Help: a question about Gilbert Mixer

I saw some papers mentioned the design of Gilbert Mixer. As the attachment, they use a parallel LC Tank below the degeneration inductors.
Paper[1] said, "a parallel LC resonator, tuned on the second RF harmonic, as an ac current source in the emitter of the input transistor pair"
Paper2] said, "The on chip LC Tank is tuned at the input RF frequency to provide the necessary common-mode rejection at RF frequency"

so i am a little confused, which harmonic ( first or second) should the Tank be tuned?
and what exactly the tank works? can anybody explain more detaily? thanks a lot.

[1]5.8GHz and 12.6GHz Si bipolar MMICs ISSCC1997
[2]Frequency-scalable SiGe bipolar RF Front-end design JSSC vol36, 2001.6

That tank circuit is tuned on 2nd RF Harmonic and is used for both, common mode rejection and harmonic rejection.

At the second harmonic of the input signal (RF?) the parallel tank is open. So the input stage transconductance is not impacted. So there is no significant mixer rejection at 2*RF. Because the conversion of gilbert core cell (4 NPNs) is low on the second harmonic but high on the third I suggest to make the parallel tank by adding a cap between the two emitters of the input stage. The bias tank could be avoided by simply using current sources.

Having an LC tank in series with the degeneration inductors is not a new approach but it is a good idea, and is used in many mixer designs.
Because the LC tank has finite impedance tuned on the second harmonic, it will help to reject common mode and even harmonics.