chinese Ann. Math. Ser. A 14 (1993), 686–691.

[7] G. M. Lieberman, On the regularity of the minimizer of a functional with exponential growth, Comment. Math. Univ. Carolin. 33 (1992), 45–49.

[8] H. Naito, On a local Hölder continuity for a minimizer of the exponential energy functional, Nagoya Math. J. 129 (1993), 97–113.

Coarse dynamics and fixed-point theorem

Tue, 31 May 2011 10:19 EDT

Tomohiro Fukaya

Source: Nagoya Math. J., Volume 202, 1--13.

Abstract:
We study semigroup actions on a coarse space and the induced actions on the Higson corona from a dynamical point of view. Our main theorem states that if an action of an abelian semigroup on a proper coarse space satisfies certain conditions, the induced action has a fixed point in the Higson corona. As a corollary, we deduce a coarse version of Brouwer’s fixed-point theorem.

References:
[1] A. N. Dranishnikov, J. Keesling, and V. V. Uspenskij, On the Higson corona of uniformly contractible spaces, Topology 37 (1998), 791–803.

[2] É. Ghys and P. de la Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov, Prog. Math. 83, Birkhäuser, Boston, 1990.

[3] A. Navas, Groups of Circle Diffeomorphisms, Chicago Lectures Math., University of Chicago Press, Chicago, 2011.

[4] J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (1993), no. 497.

[5] J. Roe, Lectures on Coarse Geometry, University Lecture Ser. 31, Am. Math. Soc., Providence, 2003.

On the residue class distribution of the number of prime divisors of an integer

Tue, 31 May 2011 10:19 EDT

Michael Coons, Sander R. Dahmen

Source: Nagoya Math. J., Volume 202, 15--22.

Abstract:
Let $\Omega(n)$ denote the number of prime divisors of $n$ counting multiplicity. One can show that for any positive integer $m$ and all $j=0,1,\ldots,m-1$ , we have \[\#\bigl\{n\leq x:\Omega(n)\equiv j(\operatorname {mod}m)\bigr\}=\frac{x}{m}+o(x^{\alpha }),\] with $\alpha =1$ . Building on work of Kubota and Yoshida, we show that for $m\textgreater 2$ and any $j=0,1,\ldots,m-1$ , the error term is not $o(x^{\alpha })$ for any $\alpha \textless 1$ .

References:
[1] T. M. Apostol, Introduction to Analytic Number Theory, Undergrad. Texts Math., Springer, New York, 1976.

[2] P. Borwein, R. Ferguson, and M. J. Mossinghoff, Sign changes in sums of the Liouville function, Math. Comp. 77 (2008), 1681–1694.

[3] R. R. Hall, A sharp inequality of Halász type for the mean value of a multiplicative arithmetic function, Mathematika 42 (1995), 144–157.

[4] T. Kubota and M. Yoshida, A note on the congruent distribution of the number of prime factors of natural numbers, Nagoya Math. J. 163 (2001), 1–11.

[5] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2 Bände, 2nd ed., with an appendix by P. T. Bateman, Chelsea, New York, 1953.

[6] J. Rivat, A. Sárközy, and C. L. Stewart, Congruence properties of the Ω-function on sumsets, Illinois J. Math. 43 (1999), 1–18.

Dilogarithm identities for conformal field theories and cluster algebras: Simply laced case

Tue, 31 May 2011 10:19 EDT

Tomoki Nakanishi

Source: Nagoya Math. J., Volume 202, 23--43.

Abstract:
The dilogarithm identities for the central charges of conformal field theories of simply laced type were conjectured by Bazhanov, Kirillov, and Reshetikhin. Their functional generalizations were conjectured by Gliozzi and Tateo. They have been partly proved by various authors. We prove these identities in full generality for any pair of Dynkin diagrams of simply laced type based on the cluster algebra formulation of the Y-systems.

References:
[BR1] V. V. Bazhanov and N. Y. Reshetikhin, Critical RSOS models and conformal field theory, Internat. J. Modern Phys. A 4 (1989), 115–142.

[BR2] V. V. Bazhanov and N. Y. Reshetikhin, Restricted solid-on-solid models connected with simply laced algebras and conformal field theory, J. Phys. A 23 (1990), 1477–1492.

[B] S. Bloch, “Applications of the dilogarithm function in algebraic K-theory and algebraic geometry” in Proceedings of the International Symposium on Algebraic Geometry (Kyoto, 1977), Kinokuniya, Tokyo, 1978, 103–114.

[CGT] R. Caracciolo, F. Gliozzi, and R. Tateo, A topological invariant of RG flows in 2D integrable quantum field theories, Internat. J. Modern Phys. 13 (1999), 2927–2932.

[C] F. Chapoton, Functional identities for the Rogers dilogarithm associated to cluster Y-systems, Bull. Lond. Math. Soc. 37 (2005), 755–760.

[DWZ] H. Derksen, J. Weyman, and A. Zelevinsky, Quivers with potentials and their representations II: Applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), 749–790.

[DS] J. L. Dupont and C.-H. Sah, Dilogarithm identities in conformal field theory and group homology, Comm. Math. Phys. 161 (1994), 265–282.

[FZ] V. A. Fateev and A. B. Zamolodchikov, Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in Z_N-symmetric statistical systems, J. Exp. Theor. Phys. 62 (1985), 215–225.

[FG] V. V. Fock and A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 39 (2009), 865–930.

[FZ1] S. Fomin and A. Zelevinsky, Cluster algebras, I: Foundations, J. Amer. Math. Soc. 15 (2002), 497–529.

[FZ2] S. Fomin and A. Zelevinsky, Cluster algebras, II: Finite type classification, Invent. Math. 154 (2003), 63–121.

[FZ3] S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), 977–1018.

[FZ4] S. Fomin and A. Zelevinsky, Cluster algebras, IV: Coefficients, Compos. Math. 143 (2007), 112–164.

[FS] E. Frenkel and A. Szenes, Thermodynamic Bethe ansatz and dilogarithm identities, I, Math. Res. Lett. 2 (1995), 677–693.

[G] D. Gepner, New conformal field theories associated with Lie algebras and their partition functions, Nuclear Phys. B 290 (1987), 10–24.

[GW] D. Gepner and E. Witten, String theory on group manifolds, Nuclear Phys. B 278 (1986), 493–549.

