What is the function of the configuration of R1, C3,C10 and R3? How does the LC tank in this circuit work?
EDIT to be more specific about the input circuit per Phil's comment:
The antenna (aerial) is connected to the the oscillator of the first transistor stage formed by VC, L and the self capacitance of T1. This forms a classic regenerative circuit where the oscillator essentially samples a portion of the signal from the antenna as the oscillator builds up its amplitude. T2 is the quenching feature of the regenerative circuit that stops the oscillations when they have reached an amplitude sufficient to turn on T2. At this time T2 shorts out the power supply to the T1 oscillator circuit, causing it to restart its oscillating cycle. R1 prevents the power supply from being directly shorted out and forms part of the detection circuit.
Note that the length of the antenna will play a critical role in the correct operation of the regenerative cycle since it is directly tied to the LC circuit.
R1, C3,C10 and R3 form a decoupling network from the battery. This is to ensure that the comparatively high load of the LM386 audio amplifier does not affect the RF detection circuits. Without this, the audio output of the circuit will couple back through the RF circuit via the DC bus causing distortion or feedback. The decoupling network also ensures that RF is not coupled back to the LM386 due to the quenching action of T2.
The detection of the FM signal is done by R2 and C2 as a low pass filter that form a basic discriminator. If the FM carrier falls on a relatively linear portion of the low pass curve, it performs an FM to AM conversion through slope conversion.