
beruhen auf
to be based on
To be based on something, be founded on something, or have something as a foundation. Used for facts, stories, principles, or assumptions.
Der Film beruht auf einer wahren Geschichte.
The film is based on a true story.
to be based
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to be based on
To be based on something, be founded on something, or have something as a foundation. Used for facts, stories, principles, or assumptions.
Der Film beruht auf einer wahren Geschichte.
The film is based on a true story.

to let something rest
A fixed expression meaning to not pursue a matter any further, to end an argument, or to close a topic.
Nach seiner Entschuldigung haben wir beschlossen, die Sache auf sich beruhen zu lassen.
After his apology, we decided to let the matter rest.
'Beruhen' (to be based) is an inseparable verb made up of the prefix 'be-' and the verb 'ruhen' (to rest). The meaning 'to be based on' comes from the idea that one thing 'rests' on another.
Die Statue ruht auf einem massiven Steinsockel.
The statue rests on a massive stone base.
The verb 'beruhen' (to be based) is always used with the preposition 'auf' and the dative. A connection with another case or another preposition is not correct.
Sein Erfolg beruht auf harter Arbeit und ein wenig Glück.
His success is based on hard work and a little luck.
The idiom 'etwas auf sich beruhen lassen' (to let something rest) has a fixed structure. 'Etwas' (something) is the accusative object of the verb 'lassen' (to let). It means deliberately not discussing a topic any further.
Wir waren uns uneinig, aber wir haben es dann auf sich beruhen lassen.
We disagreed, but then we let it rest.
to found / be based on
Both „sich gründen auf“ and „beruhen auf“ name the basis of a statement or decision. Gründen is usually used reflexively, whereas beruhen is used without a reflexive pronoun. In terms of meaning, the two expressions are largely equivalent in this area.

Unsere Entscheidung beruht auf den Ergebnissen der Umfrage.
Our decision is based on the survey results.
Unsere Entscheidung gründet sich auf die Ergebnisse der Umfrage.
Our decision is based on the survey results.