[GT1] F. Gliozzi and R. Tateo, ADE functional dilogarithm identities and integrable models, Phys. Lett. B 348 (1995), 677–693.

[GT2] F. Gliozzi and R. Tateo, Thermodynamic Bethe ansatz and three-fold triangulations, Internat. J. Modern Phys. A 11 (1996), 4051–4064.

[HW] H. C. Hutchins and H. J. Weinert, Homomorphisms and kernels of semifields, Period. Math. Hungar. 21 (1990), 113–152.

[IIKKN1] R. Inoue, O. Iyama, B. Keller, A. Kuniba, and T. Nakanishi, Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras, I: Type B_r, preprint, arXiv:1001.1880 [math.QA]

[IIKKN2] R. Inoue, O. Iyama, B. Keller, A. Kuniba, and T. Nakanishi, Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras, II: Types C_r, F₄, and G₂, preprint, arXiv:1001.1881 [math.QA]

[IIKNS] R. Inoue, O. Iyama, A. Kuniba, T. Nakanishi, and J. Suzuki, Periodicities of T-systems and Y-systems, Nagoya Math. J. 197 (2010), 59–174.

[Ke1] B. Keller, Cluster algebras, quiver representations and triangulated categories, preprint, arXiv:0807.1960 [math.RT]

[Ke2] B. Keller, The periodicity conjecture for pairs of Dynkin diagrams, preprint, arXiv:1001.1531 [math.RT]

[K1] A. N. Kirillov, Identities for the Rogers dilogarithm function connected with simple Lie algebras, J. Soviet Math. 47 (1989), 2450–2458.

[K2] A. N. Kirillov, Dilogarithm identities, Progr. Theoret. Phys. Suppl. 118 (1995), 61–142.

[KR1] A. N. Kirillov and N. Y. Reshetikhin, Exact solution of the Heisenberg XXZ model of spin s, J. Soviet Math. 35 (1986), 2627–2643.

[KR2] A. N. Kirillov and N. Y. Reshetikhin, Representations of Yangians and multiplicities of the inclusion of the irreducible components of the tensor product of representations of simple Lie algebras, J. Soviet Math. 52 (1990), 3156–3164.

[KnZ] V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nuclear Phys. B 247 (1984), 83–103.

[Ku] A. Kuniba, Thermodynamics of the U_q(X_r⁽¹⁾) Bethe ansatz system with q a root of unity, Nuclear Phys. B 389 (1993), 209–244.

[KuN] A. Kuniba and T. Nakanishi, Spectra in conformal field theories from the Rogers dilogarithm, Modern Phys. Lett. A 7 (1992), 3487–3494.

[KuNS] A. Kuniba, T. Nakanishi, and J. Suzuki, T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), 1–23.

[L] L. Lewin, Polylogarithms and Associated Functions, North-Holland, Amsterdam, 1981.

[N] W. Nahm, “Conformal field theory and torsion elements of the Bloch group” in Frontiers in Number Theory, Physics, and Geometry, II, Springer, Berlin, 2007, 67–132.

[NK] W. Nahm and S. Keegan, Integrable deformations of CFTs and the discrete Hirota equations, preprint, arXiv.0905.3776 [hep-th]

[NRT] W. Nahm, A. Recknagel, and M. Terhoeven, Dilogarithm identities in conformal field theory, Modern Phys. Lett. A 8 (1993), 1835–1847.

[RTV] F. Ravanini, R. Tateo, and A. Valleriani, Dynkin TBA’s, Internat. J. Modern Phys. A 8 (1993), 1707–1727.

[RS] B. Richmond and G. Szekeres, Some formulas related to dilogarithm, the zeta function and the Andrews-Gordon identities, J. Aust. Math. Soc. 31 (1981), 362–373.

[S] A. Szenes, Periodicity of Y-systems and flat connections, Lett. Math. Phys. 89 (2009), 217–230.

[V] A. Y. Volkov, On the periodicity conjecture for Y-systems, Comm. Math. Phys. 276 (2007), 509–517.

[Zag1] D. Zagier, “Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields” in Arithmetic Algebraic Geometry, Progr. Math. 89, Birkhäuser, Boston, 1990, 391–430.

[Zag2] D. Zagier, “The dilogarithm function” in Frontiers in Number Theory, Physics, and Geometry, II, Springer, Berlin, 2007, 3–65.

[Zam] A. B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991), 391–394.

A cohomological Tamagawa number formula

Tue, 31 May 2011 10:19 EDT

Annette Huber, Guido Kings

Source: Nagoya Math. J., Volume 202, 45--75.

Abstract:
For smooth linear group schemes over $\mathbb{Z}$ , we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori, the cohomological and the motivic Tamagawa measures coincide, which proves again the Bloch-Kato conjecture for motives associated to tori.

References:
[B] S. Bloch, A note on height pairings, Tamagawa numbers, and the Birch-Swinnerton-Dyer conjecture, Invent. Math. 58 (1980), 65–76.

[BK] S. Bloch and K. Kato, “L-functions and Tamagawa numbers of motives” in The Grothendieck Festschrift, Vol. I, Progr. Math. 86, Birkhäuser, Boston, 1990, 333–400.

[Bo] A. Borel, Cohomologie de SL_n et valeurs de fonctions zeta aux points entiers, Ann. Sc. Norm. Supér. Pisa Cl. Sci. (5) 4 (1977), 613–636.

[BLR] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin, 1990.

[CR] C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. II: With Applications to Finite Groups and Orders, Pure Appl. Math., Wiley, New York, 1987.

[F] J.-M. Fontaine, Valeurs spéciales des fonctions L de motifs, Astérisque 206 (1992), Séminaire Bourbaki, no. 751.

[G] B. H. Gross, On the motive of a reductive group, Invent. Math. 130 (1997), 287–313.

[H] A. Huber, Poincaré duality for p-adic Lie groups, Arch. Math. 95 (2010), 509–517.

[HK1] A. Huber and G. Kings, Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters, Duke Math. J. 119 (2003), 393–464.

[HK2] A. Huber and G. Kings, A p-adic analogue of the Borel regulator and the Bloch-Kato exponential map, J. Inst. Math. Jussieu 10 (2011), 149–190.

[HKN] A. Huber, G. Kings, and N. Naumann, Some complements to the Lazard isomorphism, Compos. Math. 147 (2011), 235–262.

