1,275 research outputs found
A -microscope for supercongruences
By examining asymptotic behavior of certain infinite basic (-)
hypergeometric sums at roots of unity (that is, at a "-microscopic" level)
we prove polynomial congruences for their truncations. The latter reduce to
non-trivial (super)congruences for truncated ordinary hypergeometric sums,
which have been observed numerically and proven rarely. A typical example
includes derivation, from a -analogue of Ramanujan's formula of the two supercongruences valid
for all primes , where denotes the truncation of the infinite sum
at the -th place and stands for the quadratic character
modulo .Comment: 26 page
Supercongruences and Complex Multiplication
We study congruences involving truncated hypergeometric series of the
form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} =
\sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r
are positive integers. These truncated hypergeometric series are related to the
arithmetic of a family of algebraic varieties and exhibit Atkin and
Swinnerton-Dyer type congruences. In particular, when r=3, they are related to
K3 surfaces. For special values of \lambda, with s=1 and r=3, our congruences
are stronger than what can be predicted by the theory of formal groups because
of the presence of elliptic curves with complex multiplications. They
generalize a conjecture made by Rodriguez-Villegas for the \lambda=1 case and
confirm some other supercongruence conjectures at special values of \lambda.Comment: 19 page
A regularization algorithm for matrices of bilinear and sesquilinear forms
We give an algorithm that uses only unitary transformations and for each
square complex matrix constructs a *congruent matrix that is a direct sum of a
nonsingular matrix and singular Jordan blocks.Comment: 18 page
Minimal isometric immersions into S^2 x R and H^2 x R
For a given simply connected Riemannian surface Sigma, we relate the problem
of finding minimal isometric immersions of Sigma into S^2 x R or H^2 x R to a
system of two partial differential equations on Sigma. We prove that a constant
intrinsic curvature minimal surface in S^2 x R or H^2 x R is either totally
geodesic or part of an associate surface of a certain limit of catenoids in H^2
x R. We also prove that if a non constant curvature Riemannian surface admits a
continuous one-parameter family of minimal isometric immersions into S^2 x R or
H^2 x R, then all these immersions are associate
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