Lifting newforms to vector-valued modular forms for the Weil representation

Given a discriminant form D of level N, there is a natural lifting which maps elliptic modular forms of level N to vector-valued modular forms for the Weil representation associated to D. We show that in some cases the zero component of a lifting of a newform f is just a scalar multiple of f. In order to do so, we split the lifting map into certain partial liftings corresponding to the prime powers exactly dividing N and then proceed to compute the zero components of these partial maps explicitly. As an application, we show that the L-function L𝒜(f, s) of a newform f and an ideal class 𝒜 as defined by Gross and Zagier can be written as a certain L-series of the lifting of f.

ON THE CHOICE OF THE BEST TEST FOR ATTRIBUTE DEPENDENCY IN PROGRAMS FOR LEARNING FROM EXAMPLES

The paper discusses a problem associated with learning from examples. Learning programs under consideration, LEM and LERS, were designed to automate knowledge acquisition for expert systems. Hence, both programs induce rules in the minimal discriminant form, i.e., rules based on minimal sets of relevant attributes, called coverings. The problem addressed in the paper is the selection of the best algorithm for determining coverings. Four different methods, based on indiscernibility relation, partition, characteristic set and lower boundary are compared. Both theoretical analysis and experimental results of multiple running of many sets of examples, with variable number of examples and with variable number of attributes are taken into account. As a result the partition method is determined to be the most efficient way to compute coverings.