[L] M. Lazard, Groupes analytiques p-adiques, Publ. Math. Inst. Hautes Études Sci. 26 (1965).

[M] J. S. Milne, Arithmetic Duality Theorems, Perspect. Math. 1, Academic Press, Boston, 1986.

[N] N. Naumann, Arithmetically defined dense subgroups of Morava stabilizer groups, Compos. Math. 144 (2008), 247–270.

[O] T. Ono, On the Tamagawa number of algebraic tori, Ann. of Math. (2) 78 (1963), 47–73.

[PR] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure Appl. Math. 139, Academic Press, Boston, 1994.

[S] J.-P. Serre, Lie Algebras and Lie Groups, Harvard University Lectures, W. A. Benjamin, New York, 1965.

[W] A. Weil, Adeles and Algebraic Groups, with appendices by M. Demazure and T. Ono, Progr. Math. 23, Birkhäuser, Boston, 1982.

Locally conformally Kähler metrics on Kato surfaces

Tue, 31 May 2011 10:19 EDT

Marco Brunella

Source: Nagoya Math. J., Volume 202, 77--81.

Abstract:
We show that every Kato surface admits a locally conformally Kähler metric.

References:
[Bel] F. A. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), 1–40.

[Bru] M. Brunella, Locally conformally Kähler metrics on certain non-Kählerian surfaces, Math. Ann. 346 (2010), 629–639.

[Dlo] G. Dloussky, Structure des surfaces de Kato, Mém. Soc. Math. France (N.S.), no. 14 (1984).

[Kat] M. Kato, “Compact complex manifolds containing ‘global’ spherical shells” in Proceedings of the International Symposium on Algebraic Geometry (Kyoto, 1977), Kinokuniya, Tokyo, 1978, 45–84.

The rationality problem for norm one tori

Tue, 31 May 2011 10:19 EDT

Shizuo Endo

Source: Nagoya Math. J., Volume 202, 83--106.

Abstract:
We consider the problem of whether the norm one torus defined by a finite separable field extension $K/k$ is stably (or retract) rational over $k$ . This has already been solved for the case where $K/k$ is a Galois extension. In this paper, we solve the problem for the case where $K/k$ is a non-Galois extension such that the Galois group of the Galois closure of $K/k$ is nilpotent or metacyclic.

References:
[Be] Y. Berkovich, Groups of Prime Power Order, I, de Gruyter, Berlin, 2008.

[Br] K. S. Brown, Cohomology of Groups, Grad. Texts in Math. 87, Springer, New York, 1982.

[CS1] J.-L. Colliot-Thélène and J.-J. Sansuc, La R-équivalence sur les tores, Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 175–230.

[CS2] J.-L. Colliot-Thélène and J.-J. Sansuc, Principal homogeneous spaces under flasque tori: Applications, J. Algebra 106 (1987), 148–205.

[CK] A. Cortella and B. Kunyavskiĭ, Rationality problem for generic tori in simple groups, J. Algebra 225 (2000), 771–793.

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[E] S. Endo, On the rationality of algebraic tori of norm type, J. Algebra 235 (2001), 27–35.

[EM1] S. Endo and T. Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan 25 (1973), 7–26.

[EM2] S. Endo and T. Miyata, On a classification of the function fields of algebraic tori, Nagoya Math. J. 56 (1975), 85–104; Correction, Nagoya Math. J. 79 (1980), 187–190.

[F] M. Florence, Non rationality of some norm one tori, preprint, 2006.

[lB] L. Le Bruyn, Generic norm one tori, Nieuw Arch. Wiskd. (5) 13 (1995), 401–407.

[LL] N. Lemire and M. Lorenz, On certain lattices associated with generic division algebras, J. Group Theory 3 (2000), 385–405.

[R] M. Rosen, Representations of twisted group rings, Ph.D. dissertation, Princeton University, Princeton, N. J., 1963.

[S] D. J. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math. 47 (1984), 165–215.

[V] V. E. Voskresenskiĭ, Algebraic Groups and Their Birational Invariants, Transl. Math. Monogr. 179, Amer. Math. Soc., Providence, 1998.

Harmonic morphisms applied to classical potential theory

Tue, 31 May 2011 10:19 EDT

Bent Fuglede

Source: Nagoya Math. J., Volume 202, 107--126.

Abstract:
It is shown that if $\varphi $ denotes a harmonic morphism of type Bl between suitable Brelot harmonic spaces $X$ and $Y$ , then a function $f$ , defined on an open set $V\subset Y$ , is superharmonic if and only if $f\circ \varphi $ is superharmonic on $\varphi ^{-1}(V)\subset X$ . The “only if” part is due to Constantinescu and Cornea, with $\varphi $ denoting any harmonic morphism between two Brelot spaces. A similar result is obtained for finely superharmonic functions defined on finely open sets. These results apply, for example, to the case where $\varphi $ is the projection from $\mathbb{R}^{N}$ to $\mathbb{R}^{n}$ ( $N\textgreater n\geq1$ ) or where $\varphi $ is the radial projection from $\mathbb{R}^{N}\setminus\{0\}$ to the unit sphere in $\mathbb{R}^{N}$ ( $N\ge2$ ).

References:
[AG] D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer, Berlin, 2001.

[BW] P. Baird and J. C. Wood, Harmonic Morphisms Between Riemannian Manifolds, Clarendon Press, Oxford, 2003.

[Ba] H. Bauer, Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Math. 22, Springer, Berlin, 1966.

[BH] J. Bliedtner and W. Hansen, Potential Theory — An Analytic and Probabilistic Approach to Balayage, Springer, Berlin, 1986.

[Br] M. Brelot, Lectures on Potential Theory, Tata Inst. Fund. Res., Mumbai, 1960.

[CC1] C. Constantinescu and A. Cornea, Ideale Ränder Riemannscher Flächen, Ergeb. Math. Grenzgeb. 32, Springer, Berlin, 1963.

[CC2] C. Constantinescu and A. Cornea, Compactifications of harmonic spaces, Nagoya Math. J. 25 (1965), 1–57.

[CC3] C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Grundlehren Math. Wiss., Band 158, Springer, Berlin, 1972.

[DL] J. Deny and P. Lelong, Étude des fonctions sousharmoniques dans un cylindre ou dans un cône, Bull. Soc. Math. France 75 (1947), 89–112.

