63 research outputs found
K-Rational D-Brane Crystals
In this paper the problem of constructing spacetime from string theory is
addressed in the context of D-brane physics. It is suggested that the knowledge
of discrete configurations of D-branes is sufficient to reconstruct the motivic
building blocks of certain Calabi-Yau varieties. The collections of D-branes
involved have algebraic base points, leading to the notion of K-arithmetic
D-crystals for algebraic number fields K. This idea can be tested for D0-branes
in the framework of toroidal compactifications via the conjectures of Birch and
Swinnerton-Dyer. For the special class of D0-crystals of Heegner type these
conjectures can be interpreted as formulae that relate the canonical Neron-Tate
height of the base points of the D-crystals to special values of the motivic
L-function at the central point. In simple cases the knowledge of the
D-crystals of Heegner type suffices to uniquely determine the geometry.Comment: 36 page
Selmer Groups in Twist Families of Elliptic Curves
The aim of this article is to give some numerical data related to the order
of the Selmer groups in twist families of elliptic curves. To do this we assume
the Birch and Swinnerton-Dyer conjecture is true and we use a celebrated
theorem of Waldspurger to get a fast algorithm to compute . Having
an extensive amount of data we compare the distribution of the order of the
Selmer groups by functions of type with small. We discuss how the
"best choice" of is depending on the conductor of the chosen elliptic
curves and the congruence classes of twist factors.Comment: to appear in Quaestiones Mathematicae. 16 page
On the Birch-Swinnerton-Dyer quotients modulo squares
Let A be an abelian variety over a number field K. An identity between the
L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation
between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo
finiteness of Sha, and give an analogous statement for Selmer groups. Based on
this, we develop a method for determining the parity of various combinations of
ranks of A over extensions of K. As one of the applications, we establish the
parity conjecture for elliptic curves assuming finiteness of Sha[6^\infty] and
some restrictions on the reduction at primes above 2 and 3: the parity of the
Mordell-Weil rank of E/K agrees with the parity of the analytic rank, as
determined by the root number. We also prove the p-parity conjecture for all
elliptic curves over Q and all primes p: the parities of the p^\infty-Selmer
rank and the analytic rank agree.Comment: 29 pages; minor changes; to appear in Annals of Mathematic
Tamagawa defect of Euler systems
As remarked in [Kolyvagin systems, by Barry Mazur and Karl Rubin] Proposition
6.2.6 and Buyukboduk[ arXiv:0706.0377v1 ] Remark 3.25 one does not expect the
Kolyvagin system obtained from an Euler system for a p-adic Galois
representation T to be primitive (in the sense of [Kolyvagin systems, by Barry
Mazur and Karl Rubin] Definition 4.5.5) if p divides a Tamagawa number at a
prime \ell different from p; thus fails to compute the correct size of the
relevant Selmer module. In this paper we obtain a lower bound for the size of
the cokernel of the Euler system to Kolyvagin system map (see Theorem 3.2.4 of
[Kolyvagin systems, by Barry Mazur and Karl Rubin] for a definition of this
map) in terms of the Tamagawa numbers of T, refining [Kolyvagin systems, by
Barry Mazur and Karl Rubin] Propostion 6.2.6. We show how this partially
accounts for the missing Tamagawa factors in Kato's calculations with his Euler
system.Comment: 20 page
Effective equidistribution and the Sato-Tate law for families of elliptic curves
Extending recent work of others, we provide effective bounds on the family of
all elliptic curves and one-parameter families of elliptic curves modulo p (for
p prime tending to infinity) obeying the Sato-Tate Law. We present two methods
of proof. Both use the framework of Murty-Sinha; the first involves only
knowledge of the moments of the Fourier coefficients of the L-functions and
combinatorics, and saves a logarithm, while the second requires a Sato-Tate
law. Our purpose is to illustrate how the caliber of the result depends on the
error terms of the inputs and what combinatorics must be done.Comment: Version 1.1, 24 pages: corrected the interpretation of Birch's moment
calculations, added to the literature review of previous results
Towards an 'average' version of the Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer conjecture states that the rank of the
Mordell-Weil group of an elliptic curve E equals the order of vanishing at the
central point of the associated L-function L(s,E). Previous investigations have
focused on bounding how far we must go above the central point to be assured of
finding a zero, bounding the rank of a fixed curve or on bounding the average
rank in a family. Mestre showed the first zero occurs by O(1/loglog(N_E)),
where N_E is the conductor of E, though we expect the correct scale to study
the zeros near the central point is the significantly smaller 1/log(N_E). We
significantly improve on Mestre's result by averaging over a one-parameter
family of elliptic curves, obtaining non-trivial upper and lower bounds for the
average number of normalized zeros in intervals on the order of 1/log(N_E)
(which is the expected scale). Our results may be interpreted as providing
further evidence in support of the Birch and Swinnerton-Dyer conjecture, as
well as the Katz-Sarnak density conjecture from random matrix theory (as the
number of zeros predicted by random matrix theory lies between our upper and
lower bounds). These methods may be applied to additional families of
L-functions.Comment: 20 pages, 2 figures, revised first draft (fixed some typos
Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture
Most, if not all, unconditional results towards the abc-conjecture rely
ultimately on classical Baker's method. In this article, we turn our attention
to its elliptic analogue. Using the elliptic Baker's method, we have recently
obtained a new upper bound for the height of the S-integral points on an
elliptic curve. This bound depends on some parameters related to the
Mordell-Weil group of the curve. We deduce here a bound relying on the
conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable
quantities. We then study which abc-type inequality over number fields could be
derived from this elliptic approach.Comment: 20 pages. Some changes, the most important being on Conjecture 3.2,
three references added ([Mas75], [MB90] and [Yu94]) and one reference updated
[BS12]. Accepted in Bull. Brazil. Mat. So
On the vanishing of Selmer groups for elliptic curves over ring class fields
Let E be a rational elliptic curve of conductor N without complex
multiplication and let K be an imaginary quadratic field of discriminant D
prime to N. Assume that the number of primes dividing N and inert in K is odd,
and let H be the ring class field of K of conductor c prime to ND with Galois
group G over K. Fix a complex character \chi of G. Our main result is that if
the special value of the \chi-twisted L-function of E/K is non-zero then the
tensor product (with respect to \chi) of the p-Selmer group of E/H with W over
Z[G] is 0 for all but finitely many primes p, where W is a suitable finite
extension of Z_p containing the values of \chi. Our work extends results of
Bertolini and Darmon to almost all non-ordinary primes p and also offers
alternative proofs of a \chi-twisted version of the Birch and Swinnerton-Dyer
conjecture for E over H (Bertolini and Darmon) and of the vanishing of the
p-Selmer group of E/K for almost all p (Kolyvagin) in the case of analytic rank
zero.Comment: 31 pages, minor modifications; final version, to appear in Journal of
Number Theor
Period polynomials, derivatives of L-functions, and zeros of polynomials
Period polynomials have long been fruitful tools for the study of values of L-functions in the context of major outstanding conjectures. In this paper, we survey some facets of this study from the perspective of Eichler cohomology. We discuss ways to incorporate non-cuspidal modular forms and values of derivatives of L-functions into the same framework. We further review investigations of the location of zeros of the period polynomial as well as of its analogue for L-derivatives
Mordell-Weyl and Shapharevich-Tate Groups for Elliptic Weyl Curves
Available from VNTIC / VNTIC - Scientific & Technical Information Centre of RussiaSIGLERURussian Federatio
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