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78
Billiards and Teichmüller curves on Hilbert modular surfaces
, 2007
"... This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmüller curves lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication. Explicit examples, constructed from Lshaped ..."
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Cited by 109 (9 self)
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This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmüller curves lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication. Explicit examples, constructed from Lshaped polygons, give billiard tables with optimal
Hilbert Modular Forms And pAdic Hodge Theory
 Invent. Math
, 1997
"... this paper, we show that the same is true for the places dividing p, in the sense of padic Hodge theory [Fo], as is shown for an elliptic modular form in [Sa]. We also prove that the monodromyweight conjecture holds such representations. We prove the compatibility by comparing the padic and #adi ..."
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Cited by 49 (0 self)
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this paper, we show that the same is true for the places dividing p, in the sense of padic Hodge theory [Fo], as is shown for an elliptic modular form in [Sa]. We also prove that the monodromyweight conjecture holds such representations. We prove the compatibility by comparing the padic and #adic representations for it is already established for #adic representation [C2]. More precisely, we prove it by comparing the traces of Galois action and proving the monodromyweight conjecture. The first task is to construct the Galois representation in purely geometric way in terms of etale cohomology of an analogue of KugaSato variety and algebraic correspondences acting on it. Then we apply the comparison theorem of padic Hodge theory [Tj] and weight spectral sequence [RZ], [M] to compute the traces and monodromy operaters in terms of the reduction modulo p. We obtain the required equality between traces by applying Lefschetz trace formula which has the same form for #adic and for cristalline cohomology. We deduce the monodromyweight conjecture from the Weil conjecture and a certain vanishing of global sections. The last vanishing result is an analogue of the vanishing of the fixed part (Sym k2 T # E) SL2 (Z # ) for k > 2 for the universal elliptic curve E over a modular curve in positive characteristic. We briefly recall the basic definitions on Hilbert modular forms in Section 1 and an #adic representation associated to it in Section 2. The main compatibility result, Theorem 1, and the monodromyweight conjecture, Theorem 2, are stated at the end of Section 2. We recall a cohomological construction of the #adic representation in Section 3. After introducing Shimura curves in Section 4 and recalling its modular interpretation in Section 5, we give a geometric c...
Borcherds products in the arithmetic intersection theory of Hilbert modular surfaces
, 2004
"... We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that HirzebruchZagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two. Moreover, we determine the arithmetic selfintersection number of the line bundle of modular forms e ..."
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Cited by 33 (13 self)
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We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that HirzebruchZagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two. Moreover, we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and study Faltings heights of
Arithmetic compactifications of PELtype Shimura varieties
, 2010
"... In this thesis, we constructed minimal (SatakeBailyBorel) compactifications and smooth toroidal compactifications of integral models of general PELtype Shimura varieties (defined as in Kottwitz [72]), with descriptions of stratifications and local structures on them extending the wellknown ones ..."
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Cited by 33 (3 self)
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In this thesis, we constructed minimal (SatakeBailyBorel) compactifications and smooth toroidal compactifications of integral models of general PELtype Shimura varieties (defined as in Kottwitz [72]), with descriptions of stratifications and local structures on them extending the wellknown ones in the complex analytic theory. This carries out a program initiated by Chai, Faltings, and some other people more than twenty years ago. The approach we have taken is to redo the FaltingsChai theory [39] in full generality, with as many details as possible, but without any substantial casebycase study. The essential new ingredient in our approach is the emphasis on level structures, leading to a crucial Weil pairing calculation that enables us to avoid unwanted boundary components in naive constructions.
Galois representations modulo p and cohomology of Hilbert modular varieties
 MR MR2172950 (2006k:11100
"... Abstract. The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let’s mention: − the control of the image of the Galois representation modulo p [37][35], − Hida’s congruence criterion outside an explicit set of ..."
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Cited by 29 (2 self)
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Abstract. The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let’s mention: − the control of the image of the Galois representation modulo p [37][35], − Hida’s congruence criterion outside an explicit set of primes p [21], − the freeness of the integral cohomology of the Hilbert modular variety over certain local components of the Hecke algebra and the Gorenstein property of these local algebras [30][16]. We study the arithmetic of the Hilbert modular forms by studying their modulo p Galois representations and our main tool is the action of the inertia groups at the primes above p. In order to determine this action, we compute the HodgeTate (resp. the FontaineLaffaille) weights of the padic (resp. the modulo p) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine [31, 33] on the cohomology of the Siegel modular varieties and builds upon the geometric constructions of [10, 11]. Contents
The Iwasawa main conjectures for GL2
, 2010
"... In this paper we prove the IwasawaGreenberg Main Conjecture for a large class of elliptic curves and modular forms. 1.1. The IwasawaGreenberg Main Conjecture. Let p be an odd prime. Let Q ⊂ C be the algebraic closure of Q in C. We fix an embedding Q ↩ → Q p. For simplicity we ..."
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Cited by 29 (1 self)
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In this paper we prove the IwasawaGreenberg Main Conjecture for a large class of elliptic curves and modular forms. 1.1. The IwasawaGreenberg Main Conjecture. Let p be an odd prime. Let Q ⊂ C be the algebraic closure of Q in C. We fix an embedding Q ↩ → Q p. For simplicity we
Shimura varieties and motives
, 1993
"... Deligne has expressed the hope that a Shimura variety whose weight is defined over Q is the moduli variety for a family of motives. Here we prove that this is the case for “most ” Shimura varieties. As a consequence, for these Shimura varieties, we obtain an explicit interpretation of the canonical ..."
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Cited by 26 (6 self)
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Deligne has expressed the hope that a Shimura variety whose weight is defined over Q is the moduli variety for a family of motives. Here we prove that this is the case for “most ” Shimura varieties. As a consequence, for these Shimura varieties, we obtain an explicit interpretation of the canonical model and a modular description of its points in any field containing the reflex field. Moreover, when we assume the existence of a sufficiently good theory of motives in mixed characteristic, we are able to obtain a description of the points on the Shimura variety modulo a prime of good reduction.
Overconvergent Hilbert modular forms
 AMER. J. MATH
, 2005
"... We generalize the construction of the eigencurve by ColemanMazur to the setting of totally real fields, and show that a finite slope Hilbert modular eigenform can be deformed into a one parameter family of finite slope eigenforms. The key point is to show the overconvergence of the canonical subgr ..."
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Cited by 24 (0 self)
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We generalize the construction of the eigencurve by ColemanMazur to the setting of totally real fields, and show that a finite slope Hilbert modular eigenform can be deformed into a one parameter family of finite slope eigenforms. The key point is to show the overconvergence of the canonical subgroup and the complete continuity of the Up operator. We deduce this form some general considerations in rigid analytic geometry.