I've seen the following sentence come up a few times in papers:

Let $E$ be the universal elliptic curve over the modular curve $Y_1(N)$. Then the localization of $E$ at any choice of cusp is isomorphic to the Tate curve with some suitable level structure.

Could somebody explain what exactly this sentence means? What does it mean to localize $E$ at a cusp? (Here $Y_1(N)$ is the open modular curve so it has no cusps.) And in what sense is this localization of $E$ at a cusp given by the Tate curve?

My only exposure to Tate curves has been from Silverman's Advanced Topics book, where he explains how the Tate Curve $E_q$ over $\mathbf{Q}_p$ can be $p$-adically uniformized. But I'm not so comfortable with how the Tate curve shows up when dealing with universal elliptic curves. Could someone shed some light on this?

I've seen the following sentence come up a few times in papers:

Let $E$ be the universal elliptic curve over the modular curve $Y_1(N)$. Then the localization of $E$ at any choice of cusp is isomorphic to the Tate curve with some suitable level structure.

Could somebody explain what exactly this sentence means? What does it mean to localize $E$ at a cusp? (Here $Y_1(N)$ is the open modular curve so it has no cusps.) And in what sense is this localization of $E$ at a cusp given by the Tate curve?

My only exposure to Tate curves has been from Silverman's Advanced Topics book, where he explains how the Tate Curve $E_q$ over $\mathbf{Q}_p$ can be $p$-adically uniformized. But I'm not so comfortable with how the Tate curve shows up when dealing with universal elliptic curves. Could someone shed some light on the connection between Tate curves and universal elliptic curves?