Ashe's True Ice bow is said to have belonged to her ancestor Avarosa, the ancient Iceborn queen of unified Freljord.
However, dialogue from Swain suggests Ashe has possibly lied about unearthing the bow from Avarosa's tomb.
Dialogue from Taric suggests the bow is actually a celestial weapon of Targon infused with True Ice by one of her ancestors that were "of the stars".
According to Lissandra's biography, the only ancestor stated to have ever had any interaction with Targon was Serylda, Sejuani's ancestor, which ultimately suggests the bow might actually belong to Serylda.
Alternatively, the overly idealistic Ashe naïvely assumes the True Ice bow to be Avarosa's, merely because of its presence in Avarosa's tomb; even though Serylda actually possessed it.
If Ashe, Sejuani, and/or Lissandra are played on opposing teams, Battle for Freljord will trigger. This in-game quest consists in one killing the other to earn the title of 'Queen of Freljord' (complete with a floating ice crown floating above their heads) referencing the civil war taking place there, between the three tribes each lead.
Amethyst is a violet variety of Quartz, often used in jewelry. It is known as a meditative and calming stone which works in the emotional, spiritual, and physical planes to provide calm, balance, patience, and peace.
This fits perfectly Ashe's goals of peace, unification, and prosperity for the Freljord.
The Avarosan (led by Ashe) are engaged in a civil war with the Winter's Claw (led by Sejuani) over sovereignty of the Freljord. The Frostguard (led by Lissandra) do not directly confront both sides.
Unbeknownst to all, Lissandra is the Ice Witch of Freljordian legend, who is actually waiting for the right moment to sticke Ashe's and Sejuani's tribes and wipe out any potential incarnations of Avarosa and Serylda in order to take over the Freljord.
As revealed by Lissandra, she herself is one of the Three Sisters who led the Iceborn uprising against the Watchers. Whether or not she knows Ashe and Sejuani are direct descendants of her sisters (Avarosa and Serylda, respectively) making her their great-aunt-many-times-removed is unknown.
Evidence to support Ashe's heritage stems from Ezreal's uncle Lyte stating her 'striking resemblance' to Avarosa herself, as well as the fact Ashe was coincidentally led to Avarosa's grave by a great hawk and recovered her ancestor's True Ice bow she carries with her to this day.
Anivia support her and the Avarosan's goal of peace and unification. Gragas is another supporter.
It is unknown if Nunu and Willump support Ashe, but their deeds around the Freljord lead towards the same goal.
Ashe and Tryndamere are Bloodsworn, a political marriage, but the two eventually developed genuine affection for each other. Ashe originally married Tryndamere in order to avoid having to marry multiple suitors from neighboring tribes.