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[F3] B. Fuglede, Finely harmonic mappings and finely holomorphic functions, Ann. Acad. Sci. Fenn. Math. 2 (1976), 113–127.

[F4] B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978), 107–144.

[F5] B. Fuglede, “Harmonic morphisms” in Complex Analysis (Joensuu, 1978), Lecture Notes in Math. 747, Springer, Berlin, 1979, 123–131.

[F6] B. Fuglede, Harnack sets and openness of harmonic morphisms, Math. Ann. 241 (1979), 181–186.

[G1] S. J. Gardiner, The Lusin-Primalov theorem for subharmonic functions, Proc. Amer. Math. Soc. 124 (1996), 3721–3727.

[G2] S. J. Gardiner, Growth properties of superharmonic functions along rays, Proc. Amer. Math. Soc. 128 (2000), 1963–1970.

[GH] S. J. Gardiner and W. Hansen, The Riesz decomposition of finely superharmonic functions, Adv. Math. 214 (2007), 417–436.

[Ha] W. Hansen, Abbildungen harmonischer Räume mit Anwendung auf die Laplace und Wärmeleitungsgleichung, Ann. Inst. Fourier (Grenoble) 21 (1971), 203–216.

[HKM] J. Heinonen, T. Kilpeläinen, and O. Martio, Harmonic morphisms in nonlinear potential theory, Nagoya Math. J. 125 (1992), 115–140.

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[He] R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415–571.

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[Jan] K. Janssen, A cofine domination principle for harmonic spaces, Math. Z. 141 (1975), 185–191.

[La1] I. Laine, Covering properties of harmonic Bl-mappings, II, Ann. Acad. Sci. Fenn. Math. 570 (1974), 3–13.

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[Me] C. Meghea, Compactification des espaces harmoniques, Lecture Notes in Math. 222, Springer, Berlin, 1971.

Kummer surfaces associated to $(1,2)$ -polarized abelian surfaces

Tue, 31 May 2011 10:19 EDT

Afsaneh Mehran

Source: Nagoya Math. J., Volume 202, 127--143.

Abstract:
The aim of this paper is to describe the geometry of the generic Kummer surface associated to a $(1,2)$ -polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of type $I_{2}$ .

References:
[B] A. Beauville, Préliminaires sur les périodes des surfaces K3, Astérisque 126 (1985), 91–97.

[BL] C. Birkenhake and H. Lange, Complex Abelian Varieties, 2nd ed., Grundlehren Math. Wiss. 302, Springer, Berlin, 2004.

[H] R. W. H. T. Hudson, Kummer’s Quartic Surface, revised reprint of the 1905 original, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1990.

[K] J. H. Keum, Automorphisms of Jacobian Kummer surfaces, Compos. Math. 107 (1997), 269–288.

[M] A. Mehran, Double covers of Kummer surfaces, Manuscripta Math. 123 (2007), 205–235.

[Na] I. Naruki, On metamorphosis of Kummer surfaces, Hokkaido Math. J. 20 (1991), 407–415.

[Ni1] V. V. Nikulin, Kummer surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 278–293.

[Ni2] V. V. Nikulin, Finite groups of automorphisms of Kählerian K3 surfaces, Tr. Mosk. Mat. Obs. 38 (1979), 75–137.

[PŠŠ] I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli’s theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572.

[SI] T. Shioda and H. Inose, “On singular K3 surfaces” in Complex Analysis and Algebraic Geometry, Iwanami, Tokyo, 1977, 119–136.

Normality of orbit closures in the enhanced nilpotent cone

Thu, 18 Aug 2011 11:45 EDT

Pramod N. Achar, Anthony Henderson, Benjamin F. Jones

Source: Nagoya Math. J., Volume 203, 1--45.

Abstract:
We continue the study of the closures of $\operatorname{GL}(V)$ -orbits in the enhanced nilpotent cone $V\times\mathcal{N}$ begun by the first two authors. We prove that each closure is an invariant-theoretic quotient of a suitably defined enhanced quiver variety. We conjecture, and prove in special cases, that these enhanced quiver varieties are normal complete intersections, implying that the enhanced nilpotent orbit closures are also normal.

Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero

Thu, 18 Aug 2011 11:45 EDT

Yuichiro Hoshi

Source: Nagoya Math. J., Volume 203, 47--100.

Abstract:
Let $l$ be a prime number. In this paper, we prove that the isomorphism class of an $l$ -monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro- $l$ outer Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.

On self-injective algebras of stable dimension zero

Thu, 18 Aug 2011 11:45 EDT

Michio Yoshiwaki

Source: Nagoya Math. J., Volume 203, 101--108.

Abstract:
Let $A$ be a self-injective algebra over an algebraically closed field. We study the stable dimension of $A$ , which is the dimension of the stable module category of $A$ in the sense of Rouquier. Then we prove that $A$ is representation-finite if the stable dimension of $A$ is zero.

Notes on boundedness of spectral multipliers on Hardy spaces associated to operators

Thu, 18 Aug 2011 11:45 EDT

Bui The Anh

Source: Nagoya Math. J., Volume 203, 109--122.

Abstract:
Let $L$ be a nonnegative self-adjoint operator on $L^{2}(X)$ , where $X$ is a space of homogeneous type. Assume that $L$ generates an analytic semigroup $e^{-tL}$ whose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplier $F(L)$ is bounded on $H^{p}_{L}(X)$ for $0\textless p\leq1$ , the Hardy space associated to operator $L$ , when $F$ is a suitable function.

Stickelberger elements and Kolyvagin systems

Thu, 18 Aug 2011 11:45 EDT

Kâzım Büyükboduk

Source: Nagoya Math. J., Volume 203, 123--173.

Abstract:
In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank is $r\textgreater 1$ (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.

Geometry of $G_{2}$ orbits and isoparametric hypersurfaces

Thu, 18 Aug 2011 11:45 EDT

Reiko Miyaoka

Source: Nagoya Math. J., Volume 203, 175--189.

Abstract:
We characterize the adjoint $G_{2}$ orbits in the Lie algebra $\mathfrak{g}$ of $G_{2}$ as fibered spaces over $S^{6}$ with fibers given by the complex Cartan hypersurfaces. This combines the isoparametric hypersurfaces of case $(g,m)=(6,2)$ with case $(3,2)$ . The fibrations on two singular orbits turn out to be diffeomorphic to the twistor fibrations of $S^{6}$ and $G_{2}/SO(4)$ . From the symplectic point of view, we show that there exists a 2-parameter family of Lagrangian submanifolds on every orbit.

