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18
SaitoKurokawa lifts and applications to the BlochKato conjecture
 REDUCIBLE REPRESENTATIONS 7
, 1994
"... Let f be a newform of weight 2k −2 and level 1. In this paper we provide evidence for the BlochKato conjecture for modular forms. We demonstrate an implication that under suitable hypotheses if ̟  Lalg(k,f) then p  # Hf(Q,Wf(1 − k)) where p is a suitably chosen prime and ̟ a uniformizer of a fi ..."
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Cited by 27 (10 self)
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Let f be a newform of weight 2k −2 and level 1. In this paper we provide evidence for the BlochKato conjecture for modular forms. We demonstrate an implication that under suitable hypotheses if ̟  Lalg(k,f) then p  # Hf(Q,Wf(1 − k)) where p is a suitably chosen prime and ̟ a uniformizer of a finite extension K/Qp. We demonstrate this by establishing a congruence between the SaitoKurokawa lift Ff of f and a cuspidal Siegel eigenform G that is not a SaitoKurokawa lift. We then examine what this congruence says in terms of Galois representations to produce a nontrivial ptorsion element in H 1 f (Q,Wf(1 − k)). 1.
A DEFORMATION PROBLEM FOR GALOIS REPRESENTATIONS OVER IMAGINARY QUADRATIC FIELDS
, 2009
"... We prove the modularity of minimally ramified ordinary residually reducible padic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of defo ..."
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Cited by 4 (3 self)
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We prove the modularity of minimally ramified ordinary residually reducible padic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the nonexistence of certain field extensions which in many cases can be reduced to a condition on an Lvalue) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R = T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.
Visibility and the Birch and SwinnertonDyer conjecture for analytic rank one
, 2009
"... Let E be an optimal elliptic curve over Q of conductor N having analytic rank one, i.e., such that the Lfunction LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the Lfunction of E over K vanishes to order one at ..."
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Cited by 3 (3 self)
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Let E be an optimal elliptic curve over Q of conductor N having analytic rank one, i.e., such that the Lfunction LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the Lfunction of E over K vanishes to order one at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose MordellWeil rank is greater than one and whose associated newform is congruent to the newform associated to E modulo an integer r. The theory of visibility then shows that under certain additional hypotheses, r divides the product of the order of the ShafarevichTate group of E over K and the orders of the arithmetic component groups of E. We extract an explicit integer factor from the Birch and SwinnertonDyer conjectural formula for the product mentioned above, and under some hypotheses similar to the ones made in the situation above, we show that r divides this integer factor. This provides theoretical evidence for the second part of the Birch and SwinnertonDyer conjecture in the analytic rank one case. 1
A visible factor of the Heegner index
"... Abstract. Let E be an optimal elliptic curve over Q of conductor N, such that the Lfunction of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N are split and such that the Lfunction of E over K also vanishes to order one at s = 1. In view of ..."
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Abstract. Let E be an optimal elliptic curve over Q of conductor N, such that the Lfunction of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N are split and such that the Lfunction of E over K also vanishes to order one at s = 1. In view of the GrossZagier theorem, the Birch and SwinnertonDyer conjecture says that the index in E(K) of the subgroup generated by the Heegner point is equal to the product of the Manin constant of E, the Tamagawa numbers of E, and the square root of the order of the ShafarevichTate group of E (over K). We extract an integer factor from the index mentioned above and relate this factor to certain congruences of the newform associated to E with eigenforms of analytic rank bigger than one. We use the theory of visibility to show that, under certain hypotheses (which includes the first part of the Birch and SwinnertonDyer conjecture on rank), if an odd prime q divides this factor, then q divides the order of the ShafarevichTate group, as predicted by the Birch and SwinnertonDyer conjecture. 1. Introduction and
Received (Day Month Year) Accepted (Day Month Year)
"... Communicated by xxx Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform f such that the special Lvalue of A (at s = 1) is nonzero. Suppose that the algebraic part of the special Lvalue of A is divisible by an odd prime q such that q does not divide the numerator of N− ..."
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Communicated by xxx Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform f such that the special Lvalue of A (at s = 1) is nonzero. Suppose that the algebraic part of the special Lvalue of A is divisible by an odd prime q such that q does not divide the numerator of N−1. Then the Birch and SwinnertonDyer conjecture predicts that q2 12 divides the algebraic part of special L value of A as well as the order of the ShafarevichTate group. Under a mod q nonvanishing hypothesis on special Lvalues of twists of A, we show that q2 does indeed divide the algebraic part of the special Lvalue of A and the Birch and SwinnertonDyer conjectural order of the ShafarevichTate group of A. We also give a formula for the algebraic part of the special Lvalue of A over suitable quadratic imaginary fields in terms of the free abelian group on isomorphism classes of supersingular elliptic curves in characteristic N (equivalently over conjugacy classes of maximal orders in the definite quaternion algebra over Q ramified at N and ∞) which shows that this algebraic part is a perfect square away from two.
