1. What does the prefix ad- semantically mean here?

2. How did the compounding of ad- + rogare yield 'to make great claims about oneself' and "to claim for oneself, assume"?

3. What semantic notions underlie ad- + rogare with 'to make great claims about oneself'? Doubtless, the act of asking for or proposing someone or something doesn't mean "to claim for oneself, assume".

arrogant [14] Etymologically, to be arrogant is to make great claims about oneself. It originated in the Latin compound verb arrogāre ‘claim for oneself’, formed from the prefix ad- ‘to’ and rogāre ‘ask’ (as in English interrogate). Already in Latin the present participle arrogāns was being used adjectivally, for ‘overbearing’, and this passed via Old French into English.

Word Origins (2005 2e) by John Ayto, pp 35-6. Ayto doesn't expound this semantic shift.

arrogance (n.)

c. 1300, from Old French arrogance (12c.),
from Latin arrogantia,
from arrogantem (nominative arrogans) "assuming, overbearing, insolent," present participle of [2.] arrogare "to claim for oneself, assume,"
[1.] from ad- "to" (see ad-) + rogare "ask, propose" (see rogation).

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