40 research outputs found
The modularity of certain non-rigid Calabi-Yau threefolds
Let be a Calabi--Yau threefold fibred over by
non-constant semi-stable K3 surfaces and reaching the Arakelov--Yau bound. In
[STZ], X. Sun, Sh.-L. Tan, and K. Zuo proved that is modular in a certain
sense. In particular, the base curve is a modular curve. In their result they
distinguish the rigid and the non-rigid cases. In [SY] and [V] rigid examples
were constructed. In this paper we construct explicit examples in non-rigid
cases. Moreover, we prove for our threefolds that the ``interesting'' part of
their -series is attached to an automorphic form, and hence that they are
modular in yet another sense.Comment: 19 pages; some corrections made; see also related submission by
Hulek-Verril
The Langlands Program and String Modular K3 Surfaces
A number theoretic approach to string compactification is developed for
Calabi-Yau hypersurfaces in arbitrary dimensions. The motivic strategy involved
is illustrated by showing that the Hecke eigenforms derived from Galois group
orbits of the holomorphic two-form of a particular type of K3 surfaces can be
expressed in terms of modular forms constructed from the worldsheet theory. The
process of deriving string physics from spacetime geometry can be reversed,
allowing the construction of K3 surface geometry from the string characters of
the partition function. A general argument for K3 modularity follows from
mirror symmetry, in combination with the proof of the Shimura-Taniyama
conjecture.Comment: 33 page
Three-variable Mahler measures and special values of modular and Dirichlet -series
In this paper we prove that the Mahler measures of the Laurent polynomials
, ,
and , for various values of , are of the form , where , is a CM newform of
weight 3, and is a quadratic character. Since it has been proved that
these Maher measures can also be expressed in terms of logarithms and
-hypergeometric series, we obtain several new hypergeometric evaluations
and transformations from these results
Uniformization of modular elliptic curves via p-adic periods
The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve Ef over K. In [19], we associated, building on works of Darmon [8] and Greenberg [20], a p-adic lattice Λ to f, under certain hypothesis, and implicitly conjectured that Λ is commensurable with the p -adic Tate lattice of Ef. In this paper, we present this conjecture in detail and discuss how it can be used to compute, directly from f , a Weierstrass equation for the conjectural Ef. We develop algorithms to this end and implement them in order to carry out extensive systematic computations in which we compute Weierstrass equations of hundreds of elliptic curves, some with huge heights, over dozens of number fields. The data we obtain give extensive support for the conjecture and furthermore demonstrate that the conjecture provides an efficient tool to building databases of elliptic curves over number fields