969 research outputs found
CM liftings of Supersingular Elliptic Curves
Assuming GRH, we present an algorithm which inputs a prime and outputs
the set of fundamental discriminants such that the reduction map modulo a
prime above from elliptic curves with CM by \order_{D} to supersingular
elliptic curves in characteristic . In the algorithm we first determine an
explicit constant so that implies that the map is necessarily
surjective and then we compute explicitly the cases .Comment: 26 pages, 10 table
Polar harmonic Maass forms and their applications
In this survey, we present recent results of the authors about
non-meromorphic modular objects known as polar harmonic Maass forms. These
include the computation of Fourier coefficients of meromorphic modular forms
and relations between inner products of meromorphic modular forms and higher
Green's functions evaluated at CM-points
Regularized inner products and weakly holomorphic Hecke eigenforms
We show that the image of repeated differentiation on weak cusp forms is
precisely the subspace which is orthogonal to the space of weakly holomorphic
modular forms. This gives a new interpretation of the weakly holomorphic Hecke
eigenforms
The triangular theorem of eight and representation by quadratic polynomials
We investigate here the representability of integers as sums of triangular
numbers, where the -th triangular number is given by . In
particular, we show that ,
for fixed positive integers , represents every nonnegative
integer if and only if it represents 1, 2, 4, 5, and 8. Moreover, if
`cross-terms' are allowed in , we show that no finite set of positive
integers can play an analogous role, in turn showing that there is no
overarching finiteness theorem which generalizes the statement from positive
definite quadratic forms to totally positive quadratic polynomials
- …