Could I make a reservation under the name…?

Meaning

Specifying the name for the reservation.

Examples

• John: I'd like to book a table for two.
• Alice: Certainly. <b>Could I make a reservation under the name</b> Smith?
• John: Yes, that's correct.
• Alice: Great. And what time? <b>Could I make a reservation under the name</b> Smith for 8 pm?