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Journal articles on the topic 'Hilbert modular surfaces'

Author: Grafiati

Published: 4 June 2021

Last updated: 30 July 2024

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1

McREYNOLDS, D. B. "Cusps of Hilbert modular varieties." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 3 (2008): 749–59. http://dx.doi.org/10.1017/s0305004107001004.

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AbstractMotivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifoldMto be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3–manifold is diffeo morphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3–manifolds that cannot arise as a cusp cross-section of a 1–cusped n

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2

Faltings, Gerd. "Book Review: Hilbert modular surfaces." Bulletin of the American Mathematical Society 20, no. 2 (1989): 247–52. http://dx.doi.org/10.1090/s0273-0979-1989-15783-2.

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3

McMullen, Curtis T. "Foliations of Hilbert modular surfaces." American Journal of Mathematics 129, no. 1 (2007): 183–215. http://dx.doi.org/10.1353/ajm.2007.0002.

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4

Milio, Enea, and Damien Robert. "Modular polynomials on Hilbert surfaces." Journal of Number Theory 216 (November 2020): 403–59. http://dx.doi.org/10.1016/j.jnt.2020.04.014.

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5

Hamahata, Yoshinori. "Hilbert Modular Surfaces withpg ≤ 1." Mathematische Nachrichten 173, no. 1 (1995): 193–236. http://dx.doi.org/10.1002/mana.19951730113.

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6

Elkies, Noam, and Abhinav Kumar. "K3 surfaces and equations for Hilbert modular surfaces." Algebra & Number Theory 8, no. 10 (2014): 2297–411. http://dx.doi.org/10.2140/ant.2014.8.2297.

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7

Möller, Martin, and Don Zagier. "Modular embeddings of Teichmüller curves." Compositio Mathematica 152, no. 11 (2016): 2269–349. http://dx.doi.org/10.1112/s0010437x16007636.

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Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmüller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apéry-like integrality statement for solutions of Pica

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8

Kumar, Abhinav, and Ronen E. Mukamel. "Real multiplication through explicit correspondences." LMS Journal of Computation and Mathematics 19, A (2016): 29–42. http://dx.doi.org/10.1112/s1461157016000188.

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We compute equations for real multiplication on the divisor classes of genus-2 curves via algebraic correspondences. We do so by implementing van Wamelen’s method for computing equations for endomorphisms of Jacobians on examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies and Kumar. We also compute a correspondence over the universal family for the Hilbert modular surface of discriminant $5$ and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.

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9

Gon, Yasuro. "Determinants of Laplacians on Hilbert modular surfaces." Publicacions Matemàtiques 62 (July 1, 2018): 615–39. http://dx.doi.org/10.5565/publmat6221808.

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10

EHLEN, STEPHAN. "TWISTED BORCHERDS PRODUCTS ON HILBERT MODULAR SURFACES AND THE REGULARIZED THETA LIFT." International Journal of Number Theory 06, no. 07 (2010): 1473–89. http://dx.doi.org/10.1142/s1793042110003642.

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We construct a lifting from weakly holomorphic modular forms of weight 0 for SL 2(ℤ) with integral Fourier coefficients to meromorphic Hilbert modular forms of weight 0 for the full Hilbert modular group of a real quadratic number field with an infinite product expansion and a divisor given by a linear combination of twisted Hirzebruch–Zagier divisors. The construction uses the singular theta lifting by considering a suitable twist of a Siegel theta function. We generalize the work by Bruinier and Yang who showed the existence of the lifting for prime discriminants using a different approach.

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11

Yang, Tonghai. "The Chowla–Selberg Formula and The Colmez Conjecture." Canadian Journal of Mathematics 62, no. 2 (2010): 456–72. http://dx.doi.org/10.4153/cjm-2010-028-x.

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AbstractIn this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.

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12

Tonghai Yang. "An arithmetic intersection formula on Hilbert modular surfaces." American Journal of Mathematics 132, no. 5 (2010): 1275–309. http://dx.doi.org/10.1353/ajm.2010.0002.

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13

McMullen, Curtis T. "Billiards and Teichmüller curves on Hilbert modular surfaces." Journal of the American Mathematical Society 16, no. 4 (2003): 857–85. http://dx.doi.org/10.1090/s0894-0347-03-00432-6.

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14

Funke, Jens, and John Millson. "The geometric theta correspondence for Hilbert modular surfaces." Duke Mathematical Journal 163, no. 1 (2014): 65–116. http://dx.doi.org/10.1215/00127094-2405279.

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15

Berman, Robert J., and Gerard Freixas i Montplet. "An arithmetic Hilbert–Samuel theorem for singular hermitian line bundles and cusp forms." Compositio Mathematica 150, no. 10 (2014): 1703–28. http://dx.doi.org/10.1112/s0010437x14007325.

