Can an Arcane Trickster copy a Wizard spell from a scroll into their "spellbook"?

Question

Page 200 of the DMG says that a wizard scroll can be copied into a spellbook

A wizard spell on a spell scroll can be copied just as spells in spellbooks can be copied. When a spell is copied from a spell scroll, the copier must succeed on an Intelligence (Arcana) check with a DC equal to 10 + the spell's level. If the check succeeds, the spell is successfully copied. Whether the check succeeds or fails, the spell scroll is destroyed (DMG)

1. Does an Arcane Trickster have a spellbook to copy a scroll into?

2. If they do have a spellbook, is the entire Wizard list open to them, or only enchantment and illusion spells? i.e. could they copy Absorb Elements (a 1st level Wizard Abjuration spell)

   You know three 1st-level wizard spells of your choice, two of which you must choose from the enchantment and illusion spells on the wizard spell list. (PHB)

3. If they don't have a spellbook, is there an acceptable mechanic for learning or memorizing the spell on the scroll that is similar to how a wizard would be able to learn it? (And is the entire Wizard list open to them)

   Intelligence is your spellcasting ability for your wizard spells, since you learn your spells through dedicated study and memorization. (PHB)

Answer 1

1. No, they don't have a spellbook

Jeremy Crawford (official rules guru for D&D) has said this very clearly:

An Arcane Trickster casts wizard spells but doesn't use a spellbook. The same is true of the Eldritch Knight.

The only class with a spellbook is the wizard. There is nothing in the Arcane Trickster's description that mentions anything about a spellbook so there is no reason to assume it would get one.

Arcane Trickers may learn wizard spells but they are not wizards.

For example the Arcane Trickster feature says that they know spells:

You know three 1st-level wizard spells of your choice...

In essence, the Arcane Trickster knows the few spells so well that they have them always in mind. Knowing spells mean there is no need for a spellbook and no way to use one even if they did have one.

This is directly opposed to way the wizard must prepare their spells from their spellbook.

2. No, they don't have a spellbook

(see 1.)

3. No, they cannot learn spells like a wizard

See this excellent answer for a full discussion.

The Wizard is the only class to have a method to learn new spells aside from level-up or feats. And again the Arcane Trickster is not a wizard in any way.

The Arcane Trickster only lists one way to learn new spells: leveling up.

The Spells Known column of the Arcane Trickster Spellcasting table shows when you learn more wizard spells...For instance, when you reach 7th level in this class, you can learn one new spell of 1st or 2nd level.

Note: Arcane Tricksters can use spell scrolls whose spells are on the wizard list

Jeremy Crawford confirmed this:

The intent is yes, the Arcane Trickster and the Eldritch Knight can use scrolls with wizard spells on them.

But they definitely cannot learn spells from them.

Answer 2

No. Arcane Tricksters (and Eldritch Knights) do not have spellbooks and cannot add spells from scrolls to their known spells. The scroll IS the mechanism for adding and using such spells, and must be created for each use, and is destroyed in its use.

Arcane Tricksters and Eldritch Knights are special limited forms of wizardry, and they learn their spells essentially like bards, rangers, sorcerors, and warlocks -- a limited number that they keep in their head, that have limited options to add to or change.

Answer 3

NO.

Arcane Tricksters do not have a Spellbook, they instead "know" their spells, much like Sorcerers do. Compare the Arcane Trickster Spellcasting feature with the Wizards', and note there is no mention of a Spellbook in the former's description. That means there can be no copying of spells you find in other spellbooks or scrolls into your spellbook, as you have no spellbook.