Isoparametric hypersurfaces with four principal curvatures, II

Mon, 05 Dec 2011 12:57 EST

Quo-Shin Chi

Source: Nagoya Math. J., Volume 204, 1--18.

Abstract:
In this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $(3,4)$ in $S^{15}$ is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively as of OT-FKM type . In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint $m_{2}\geq 2m_{1}-1$ is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs $(4,5)$ , $(3,4)$ , $(7,8)$ , and $(6,9)$ , where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs $(4,5)$ , $(6,9)$ , and $(7,8)$ remain open now.

Variation formulas for principal functions, II: Applications to variation for harmonic spans

Mon, 05 Dec 2011 12:57 EST

Sachiko Hamano, Fumio Maitani, Hiroshi Yamaguchi

Source: Nagoya Math. J., Volume 204, 19--56.

Abstract:
A domain $D\subset\mathbb{C}_{z}$ admits the circular slit mapping $P(z)$ for $a,b\in D$ such that $P(z)-1/(z-a)$ is regular at $a$ and $P(b)=0$ . We call $p(z)=\log|P(z)|$ the $L_{1}$ - principal function and $\alpha =\log|P'(b)|$ the $L_{1}$ - constant , and similarly, the radial slit mapping $Q(z)$ implies the $L_{0}$ -principal function $q(z)$ and the $L_{0}$ -constant $\beta$ . We call $s=\alpha-\beta $ the harmonic span for $(D,a,b)$ . We show the geometric meaning of $s$ . Hamano showed the variation formula for the $L_{1}$ -constant $\alpha(t)$ for the moving domain $D(t)$ in $\mathbb{C}_{z}$ with $t\in B:=\{t\in \mathbb{C}:|t|\textless \rho\}$ . We show the corresponding formula for the $L_{0}$ -constant $\beta(t)$ for $D(t)$ and combine these to prove that, if the total space $\mathcal{D}=\bigcup_{t\in B}(t,D(t))$ is pseudoconvex in $B\times\mathbb{C}_{z}$ , then $s(t)$ is subharmonic on $B$ . As a direct application, we have the subharmonicity of $\log \cosh d(t)$ on $B$ , where $d(t)$ is the Poincaré distance between $a$ and $b$ on $D(t)$ .

Cohen-Macaulay binomial edge ideals

Mon, 05 Dec 2011 12:57 EST

Viviana Ene, Jürgen Herzog, Takayuki Hibi

Source: Nagoya Math. J., Volume 204, 57--68.

Abstract:
We study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen-Macaulay.

Quantum $(\mathfrak{sl}_{n},\wedge V_{n})$ link invariant and matrix factorizations

Mon, 05 Dec 2011 12:57 EST

Yasuyoshi Yonezawa

Source: Nagoya Math. J., Volume 204, 69--123.

Abstract:
In this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum $(\mathfrak{sl}_{n},\wedge V_{n})$ link invariant, where $\wedge V_{n}$ is the set of fundamental representations of $U_{q}(\mathfrak{sl}_{n})$ . In the case of an oriented link diagram composed of $[k,1]$ -crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general $[i,j]$ -crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.

Ordinary varieties and the comparison between multiplier ideals and test ideals

Mon, 05 Dec 2011 12:57 EST

Mircea Mustaţă, Vasudevan Srinivas

Source: Nagoya Math. J., Volume 204, 125--157.

Abstract:
We consider the following conjecture: if $X$ is a smooth and irreducible $n$ -dimensional projective variety over a field $k$ of characteristic zero, then there is a dense set of reductions $X_{s}$ to positive characteristic such that the action of the Frobenius morphism on $H^{n}(X_{s},\mathcal {O}_{X_{s}})$ is bijective. There is another conjecture relating certain invariants of singularities in characteristic zero (the multiplier ideals) with invariants in positive characteristic (the test ideals). We prove that the former conjecture implies the latter one in the case of ambient nonsingular varieties.

How not to prove the Alon-Tarsi conjecture

Thu, 01 Mar 2012 09:10 EST

Douglas S. Stones, Ian M. Wanless

Source: Nagoya Math. J., Volume 205, 1--24.

Abstract:
The sign of a Latin square is $-1$ if it has an odd number of rows and columns that are odd permutations; otherwise, it is $+1$ . Let $L^{\text{\textsc{e}}}_{n}$ and $L^{\text{\textsc{o}}}_{n}$ be, respectively, the number of Latin squares of order $n$ with sign $+1$ and $-1$ . The Alon-Tarsi conjecture asserts that $L^{\text{\textsc{e}}}_{n}\neq L^{\text{\textsc{o}}}_{n}$ when $n$ is even. Drisko showed that $L^{\text{\textsc{e}}}_{p+1}\notequiv L^{\text{\textsc{o}}}_{p+1}\pmod{p^{3}}$ for prime $p\geq3$ and asked if similar congruences hold for orders of the form $p^{k}+1$ , $p+3$ , or $pq+1$ . In this article we show that if $t\leq n$ , then $L^{\text{\textsc{e}}}_{n+1}\notequiv L^{\text{\textsc{o}}}_{n+1}\pmod{t^{3}}$ only if $t=n$ and $n$ is an odd prime, thereby showing that Drisko’s method cannot be extended to encompass any of the three suggested cases. We also extend exact computation to $n\leq9$ , discuss asymptotics for $L^{\text{\textsc{o}}}/L^{\text{\textsc{e}}}$ , and propose a generalization of the Alon-Tarsi conjecture.

Quantitative extensions of pluricanonical forms and closed positive currents

Thu, 01 Mar 2012 09:10 EST

Bo Berndtsson, Mihai Păun

Source: Nagoya Math. J., Volume 205, 25--65.

Abstract:
We establish here several “invariance of plurigenera type” theorems for twisted pluricanonical forms and metrics of adjoint $\mathbb{R}$ -bundles.

$q$ -Titchmarsh-Weyl theory: Series expansion

Thu, 01 Mar 2012 09:10 EST

M. H. Annaby, Z. S. Mansour, I. A. Soliman

Source: Nagoya Math. J., Volume 205, 67--118.