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AbstractWe prove arithmetic Hilbert–Samuel type theorems for semi-positive singular hermitian line bundles of finite height. This includes the log-singular metrics of Burgos–Kramer–Kühn. The results apply in particular to line bundles of modular forms on some non-compact Shimura varieties. As an example, we treat the case of Hilbert modular surfaces, establishing an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions.

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16

Luethi, Manuel. "Primitive rational points on expanding horospheres in Hilbert modular surfaces." Journal of Number Theory 225 (August 2021): 327–59. http://dx.doi.org/10.1016/j.jnt.2021.02.009.

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17

Virdol, Cristian. "Algebraic cycles on a product of two Hilbert modular surfaces." Transactions of the American Mathematical Society 362, no. 07 (2010): 3691–703. http://dx.doi.org/10.1090/s0002-9947-10-05116-0.

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18

Murty, V. Kumar, and Dinakar Ramakrishnan. "Period relations and the Tate conjecture for Hilbert modular surfaces." Inventiones Mathematicae 89, no. 2 (1987): 319–45. http://dx.doi.org/10.1007/bf01389081.

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19

Bruinier, Jan H., José I. Burgos Gil, and Ulf Kühn. "Borcherds products and arithmetic intersection theory on Hilbert modular surfaces." Duke Mathematical Journal 139, no. 1 (2007): 1–88. http://dx.doi.org/10.1215/s0012-7094-07-13911-5.

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20

Murty, V. Kumar, and Dipendra Prasad. "Tate Cycles on a Product of Two Hilbert Modular Surfaces." Journal of Number Theory 80, no. 1 (2000): 25–43. http://dx.doi.org/10.1006/jnth.1999.2446.

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21

Scholl, A. J. "HILBERT MODULAR SURFACES (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 16)." Bulletin of the London Mathematical Society 24, no. 1 (1992): 92–93. http://dx.doi.org/10.1112/blms/24.1.92.

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22

Bruinier, Jan H. (Jan Hendrik), and Tonghai Yang. "Twisted Borcherds products on Hilbert modular surfaces and their CM values." American Journal of Mathematics 129, no. 3 (2007): 807–41. http://dx.doi.org/10.1353/ajm.2007.0019.

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23

Möller, Martin. "Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces." American Journal of Mathematics 136, no. 4 (2014): 995–1021. http://dx.doi.org/10.1353/ajm.2014.0026.

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24

Kings, Guido. "Higher regulators, Hilbert modular surfaces, and special values of $L$ -functions." Duke Mathematical Journal 92, no. 1 (1998): 61–127. http://dx.doi.org/10.1215/s0012-7094-98-09202-x.

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25

Virdol, Cristian. "On l-adic representations attached to Hilbert and Picard modular surfaces." Journal of Number Theory 130, no. 5 (2010): 1197–211. http://dx.doi.org/10.1016/j.jnt.2009.10.006.

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26

Freixas i Montplet, Gerard, and Siddarth Sankaran. "Twisted Hilbert modular surfaces, arithmetic intersections and the Jacquet–Langlands correspondence." Advances in Mathematics 329 (April 2018): 1–84. http://dx.doi.org/10.1016/j.aim.2018.01.025.

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27

GRITSENKO, VALERI, and KLAUS HULEK. "Minimal Siegel modular threefolds." Mathematical Proceedings of the Cambridge Philosophical Society 123, no. 3 (1998): 461–85. http://dx.doi.org/10.1017/s0305004197002259.

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The starting point of this paper is the maximal extension Γ*t of Γt, the subgroup of Sp4(ℚ) which is conjugate to the paramodular group. Correspondingly we call the quotient [Ascr ]*t=Γ*t\ℍ2 the minimal Siegel modular threefold. The space [Ascr ]*t and the intermediate spaces between [Ascr ]t=Γt\ℍ2 which is the space of (1, t)-polarized abelian surfaces and [Ascr ]*t have not yet been studied in any detail. Using the Torelli theorem we first prove that [Ascr ]*t can be interpreted as the space of Kummer surfaces of (1, t)-polarized abelian surfaces and that a certain degree 2 quotient of [Ascr

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28

Banagl, Markus, and Rajesh S. Kulkarni. "Self-dual Sheaves on Reductive Borel–Serre Compactifications of Hilbert Modular Surfaces." Geometriae Dedicata 105, no. 1 (2004): 121–41. http://dx.doi.org/10.1023/b:geom.0000024686.51668.c7.

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29

Daw, Christopher, and Andrei Yafaev. "An unconditional proof of the André–Oort conjecture for Hilbert modular surfaces." Manuscripta Mathematica 135, no. 1-2 (2011): 263–71. http://dx.doi.org/10.1007/s00229-011-0445-x.

