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14
Nontriviality of Rankin-Selberg L-functions and CM points
, 2004
"... 1.1 Rankin-Selberg L-functions................... 2 ..."
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Cited by 25 (3 self)
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1.1 Rankin-Selberg L-functions................... 2
Equidistribution, L-functions and ergodic theory: on some problems of Yu. Linnik
- LINNIK, INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II, EUR. MATH. SOC., ZÜRICH
"... An old question of Linnik asks about the equidistribution of integral points on a large sphere. This question proved to be very rich: it is intimately linked to modular forms, to subconvex estimates for L-functions, and to dynamics of torus actions on homogeneous spaces. Indeed, Linnik gave a parti ..."
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Cited by 23 (6 self)
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An old question of Linnik asks about the equidistribution of integral points on a large sphere. This question proved to be very rich: it is intimately linked to modular forms, to subconvex estimates for L-functions, and to dynamics of torus actions on homogeneous spaces. Indeed, Linnik gave a partial answer using ergodic methods, and his question was completely answered by Duke using harmonic analysis and modular forms. We survey the context of these ideas and their developments over the last decades.
CM Points and Quaternion Algebras
- DOCUMENTA MATH.
, 2005
"... This paper provides a proof of a technical result (Corollary 2.10 of Theorem 2.9) which is an essential ingredient in our proof of Mazur’s conjecture over totally real number fields [3]. ..."
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Cited by 19 (4 self)
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This paper provides a proof of a technical result (Corollary 2.10 of Theorem 2.9) which is an essential ingredient in our proof of Mazur’s conjecture over totally real number fields [3].
Equidistribution of CM-points on quaternion Shimura varieties
"... The aim of this paper is to show some equidistribution statements of Galois orbits of CM-points for quaternion Shimura varieties. These equidistribution statements will imply the Zariski densities of CM-points as predicted by André-Oort conjecture (see Section 2). Our main result (Corollary 3.7) say ..."
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Cited by 16 (1 self)
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The aim of this paper is to show some equidistribution statements of Galois orbits of CM-points for quaternion Shimura varieties. These equidistribution statements will imply the Zariski densities of CM-points as predicted by André-Oort conjecture (see Section 2). Our main result (Corollary 3.7) says that the Galois orbits of CM-points with the maximal
The André-Oort conjecture
"... Abstract. In this paper we prove, assuming the Generalized Riemann Hypothesis, the André-Oort conjecture on the Zariski closure of sets of special points in a Shimura variety. In the case of sets of special points satisfying an additional assumption, we prove the conjecture without assuming the GRH ..."
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Cited by 11 (0 self)
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Abstract. In this paper we prove, assuming the Generalized Riemann Hypothesis, the André-Oort conjecture on the Zariski closure of sets of special points in a Shimura variety. In the case of sets of special points satisfying an additional assumption, we prove the conjecture without assuming the GRH. Contents
Non-triviality of CM points in ring class field towers
"... In [Co-Va 1], Cornut and Vatsal proved a generalisation of Mazur’s conjecture on higher Heegner points, valid for CM points on Shimura curves with unramified central character over totally real number fields. In the present work, which grew out of the thesis of the first author at University Paris 6 ..."
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Cited by 4 (0 self)
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In [Co-Va 1], Cornut and Vatsal proved a generalisation of Mazur’s conjecture on higher Heegner points, valid for CM points on Shimura curves with unramified central character over totally real number fields. In the present work, which grew out of the thesis of the first author at University Paris 6, we extend the results of [Co-Va 1] to more general families of CM points on slightly less general Shimura curves (those with trivial
Special subvarieties of Drinfeld modular varieties
, 2009
"... We explore an analogue of the André-Oort conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety X of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM) points if and only if X is a “special ” subvariety (i.e. X is define ..."
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Cited by 4 (2 self)
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We explore an analogue of the André-Oort conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety X of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM) points if and only if X is a “special ” subvariety (i.e. X is defined by requiring additional endomorphisms). We prove this conjecture in two cases. Firstly when X contains a Zariski-dense set of CM points with a certain behaviour above a fixed prime (which is the case if these CM points lie in one Hecke orbit), and secondly when X is a curve containing infinitely many CM points without any additional assumptions.