Abstract:
We establish a $q$ -Titchmarsh-Weyl theory for singular $q$ -Sturm-Liouville problems. We define $q$ -limit-point and $q$ -limit circle singularities, and we give sufficient conditions which guarantee that the singular point is in a limit-point case. The resolvent is constructed in terms of Green’s function of the problem. We derive the eigenfunction expansion in its series form. A detailed worked example involving Jackson $q$ -Bessel functions is given. This example leads to the completeness of a wide class of $q$ -cylindrical functions.

Schatten $p$ -class property of pseudodifferential operators with symbols in modulation spaces

Thu, 01 Mar 2012 09:10 EST

Masaharu Kobayashi, Akihiko Miyachi

Source: Nagoya Math. J., Volume 205, 119--148.

Abstract:
It is proved that the pseudodifferential operators $\sigma_{t}(X,D)$ belong to the Schatten $p$ -class $C_{p}$ , $0\textless p\leq2$ , if the symbol $\sigma(x,\omega)$ is in certain modulation spaces on ${\mathbf{R}}^{d}_{x}\times{\mathbf{R}}^{d}_{\omega}$ .

Bounds on the Hilbert-Kunz multiplicity

Thu, 01 Mar 2012 09:10 EST

Olgur Celikbas, Hailong Dao, Craig Huneke, Yi Zhang

Source: Nagoya Math. J., Volume 205, 149--165.

Abstract:
In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed nonregular local rings, bounding them uniformly away from 1. Our results improve previous work of Aberbach and Enescu.

Lowest weights in cohomology of variations of Hodge structure

Tue, 22 May 2012 08:34 EDT

Chris Peters, Morihiko Saito

Source: Nagoya Math. J., Volume 206, 1--24.

Abstract:
Let $X$ be an irreducible complex analytic space with $j:U\hookrightarrow X$ an immersion of a smooth Zariski-open subset, and let ${\mathbb {V}}$ be a variation of Hodge structure of weight $n$ over $U$ . Assume that $X$ is compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent, $IH^{k}(X,{\mathbb {V}})$ is known to carry a pure Hodge structure of weight $k+n$ , while $H^{k}(U,{\mathbb {V}})$ carries a mixed Hodge structure of weight at least $k+n$ . In this note it is shown that the image of the natural map $IH^{k}(X,{\mathbb {V}})\to H^{k}(U,{\mathbb {V}})$ is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complement $X-U$ is not a hypersurface.

Vanishing theorems on complete manifolds with weighted Poincaré inequality and applications

Tue, 22 May 2012 08:34 EDT

Hai-Ping Fu, Deng-Yun Yang

Source: Nagoya Math. J., Volume 206, 25--37.

Abstract:
Two vanishing theorems for harmonic map and $L^{2}$ harmonic $1$ -form on complete noncompact manifolds are proved under certain geometric assumptions, which generalize results of [13], [15], [18], [19], and [20]. As applications, we improve some main results in [2], [4], [6], [9], [12], [20], [22], [24], and [25].

A Noether-Lefschetz theorem for varieties of $r$ -planes in complete intersections

Tue, 22 May 2012 08:34 EDT

Zhi Jiang

Source: Nagoya Math. J., Volume 206, 39--66.

Abstract:
We prove a Noether-Lefschetz type theorem for varieties of $r$ -planes in complete intersections. We then use it to study the Abel-Jacobi map of planes on a smooth cubic fivefold.

Supplement to classification of threefold divisorial contractions

Tue, 22 May 2012 08:34 EDT

Masayuki Kawakita

Source: Nagoya Math. J., Volume 206, 67--73.

Abstract:
Every threefold divisorial contraction to a non-Gorenstein point is a weighted blowup.

Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

Tue, 22 May 2012 08:34 EDT

Alexandru Dimca

Source: Nagoya Math. J., Volume 206, 75--97.

Abstract:
The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given. It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over $\mathbb{Q}$ , has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial. We construct a hyperplane arrangement defined over $\mathbb{Q}$ , whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

Finite symplectic actions on the $K3$ lattice

Tue, 22 May 2012 08:34 EDT

Kenji Hashimoto

Source: Nagoya Math. J., Volume 206, 99--153.

Abstract:
In this paper, we study finite symplectic actions on $K3$ surfaces $X$ , that is, actions of finite groups $G$ on $X$ which act on $H^{2,0}(X)$ trivially. We show that the action on the $K3$ lattice $H^{2}(X,\mathbb {Z})$ induced by a symplectic action of $G$ on $X$ depends only on $G$ up to isomorphism, except for five groups.

Symplectic fillings of links of quotient surface singularities

Thu, 26 Jul 2012 09:36 EDT

Mohan Bhupal, Kaoru Ono

Source: Nagoya Math. J., Volume 207, 1--45.

Abstract:
We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

The Calogero-Moser partition for $G(m,d,n)$

Thu, 26 Jul 2012 09:36 EDT

Gwyn Bellamy

Source: Nagoya Math. J., Volume 207, 47--77.

Abstract:
We show that it is possible to deduce the Calogero-Moser partition of the irreducible representations of the complex reflection groups $G(m,d,n)$ from the corresponding partition for $G(m,1,n)$ . This confirms, in the case $W=G(m,d,n)$ , a conjecture of Gordon and Martino relating the Calogero-Moser partition to Rouquier families for the corresponding cyclotomic Hecke algebra.

Alternative polarizations of Borel fixed ideals

Thu, 26 Jul 2012 09:36 EDT

Kohji Yanagawa

Source: Nagoya Math. J., Volume 207, 79--93.

Abstract:
For a monomial ideal $I$ of a polynomial ring $S$ , a polarization of $I$ is a square-free monomial ideal $J$ of a larger polynomial ring $\widetilde {S}$ such that $S/I$ is a quotient of $\widetilde {S}/J$ by a (linear) regular sequence. We show that a Borel fixed ideal admits a nonstandard polarization. For example, while the usual polarization sends $xy^{2}\in S$ to $x_{1}y_{1}y_{2}\in \widetilde {S}$ , ours sends it to $x_{1}y_{2}y_{3}$ . Using this idea, we recover/refine the results on square-free operation in the shifting theory of simplicial complexes. The present paper generalizes a result of Nagel and Reiner, although our approach is very different.

Functions with finite Dirichlet sum of order $p$ and quasi-monomorphisms of infinite graphs

Thu, 26 Jul 2012 09:36 EDT

Tae Hattori, Atsushi Kasue

Source: Nagoya Math. J., Volume 207, 95--138.