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30

Howard, Benjamin. "Intersection theory on Shimura surfaces." Compositio Mathematica 145, no. 2 (2009): 423–75. http://dx.doi.org/10.1112/s0010437x09003935.

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AbstractKudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla, Rapoport and Yang. In the present paper we prove results in a higher dimensional setting. On the integral model of a Shimura surface we consider the intersection of a Shimura curve with a codimension two cycle of complex multiplication points, and relate the intersectio

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31

Goldring, Wushi, and Jean-Stefan Koskivirta. "Automorphic vector bundles with global sections on -schemes." Compositio Mathematica 154, no. 12 (2018): 2586–605. http://dx.doi.org/10.1112/s0010437x18007467.

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A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type $A_{1}^{n}$, $C_{2}$, and $\mathbf{F}_{p}$-split groups of type $A_{2}$ (this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular $3$-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for z

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32

Gon, Yasuro. "Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces." Journal of Number Theory 147 (February 2015): 396–453. http://dx.doi.org/10.1016/j.jnt.2014.07.019.

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33

Nagano, Atsuhira. "A theta expression of the Hilbert modular functions for $\sqrt{5}$ via the periods of $K3$ surfaces." Kyoto Journal of Mathematics 53, no. 4 (2013): 815–43. http://dx.doi.org/10.1215/21562261-2366102.

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34

Virdol, Cristian. "The meromorphic continuation of the zeta function of a product of Hilbert and Picard modular surfaces over CM-fields." Journal of Number Theory 133, no. 1 (2013): 123–30. http://dx.doi.org/10.1016/j.jnt.2012.07.003.

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35

OULED AZAIEZ, NAJIB. "RESTRICTIONS OF HILBERT MODULAR FORMS." International Journal of Number Theory 05, no. 01 (2009): 67–80. http://dx.doi.org/10.1142/s1793042109001931.

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Let Γ ⊂ PSL (2, ℝ) be a discrete and finite covolume subgroup. We suppose that the modular curve [Formula: see text] is "embedded" in a Hilbert modular surface [Formula: see text], where ΓK is the Hilbert modular group associated to a real quadratic field K. We define a sequence of restrictions (ρn)n∈ℕ of Hilbert modular forms for ΓK to modular forms for Γ. We denote by Mk1, k2(ΓK) the space of Hilbert modular forms of weight (k1, k2) for ΓK. We prove that ∑n∈ℕ ∑k1, k2∈ℕ ρn(Mk1, k2(ΓK)) is a subalgebra closed under Rankin–Cohen brackets of the algebra ⊕k∈ℕ Mk(Γ) of modular forms for Γ.

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36

Sokolovskaya, L. S. "Equation describing Hilbert's modular surface." Journal of Soviet Mathematics 29, no. 2 (1985): 1230–33. http://dx.doi.org/10.1007/bf02106876.

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37

VAN GEEMEN, BERT, and MATTHIAS SCHÜTT. "TWO MODULI SPACES OF ABELIAN FOURFOLDS WITH AN AUTOMORPHISM OF ORDER FIVE." International Journal of Mathematics 23, no. 10 (2012): 1250108. http://dx.doi.org/10.1142/s0129167x1250108x.

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We find explicit projective models of a compact Shimura curve and of a (non-compact) surface which are the moduli spaces of principally polarized abelian fourfolds with an automorphism of order five. The surface has a 24-nodal canonical model in P4 which is the complete intersection of two S5-invariant cubics. It is dominated by a Hilbert modular surface and we give a modular interpretation for this. We also determine the L-series of these varieties as well as those of several modular covers of the Shimura curve.

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38

Lee, Min Ho. "Mixed Hilbert modular forms and families of abelian varieties." Glasgow Mathematical Journal 39, no. 2 (1997): 131–40. http://dx.doi.org/10.1017/s001708950003202x.

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In [18] Shioda proved that the space of holomorphic 2-forms on a certain type of elliptic surface is canonically isomorphic to the space of modular forms of weight three for the associated Fuchsian group. Later, Hunt and Meyer [6] made an observation that the holomorphic 2-forms on a more general elliptic surface should in fact be identified with mixed automorphic forms associated to an automorphy factor of the formfor z in the Poincaré upper half plane ℋ, g = and χ(g) = , where g is an element of the fundamental group Γ⊂PSL(2, R) of the base space of the elliptic fibration, χ-Γ→SL(2, R) the m

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39

Yang, Tonghai. "Arithmetic intersection on a Hilbert modular surface and the Faltings height." Asian Journal of Mathematics 17, no. 2 (2013): 335–82. http://dx.doi.org/10.4310/ajm.2013.v17.n2.a4.

