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My character has 3 levels in Fighter and chose the Arcane Archer archetype. That allows him to use the Arcane Shot option Piercing Arrow (XGtE, p. 29):

Piercing Arrow. You use transmutation magic to give your arrow an ethereal quality. When you use this option, you don't make an attack roll for the attack. Instead, the arrow shoots forward in a line, which is 1 foot wide and 30 feet long, before disappearing. The arrow passes harmlessly through objects, ignoring cover. Each creature in that line must make a Dexterity saving throw. On a failed save, a creature takes damage as if it were hit by the arrow, plus an extra 1d6 piercing damage. On a successful save, a target takes half as much damage.

He also has 3 levels in Rogue and chose the Assassin archetype. That gives him the Assassinate feature (PHB, p. 97):

Starting at 3rd level, you are at your deadliest when you get the drop on your enemies. You have advantage on attack rolls against any creature that hasn't taken a turn in the combat yet. In addition, any hit you score against a creature that is surprised is a critical hit.

Does that mean that against surprised opponents, the Piercing Arrow damage would be critical and rolling double the dice damage?

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Does that mean that against surprised opponents, the Piercing Arrow damage would be critical and rolling double the dice damage?

Yes.

Piercing Arrow's description specifically states "you don't make an attack roll for the attack", so it is an attack.

Any hit onto a surprised creature would be a critical hit. Assassinate doesn't care if an attack roll is made, it just has to be a hit.

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    \$\begingroup\$ Can you support this with rules references? By convention only things with attack rolls (which is sort of supported by the "advantage on attack rolls" inferring that you had to hit with an attach not spell or ability) score critical hits and Piercing Arrow, by your logic if I were a Wizard Assassin I would be able to score critical hits with Fireball and Lightning Bolt. \$\endgroup\$ – Slagmoth Aug 31 '20 at 13:51
  • \$\begingroup\$ @Slagmoth The rules for "critical hits" don't specify an attack roll must be made, and the rules for "Assassinate" also don't specify an attack roll must be made. I can't quote the rules to support my answer, but I can say there's nothing to quote that supports the opposite. \$\endgroup\$ – JRodge01 Aug 31 '20 at 16:05
  • \$\begingroup\$ @JRodge01 so would you extend that to say that fireball and lightning bolt would score critical hits as well? \$\endgroup\$ – smbailey Aug 31 '20 at 16:10
  • \$\begingroup\$ @smbailey I'd argue no because the abilities are fundamentally different than PA. If you ask in its own question, I can expand my answer there. \$\endgroup\$ – JRodge01 Aug 31 '20 at 16:18
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    \$\begingroup\$ @JRodge01 I may write out the question, though I don't see how they are fundamentally different. All three are AoE abilities requiring a DEX saving throw with half damage on a successful save. One shoots fire, one shoots lightning, one shoots an ethereal arrow. \$\endgroup\$ – smbailey Aug 31 '20 at 16:20
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Piercing Arrow's damage can be critical in this situation.

On a failed save, a creature takes damage as if it were hit by the arrow

"As if it were X" means we treat the situation like X actually happened, so we treat this situation like the target was hit by the arrow. In fact, how else could we calculate the damage that this would deal? An arrow doesn't have a defined damage die, and from various sources could deal 1d4 (improvised), 1d6 (shortbow), 1d8 (longbow), 1d10 (monk, deflect arrows), or 2d6 (Oversized Longbow wielded by Ziraj from Dragon Heist). Thus, damage dealt is equal to what would be dealt if the creature were conventionally hit by this arrow, if this arrow were conventionally fired from the bow you are using.

And if you conventially hit a creature with this arrow when they are surprised and you are an Assassin, it is a critical hit and the dice are doubled.
The Piercing Arrow's damage must match that conventional damage, and the critical damage must be dealt.

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