Attenuated Lower Envelope from Colpits Oscillator
L1 and C3 form (or can form) a highly resosnt series pass tuned circuit and you can expect voltage amplification of a sinewave (at the collector) to the output across C3. Normal stuff for tuned circuits.

The problem with the asymmetry is that you are also feeding the modulating signal through L1 as well and so you are seeing, on the "o-scope" the addition of a perfectly normal modulated waveform and the modulation signal.

There may be other artefacts too - it may not be a perfectly symmetrical modulation due to the bias onto the base receiving the modulated signal. Ideally you need to be only modulating the collector to obtain a more symmetrical output signal. Try isolating the base bias from V1 and see what happens then.