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40

Bonanno, Claudio. "On the Generalised Transfer Operators of the Farey Map with Complex Temperature." Mathematics 11, no. 1 (2022): 134. http://dx.doi.org/10.3390/math11010134.

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We consider the problem of showing that 1 is an eigenvalue for a family of generalised transfer operators of the Farey map. This is an important problem in the thermodynamic formalism approach to dynamical systems, which in this particular case is related to the spectral theory of the modular surface via the Selberg Zeta function and the theory of dynamical zeta functions of maps. After briefly recalling these connections, we show that the problem can be formulated for operators on an appropriate Hilbert space and translated into a linear algebra problem for infinite matrices. This formulation

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41

Joyner, David. "On the Kuznetsov-Bruggeman formula for a Hilbert modular surface having one cusp." Mathematische Zeitschrift 203, no. 1 (1990): 59–104. http://dx.doi.org/10.1007/bf02570723.

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42

Joyner, D. "On the Kuznetsov-Bruggeman formula for a Hilbert modular surface having one cusp." Mathematische Zeitschrift 205, no. 1 (1990): 163. http://dx.doi.org/10.1007/bf02571232.

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43

Hulek, K., and H. Lange. "The Hilbert modular surface for the ideal $$(\sqrt 5 )$$ and the Horrocks-Mumford bundle." Mathematische Zeitschrift 198, no. 1 (1988): 95–116. http://dx.doi.org/10.1007/bf01183042.

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44

LEI, ANTONIO, DAVID LOEFFLER, and SARAH LIVIA ZERBES. "EULER SYSTEMS FOR HILBERT MODULAR SURFACES." Forum of Mathematics, Sigma 6 (2018). http://dx.doi.org/10.1017/fms.2018.23.

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We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is nontrivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.

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45

Tiba, Yusaku. "The second main theorem for entire curves into Hilbert modular surfaces." Forum Mathematicum 27, no. 4 (2015). http://dx.doi.org/10.1515/forum-2013-6002.

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AbstractThe main goal of this article is to prove a second main theorem for entire curves into Hilbert modular surfaces. As an application of our main theorem, we obtain a condition ensuring that entire curves in a Hilbert modular surface of general type are contained in the exceptional divisors. We also show a second main theorem for a simple normal crossing divisor which is tangent to a holomorphic distribution of codimension one on a smooth projective algebraic manifold.

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46

Asvin, G., Qiao He, and Ananth N. Shankar. "Just-likely intersections on Hilbert modular surfaces." Mathematische Annalen, January 11, 2024. http://dx.doi.org/10.1007/s00208-023-02793-6.

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47

EMERTON, MATTHEW, DAVIDE REDUZZI, and LIANG XIAO. "UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ARISING FROM HILBERT MODULAR SURFACES." Forum of Mathematics, Sigma 5 (2017). http://dx.doi.org/10.1017/fms.2017.26.

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Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_{\mathfrak{p}}$ acting on $(\text{mod}\,p^{m})$ Katz Hilbert modular classes which agrees with the classical Hecke operator at $\mathfrak{p}$ for global sections that lift to characteristic zero. Using these operators and the techniques of patching complexes of Calegari and Geraghty we prove that the Galois representations arising from torsion Hilbert modular classes of parallel weight $\mathbf{1}$ are unramified at $p$ when $[F:\mathbb{Q}]=2$. Some par

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48

Jin, Seokho, and Sihun Jo. "On the algebraicity about the Hodge numbers of the Hilbert schemes of algebraic surfaces." Proceedings of the Edinburgh Mathematical Society, April 19, 2022, 1–12. http://dx.doi.org/10.1017/s0013091522000141.

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Abstract Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of $n$ points of an algebraic surface is algebraic at a CM point $\tau$ and rational numbers $z_1$ and $z_2$ . Our result gives a refinement of the algebraicity on Betti numbers.

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49

Langer, Andreas. "On the Tate-conjecture for Hilbert modular surfaces in finite characteristic." Journal für die reine und angewandte Mathematik (Crelles Journal) 2004, no. 570 (2004). http://dx.doi.org/10.1515/crll.2004.033.

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50

Nelson, Paul D. "Quadratic Hecke Sums and Mass Equidistribution." International Mathematics Research Notices, May 27, 2021. http://dx.doi.org/10.1093/imrn/rnab093.

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Abstract We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over quadratic progressions. Our reduction provides an analogue for the compact case of a criterion established by Luo–Sarnak for the case of the non-compact modular surface. The novelty is that known proofs of such criteria have depended crucially upon Fourier expansions, which are not available in the compact case. Unconditionally, we establish a twiste

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