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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

If an exerted permanent is already untapped during your next untap step (most likely because an effect untapped it), exert's effect preventing it from untapping expires without having done anything.

If the mana generated with Arena of Glory's last ability is spent on two different creature spells, each of those spells (and resulting creatures) will gain haste until end of turn.

If mana generated with Arena of Glory's last ability is spent to cast a noncreature spell that creates tokens, those tokens won't gain haste.

The mana generated with Arena of Glory's last ability can be spent on anything, not just creature spells.

If you gain control of another player's permanent until end of turn and exert it, and then that player regains control of it, it will untap during that player's untap step.

If mana generated with Arena of Glory's last ability is spent to pay any part of a creature spell's cost, including an alternative or additional cost, that creature spell (and the resulting creature) will gain haste until end of turn.

If mana generated with Arena of Glory's last ability is spent on a noncreature spell that becomes a creature later in the turn, that creature won't have haste.

You must already control a Mountain as Arena of Glory enters the battlefield for it to enter untapped. If it enters the battlefield at the same time as a Mountain when you control no other Mountains, it will enter tapped.