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Modular forms and special cycles on Shimura curves
 Annals of Math. Studies series, vol 161, Princeton Univ. Publ
, 2006
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Cited by 34 (7 self)
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by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher, except for reading and browsing via the World Wide Web. Users are not permitted to mount this file on any network servers. Follow links Class Use and other Permissions. For more information, send email to:
Nontriviality of RankinSelberg Lfunctions and CM points
, 2004
"... 1.1 RankinSelberg Lfunctions................... 2 ..."
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Cited by 25 (3 self)
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1.1 RankinSelberg Lfunctions................... 2
Periods of Hilbert modular forms and rational points on elliptic curves
"... Let E be a modular elliptic curve over a totally real field. Chapter 8 of [Dar2] formulates a conjecture allowing the construction of canonical algebraic points on E by suitably integrating the associated Hilbert modular form. The main goal of the present work is to obtain numerical evidence for thi ..."
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Cited by 21 (9 self)
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Let E be a modular elliptic curve over a totally real field. Chapter 8 of [Dar2] formulates a conjecture allowing the construction of canonical algebraic points on E by suitably integrating the associated Hilbert modular form. The main goal of the present work is to obtain numerical evidence for this conjecture in the first case where it asserts something nontrivial, namely, when E has everywhere good reduction over a real quadratic field.
Equidistribution of CMpoints on quaternion Shimura varieties
"... The aim of this paper is to show some equidistribution statements of Galois orbits of CMpoints for quaternion Shimura varieties. These equidistribution statements will imply the Zariski densities of CMpoints as predicted by AndréOort conjecture (see Section 2). Our main result (Corollary 3.7) say ..."
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Cited by 18 (1 self)
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The aim of this paper is to show some equidistribution statements of Galois orbits of CMpoints for quaternion Shimura varieties. These equidistribution statements will imply the Zariski densities of CMpoints as predicted by AndréOort conjecture (see Section 2). Our main result (Corollary 3.7) says that the Galois orbits of CMpoints with the maximal
Elliptic curves and analogies between number fields and function fields
 HEEGNER POINTS AND RANKIN LSERIES, EDITED BY HENRI DARMON AND SHOUWU
, 2004
"... Wellknown analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we discuss traffic of this sort, in both directions, in the theory of elliptic curves. In the first part of the paper, we consider various works ..."
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Cited by 14 (0 self)
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Wellknown analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we discuss traffic of this sort, in both directions, in the theory of elliptic curves. In the first part of the paper, we consider various works on Heegner points and Gross–Zagier formulas in the function field context; these works lead to a complete proof of the conjecture of Birch and SwinnertonDyer for elliptic curves of analytic rank at most 1 over function fields of characteristic> 3. In the second part of the paper, we review the fact that the rank conjecture for elliptic curves over function fields is now known to be true, and that the curves which prove this have asymptotically maximal rank for their conductors. The fact that these curves meet rank bounds suggests interesting problems on elliptic curves over number fields, cyclotomic fields, and function fields over number fields. These
Iwasawa theory of Heegner points on abelian varieties of GL2 type
 Duke Math. J
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AN ARITHMETIC INTERSECTION FORMULA ON HILBERT MODULAR SURFACES
"... Abstract. In this paper, we obtain an explicit arithmetic intersection formula on a Hilbert modular surface between the diagonal embedding of the modular curve and a CM cycle associated to a nonbiquadratic CM quartic field. This confirms a special case of the author’s conjecture with J. Bruinier, an ..."
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Cited by 10 (5 self)
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Abstract. In this paper, we obtain an explicit arithmetic intersection formula on a Hilbert modular surface between the diagonal embedding of the modular curve and a CM cycle associated to a nonbiquadratic CM quartic field. This confirms a special case of the author’s conjecture with J. Bruinier, and is a generalization of the beautiful factorization formula of Gross and Zagier on singular moduli. As an application, we proved the first nontrivial nonabelian ChowlaSelberg formula, a special case of Colmez conjecture. 1. Introduction. Intersection theory and Arakelov
Arithmetic Intersection on a Hilbert Modular Surface and Faltings
, 2007
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