VISIBILITY FOR ANALYTIC RANK ONE or A VISIBLE FACTOR FOR ANALYTIC RANK ONE ∗
, 2008
"... Let E be an optimal elliptic curve of conductor N, such that the Lfunction of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the Lfunction of E over K also vanishes to order one at s = 1. In view of the GrossZagier the ..."
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Let E be an optimal elliptic curve of conductor N, such that the Lfunction of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the Lfunction of E over K also vanishes to order one at s = 1. In view of the GrossZagier theorem, the second part of the Birch and SwinnertonDyer conjecture says that the index in E(K) of the subgroup generated by the Heegner point is equal to the product of the Manin constant of E, the Tamagawa numbers of E, and the square root of the order of the ShafarevichTate group of E (over K). We extract an integer factor from the index mentioned above and relate this factor to certain congruences of the newform associated to E with eigenforms of analytic rank bigger than one. We use the theory of visibility to show that, under certain hypotheses (which includes the first part of the Birch and SwinnertonDyer conjecture on rank), if an odd prime q divides this factor, then q divides the order of the ShafarevichTate group or the order of an arithmetic component group of E, as predicted by the second part of the Birch and SwinnertonDyer conjecture. The referee asked for a new title, so I am including a second title to see if the referee agrees with it. I have included the original title above since it may have been used before to identify the paper.
Squareness in the special Lvalue
, 2007
"... Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform f such that the special Lvalue of A (at s = 1) is nonzero. Suppose that the algebraic part of special Lvalue is divisible by an odd prime q such that q does not divide the numerator of N−1 12. Then the Birch and Swin ..."
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Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform f such that the special Lvalue of A (at s = 1) is nonzero. Suppose that the algebraic part of special Lvalue is divisible by an odd prime q such that q does not divide the numerator of N−1 12. Then the Birch and SwinnertonDyer conjecture predicts that q2 divides the algebraic part of special L value of A, as well as the order of the ShafarevichTate group. Under a mod q nonvanishing hypothesis on special Lvalues of twists of A, we show that q2 does divide the algebraic part of the special Lvalue of A and the Birch and SwinnertonDyer conjectural order of the ShafarevichTate group of A. This gives theoretical evidence towards the second part of the Birch and SwinnertonDyer conjecture. We also give a formula for the algebraic part of the special Lvalue of A over suitable quadratic imaginary fields which shows that this algebraic part is a perfect square away from two. 1 Introduction and
Visibility of MordellWeil Groups
 DOCUMENTA MATH.
, 2007
"... We introduce a notion of visibility for MordellWeil groups, make a conjecture about visibility, and support it with theoretical evidence and data. These results shed new light on relations between MordellWeil and ShafarevichTate groups. ..."
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We introduce a notion of visibility for MordellWeil groups, make a conjecture about visibility, and support it with theoretical evidence and data. These results shed new light on relations between MordellWeil and ShafarevichTate groups.
Research statement
"... In this document, I briefly describe the main themes of my past and current research. The first section sets up some notation that is used in the rest of the article. After that, the other sections can be read more or less independently. Each of these latter sections discusses a particular aspect of ..."
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In this document, I briefly describe the main themes of my past and current research. The first section sets up some notation that is used in the rest of the article. After that, the other sections can be read more or less independently. Each of these latter sections discusses a particular aspect of my research. 1 Background and notation Let N be a positive integer. Let X0(N) denote the modular curve over Q associated to Γ0(N), and let J0(N) be its Jacobian. Let T denote the subring of endomorphisms of J0(N) generated by the Hecke operators (usually denoted Tℓ for ℓ∤N and Up for pN). Let f be a newform in S2(Γ0(N), C). Let If = AnnTf and let A = Af denote associated newform quotient J0(N)/If J0(N) over Q. If f has integer Fourier coefficients, then A is just an elliptic curve, and every elliptic curve over Q is isogenous to some such newform quotient. Most of my research concerns the arithmetic of newform quotients (which includes elliptic curves), especially in relation to the second part of Birch and SwinnertonDyer (BSD) conjecture, which I will now recall briefly. Let LA(s) denote the Lfunction associated to A. The analytic rank r of A is the order of vanishing of LA(s) at s = 1. Let A denote the Néron model of A over Z and let A 0 denote the largest open subgroup scheme of A in which all the fibers are connected. Let d = dim A, and let D be a generator of the dth exterior power of the group of invariant differentials on A. Let ΩA denote the volume of A(R) with respect to the measure given by D. If p is a prime number, then the (arithmetic) component group of A at p is the group of Fpvalued points of the quotient AFp/A 0 Fp; its order is denoted cp(A). Let RA denote the regulator of A. If B is an abelian variety, then we denote by B ∨ the dual abelian variety of B. If B is an abelian variety over a number field F, then X(B/F) denotes the ShafarevichTate group of B over F; if F = Q, then we write just X(B) for X(B/F). The second part of the BSD conjecture asserts the formula: lims→1{(s − 1) −rLA(s)}