Abstract:
In this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of $p$ -Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension $n$ and it is not $p$ -parabolic for $p\textgreater n-1$ , then it admits a lot of $p$ -harmonic functions with finite Dirichlet sum of order $p$ .

The first line of the Bockstein spectral sequence on a monochromatic spectrum at an odd prime

Thu, 26 Jul 2012 09:36 EDT

Ryo Kato, Katsumi Shimomura

Source: Nagoya Math. J., Volume 207, 139--157.

Abstract:
The chromatic spectral sequence was introduced by Miller, Ravenel, and Wilson to compute the $E_{2}$ -term of the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres. The $E_{1}$ -term $E_{1}^{s,t}(k)$ of the spectral sequence is an Ext group of $BP_{*}BP$ -comodules. There is a sequence of Ext groups $E_{1}^{s,t}(n-s)$ for nonnegative integers $n$ with $E_{1}^{s,t}(0)=E_{1}^{s,t}$ , and there are Bockstein spectral sequences computing a module $E_{1}^{s,*}(n-s)$ from $E_{1}^{s-1,*}(n-s+1)$ . So far, a small number of the $E_{1}$ -terms are determined. Here, we determine the $E_{1}^{1,1}(n-1)=\operatorname {Ext}^{1}M^{1}_{n-1}$ for $p\textgreater 2$ and $n\textgreater 3$ by computing the Bockstein spectral sequence with $E_{1}$ -term $E_{1}^{0,s}(n)$ for $s=1,2$ . As an application, we study the nontriviality of the action of $\alpha_{1}$ and $\beta_{1}$ in the homotopy groups of the second Smith-Toda spectrum $V(2)$ .

Lefschetz operator and local Langlands mod $\ell$ : The regular case

Wed, 05 Dec 2012 09:09 EST

Jean-François Dat

Source: Nagoya Math. J., Volume 208, 1--38.

Abstract:
Let $p$ and $\ell$ be two distinct primes. The aim of this paper is to show how, under a certain congruence hypothesis, the mod $\ell$ cohomology complex of the Lubin-Tate tower, together with a natural Lefschetz operator, provides a geometric interpretation of Vignéras’s local Langlands correspondence modulo $\ell$ for unipotent representations.

Fonctions zêta $\ell$ -modulaires

Wed, 05 Dec 2012 09:09 EST

Alberto Mínguez

Source: Nagoya Math. J., Volume 208, 39--65.

Abstract:
Let $\mathrm{F}$ be a non-Archimedean locally compact field, of residual characteristic $p$ , and let $\mathrm{D}$ be a finite-dimensional central division $\mathrm{F}$ -algebra. Let $\ell$ be a prime number different from $p$ . In this article, generalizing the results of [GJ], we associate, to each $\ell$ -modular smooth irreducible representation $\pi$ of $\mathrm{GL}_{m}(\mathrm{D})$ , two invariants $L(T,\pi)$ , $\varepsilon(T,\pi,\psi)$ , where $T$ is an indeterminate and $\psi$ is a nontrivial character of $\mathrm{F}$ .

Doubling zeta integrals and local factors for metaplectic groups

Wed, 05 Dec 2012 09:09 EST

Wee Teck Gan

Source: Nagoya Math. J., Volume 208, 67--95.

Abstract:
We develop the theory of the doubling zeta integral of Piatetski-Shapiro and Rallis for metaplectic groups $\operatorname{Mp}_{2n}$ , and we use it to give precise definitions of the local $\gamma$ -factors, $L$ -factors, and $\epsilon$ -factors for irreducible representations of $\operatorname{Mp}_{2n}\times\operatorname{GL}_{1}$ , following the footsteps of Lapid and Rallis.

On the Kottwitz-Shelstad normalization of transfer factors for automorphic induction for $\mathrm{GL}_{n}$

Wed, 05 Dec 2012 09:09 EST

Kaoru Hiraga, Atsushi Ichino

Source: Nagoya Math. J., Volume 208, 97--144.

Abstract:
Automorphic induction for $\mathrm{GL}_{n}$ is a case of endoscopic transfer, and its character identity was established by Henniart and Herb, up to a constant of proportionality. We determine this constant in terms of the Kottwitz-Shelstad normalization of transfer factors, which involves certain $\varepsilon$ -factors.

Some conjectures on endoscopic representations in odd orthogonal groups

Wed, 05 Dec 2012 09:09 EST

David Ginzburg, Dihua Jiang

Source: Nagoya Math. J., Volume 208, 145--170.

Abstract:
In this paper, we introduce two conjectures on characterizations of endoscopy structures of irreducible generic cuspidal automorphic representations of odd special orthogonal groups in terms of nonvanishing of certain period of automorphic forms. We discuss a relation between the two conjectures and prove that a special case of Conjecture 1 (and hence Conjecture 2) is true.

Extensions of representations of $p$ -adic groups

Wed, 05 Dec 2012 09:09 EST

Jeffrey D. Adler, Dipendra Prasad

Source: Nagoya Math. J., Volume 208, 171--199.

Abstract:
We calculate extensions between certain irreducible admissible representations of $p$ -adic groups.

Matrix coefficients of the large discrete series representations of $\operatorname{Sp}(2;\mathbf {R})$ as hypergeometric series of two variables

Wed, 05 Dec 2012 09:09 EST

Takayuki Oda

Source: Nagoya Math. J., Volume 208, 201--263.

Abstract:
We investigate the radial part of the matrix coefficients with minimal $K$ -types of the large discrete series representations of $\operatorname{Sp}(2;\mathbf {R})$ . They satisfy certain difference-differential equations derived from Schmid operators. This system is reduced to a holonomic system of rank 4, which is finally found to be equivalent to higher-order hypergeometric series in the sense of Appell and Kampé de Fériet.

On zeta functions associated to symmetric matrices, II: Functional equations and special values

Wed, 05 Dec 2012 09:09 EST

Tomoyoshi Ibukiyama, Hiroshi Saito

Source: Nagoya Math. J., Volume 208, 265--316.

