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James Stankewicz - One of the best experts on this subject based on the ideXlab platform.

  • computation on elliptic curves with Complex Multiplication
    arXiv: Number Theory, 2013
    Co-Authors: Pete L Clark, Patrick Corn, Alex Rice, James Stankewicz
    Abstract:

    We give the complete list of possible torsion subgroups of elliptic curves with Complex Multiplication over number fields of degree 1-13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.

  • torsion points on elliptic curves with Complex Multiplication with an appendix by alex rice
    International Journal of Number Theory, 2013
    Co-Authors: Pete L Clark, James Stankewicz, Brian Cook
    Abstract:

    We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad–Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CM-point on X1(N). Upon comparison to bounds for the least degree for which there exist infinitely many rational points on X1(N), we deduce that, for sufficiently large N, X1(N) will have a rational CM point of degree smaller than the degrees of at least all but finitely many non-CM points.

Chiyun Hsu - One of the best experts on this subject based on the ideXlab platform.

Cumrun Vafa - One of the best experts on this subject based on the ideXlab platform.

  • rational conformal field theories and Complex Multiplication
    Communications in Mathematical Physics, 2004
    Co-Authors: Sergei Gukov, Cumrun Vafa
    Abstract:

    We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFT’s corresponding to the T2 target and identify the Cardy branes with geometric branes. The T2’s leading to RCFT’s admit ‘‘Complex Multiplication’’ which characterizes Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for Calabi-Yau n-folds for n>2. RCFT’s on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in connection with freezing geometric moduli.

  • rational conformal field theories and Complex Multiplication
    arXiv: High Energy Physics - Theory, 2002
    Co-Authors: Sergei Gukov, Cumrun Vafa
    Abstract:

    We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFT's corresponding to T^2 target and identify the Cardy branes with geometric branes. The T^2's leading to RCFT's admit ``Complex Multiplication'' which characterizes Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for Calabi-Yau n-folds for n>2. RCFT's on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in connection with freezing geometric moduli.

John Coates - One of the best experts on this subject based on the ideXlab platform.

  • non vanishing theorems for central l values of some elliptic curves with Complex Multiplication ii
    arXiv: Number Theory, 2018
    Co-Authors: John Coates
    Abstract:

    Let $q$ be any prime $\equiv 7 \mod 16$, $K = \mathbb{Q}(\sqrt{-q})$, and let $H$ be the Hilbert class field of $K$. Let $A/H$ be the Gross elliptic curve defined over $H$ with Complex Multiplication by the ring of integers of $K$. We prove the existence of a large explicit infinite family of quadratic twists of $A$ whose Complex $L$-series does not vanish at $s=1$. This non-vanishing theorem is completely new when $q > 7$. Its proof depends crucially on the results established in our earlier paper for the Iwasawa theory at the prime $p=2$ of the abelian variety $B/K$, which is the restriction of scalars from $H$ to $K$ of the elliptic curve $A$.

  • on the 2 part of the birch swinnerton dyer conjecture for elliptic curves with Complex Multiplication
    Münster Journal of Mathematics, 2013
    Co-Authors: John Coates, Zhibin Liang, Minhyong Kim, Chunlai Zhao
    Abstract:

    Given an elliptic curve E over Q with Complex Multiplication having good reduction at 2, we investigate the 2-adic valuation of the algebraic part of the L-value at 1 for a family of quadratic twists. In particular, we prove a lower bound for this valuation in terms of the Tamagawa number in a form predicted by the conjecture of Birch and Swinnerton-Dyer.

  • the tate shafarevich group for elliptic curves with Complex Multiplication
    Journal of Algebra, 2009
    Co-Authors: John Coates, Zhibin Liang, R Sujatha
    Abstract:

    Let E be an elliptic curve over Q with Complex Multiplication. We give an explicit upper bound for the number of copies of Qp/Zp which can occur in the Tate–Shafarevich group of E for all sufficiently large good ordinary primes p.

  • the gl2 main conjecture for elliptic curves without Complex Multiplication
    Publications Mathématiques de l'IHÉS, 2005
    Co-Authors: John Coates, R Sujatha, Takako Fukaya, Kazuya Kato, Otmar Venjakob
    Abstract:

    Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is isomorphic to Zp. We prove the existence of a canonical Ore set S* of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S*, we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its p-primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an elliptic curve E over Q, without Complex Multiplication, over the field F generated by the coordinates of all its p-power division points; here p is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over Q.

  • the gl 2 main conjecture for elliptic curves without Complex Multiplication
    Publications Mathématiques de l'IHÉS, 2005
    Co-Authors: John Coates, R Sujatha, Takako Fukaya, Kazuya Kato, Otmar Venjakob
    Abstract:

    Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is isomorphic to Z p. We prove the existence of a canonical Ore set S* of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S*, we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its p-primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an elliptic curve E over Q, without Complex Multiplication, over the field F generated by the coordinates of all its p-power division points; here p is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over Q.

Minhyong Kim - One of the best experts on this subject based on the ideXlab platform.