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38
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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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
The parity conjecture for elliptic curves at supersingular reduction primes
, 2007
"... In number theory, the Birch and SwinnertonDyer (BSD) conjecture for a Selmer group relates the corank of a Selmer group of an elliptic curve over a number field to the order of zero of the associated Lfunction L(E,s) ats = 1. We study its modulo two version called the parity conjecture. The parity ..."
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Cited by 14 (4 self)
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In number theory, the Birch and SwinnertonDyer (BSD) conjecture for a Selmer group relates the corank of a Selmer group of an elliptic curve over a number field to the order of zero of the associated Lfunction L(E,s) ats = 1. We study its modulo two version called the parity conjecture. The parity conjecture when a prime number p is a good ordinary reduction prime was proven by Nekovar. We prove it when a prime number p>3isa good supersingular reduction prime.
Elliptic Curves and Class Field Theory
 ICM 2002 · VOL. III · 1–3
, 2002
"... Suppose E is an elliptic curve defined over Q. At the 1983 ICM the first author formulated some conjectures that propose a close relationship between the explicit class field theory construction of certain abelian extensions of imaginary quadratic fields and an explicit construction that (conjectura ..."
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Cited by 12 (2 self)
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Suppose E is an elliptic curve defined over Q. At the 1983 ICM the first author formulated some conjectures that propose a close relationship between the explicit class field theory construction of certain abelian extensions of imaginary quadratic fields and an explicit construction that (conjecturally) produces almost all of the rational points on E over those fields. Those conjectures are to a large extent settled by recent work of Vatsal and of Cornut, building on work of Kolyvagin and others. In this paper we describe a collection of interrelated conjectures still open regarding the variation of MordellWeil groups of E over abelian extensions of imaginary quadratic fields, and suggest a possible algebraic framework to organize them.
Iwasawa Theory of Elliptic Curves at Supersingular Primes over Zpextensions of Number Fields
, 2008
"... In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Zpextension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi [8] and PerrinRiou [16], we define restricted Selmer groups and λ ± , µ ..."
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Cited by 12 (2 self)
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In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Zpextension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi [8] and PerrinRiou [16], we define restricted Selmer groups and λ ± , µ ±invariants; we then derive asymptotic formulas describing the growth of the Selmer group in terms of these invariants. To be able to work with noncyclotomic Zpextensions, a new local result is proven that gives a complete description of the formal group of an elliptic curve at a supersingular prime along any ramified Zpextension of Qp.
Heegner points and nonvanishing of Rankin/Selberg Lfunctions. Analytic number theory
, 2007
"... Abstract. We discuss the nonvanishing of the family of central values L( , f ⊗ χ), where f is a fixed automorphic form on GL(2) and χ varies through class group characters of an imaginary quadratic field K = Q( √ −D), as D varies; we prove results of the nature that at least D 1/5000 such twists ar ..."
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Cited by 8 (1 self)
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Abstract. We discuss the nonvanishing of the family of central values L( , f ⊗ χ), where f is a fixed automorphic form on GL(2) and χ varies through class group characters of an imaginary quadratic field K = Q( √ −D), as D varies; we prove results of the nature that at least D 1/5000 such twists are nonvanishing. We also discuss the related question of the rank of a fixed elliptic curve E/Q over the Hilbert class field of Q( √ −D), as D varies. The tools used are results about the distribution of Heegner points, as well as subconvexity bounds for Lfunctions.
CONSEQUENCES OF THE GROSS/ZAGIER FORMULAE: STABILITY OF AVERAGE LVALUES, SUBCONVEXITY, AND NONVANISHING MOD p
, 709
"... In memory of Serge Lang In this paper we investigate some consequences of the Gross/Zagier type formulae which were introduced by Gross and Zagier and then generalized in various directions by Hatcher, Zhang, Kudla and others ..."
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In memory of Serge Lang In this paper we investigate some consequences of the Gross/Zagier type formulae which were introduced by Gross and Zagier and then generalized in various directions by Hatcher, Zhang, Kudla and others