Abstract:
New simple functional equations of zeta functions of the prehomogeneous vector spaces consisting of symmetric matrices are obtained, using explicit forms of zeta functions in the previous paper, Part I, and real analytic Eisenstein series of half-integral weight. When the matrix size is 2, our functional equations are identical with the ones by Shintani, but we give here an alternative proof. The special values of the zeta functions at nonpositive integers and the residues are also explicitly obtained. These special values, written by products of Bernoulli numbers, are used to give the contribution of “central” unipotent elements in the dimension formula of Siegel cusp forms of any degree. These results lead us to a conjecture on explicit values of dimensions of Siegel cusp forms of any torsion-free principal congruence subgroups of the symplectic groups of general degree.

On low-dimensional Ricci limit spaces

Wed, 27 Feb 2013 09:46 EST

Shouhei Honda

Source: Nagoya Math. J., Volume 209, 1--22.

Abstract:
We call a Gromov–Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space . Furthermore, we prove that any Ricci limit space has integral Hausdorff dimension, provided that its Hausdorff dimension is not greater than 2. We also classify $1$ -dimensional Ricci limit spaces.

The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces

Wed, 27 Feb 2013 09:46 EST

Minoru Tanaka, Kei Kondo

Source: Nagoya Math. J., Volume 209, 23--34.

Abstract:
We construct distinctive surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we prove that a complete noncompact Riemannian manifold $M$ is homeomorphic to the interior of a compact manifold with boundary if the manifold $M$ is not less curved than a noncompact model surface $\widetilde {M}$ of revolution and if the total curvature of the model surface $\widetilde {M}$ is finite and less than $2\pi$ . By the first result mentioned above, the second result covers a much wider class of manifolds than that of complete noncompact Riemannian manifolds whose sectional curvatures are bounded from below by a constant.

On the uniform spread of almost simple linear groups

Wed, 27 Feb 2013 09:46 EST

Timothy C. Burness, Simon Guest

Source: Nagoya Math. J., Volume 209, 35--109.

Abstract:
Let $G$ be a finite group, and let $k$ be a nonnegative integer. We say that $G$ has uniform spread $k$ if there exists a fixed conjugacy class $C$ in $G$ with the property that for any $k$ nontrivial elements $x_{1},\ldots,x_{k}$ in $G$ there exists $y\in C$ such that $G=\langle x_{i},y\rangle $ for all $i$ . Further, the exact uniform spread of $G$ , denoted by $u(G)$ , is the largest $k$ such that $G$ has the uniform spread $k$ property. By a theorem of Breuer, Guralnick, and Kantor, $u(G)\ge2$ for every finite simple group $G$ . Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if $G=\langle \operatorname {PSL}_{n}(q),g\rangle $ is almost simple, then $u(G)\ge2$ (unless $G\cong S_{6}$ ), and we determine precisely when $u(G)$ tends to infinity as $|G|$ tends to infinity.

Nonarchimedean geometry of Witt vectors

Wed, 27 Feb 2013 09:46 EST

Kiran S. Kedlaya

Source: Nagoya Math. J., Volume 209, 111--165.

Abstract:
Let $R$ be a perfect $\mathbb{F}_{p}$ -algebra equipped with the trivial norm. Let $W(R)$ be the ring of $p$ -typical Witt vectors over $R$ equipped with the $p$ -adic norm. At the level of nonarchimedean analytic spaces (in the sense of Berkovich), we demonstrate a close analogy between $W(R)$ and the polynomial ring $R[T]$ equipped with the Gauss norm, in which the role of the structure morphism from $R$ to $R[T]$ is played by the Teichmüller map. For instance, we show that the analytic space associated to $R$ is a strong deformation retract of the space associated to $W(R)$ . We also show that each fiber forms a tree under the relation of pointwise comparison, and we classify the points of fibers in the manner of Berkovich’s classification of points of a nonarchimedean disk. Some results pertain to the study of $p$ -adic representations of étale fundamental groups of nonarchimedean analytic spaces (i.e., relative $p$ -adic Hodge theory).

Kulikov surfaces form a connected component of the moduli space

Mon, 20 May 2013 09:54 EDT

Tsz On Mario Chan, Stephen Coughlan

Source: Nagoya Math. J., Volume 210, 1--27.

Abstract:
We show that the Kulikov surfaces form a connected component of the moduli space of surfaces of general type with $p_{g}=0$ and $K^{2}=6$ . We also give a new description for these surfaces, extending ideas of Inoue. Finally, we calculate the bicanonical degree of Kulikov surfaces and prove that they verify the Bloch conjecture.

The Brauer–Manin pairing, class field theory, and motivic homology

Mon, 20 May 2013 09:54 EDT

Takao Yamazaki

Source: Nagoya Math. J., Volume 210, 29--58.

Abstract:
For a smooth proper variety over a $p$ -adic field, its Brauer group and abelian fundamental group are related to higher Chow groups by the Brauer–Manin pairing and class field theory. We generalize this relation to smooth (possibly nonproper) varieties, using motivic homology and a variant of Wiesend’s ideal class group. Several examples are discussed.

Logarithmic abelian varieties, III: Logarithmic elliptic curves and modular curves

Mon, 20 May 2013 09:54 EDT

Takeshi Kajiwara, Kazuya Kato, Chikara Nakayama

Source: Nagoya Math. J., Volume 210, 59--81.

Abstract:
We illustrate the theory of log abelian varieties and their moduli in the case of log elliptic curves.

Products of pairs of Dehn twists and maximal real Lefschetz fibrations

Mon, 20 May 2013 09:54 EDT

Alex Degtyarev, Nermin Salepci

Source: Nagoya Math. J., Volume 210, 83--132.

Abstract:
We address the problem of existence and uniqueness of a factorization of a given element of the modular group into a product of two Dehn twists. As a geometric application, we conclude that any maximal real elliptic Lefschetz fibration is algebraic.

Estimates for $F$ -jumping numbers and bounds for Hartshorne–Speiser–Lyubeznik numbers

Mon, 20 May 2013 09:54 EDT

Mircea Mustaţă, Wenliang Zhang

Source: Nagoya Math. J., Volume 210, 133--160.

Abstract:
Given an ideal $\mathfrak{a}$ on a smooth variety in characteristic zero, we estimate the $F$ -jumping numbers of the reductions of $\mathfrak{a}$ to positive characteristic in terms of the jumping numbers of $\mathfrak{a}$ and the characteristic. We apply one of our estimates to bound the Hartshorne–Speiser–Lyubeznik invariant for the reduction to positive characteristic of a hypersurface singularity.