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You can also do this "topologically". The idea is to separately calculate the (l-adic etale) Euler characteristics of the stack Y_0(p) in char. 0 and in char. p, then see how they must relate. Here we go:

Over C, and hence over Q_p-bar as well, the Euler characteristic of Y_0(p) is (p+1)*(-1/12), since Y_0(p) is a (p+1)-fold cover of Y=M_{ell}.

On the other hand, over F_p-bar, up to "homeomorphism" Y_0(p) is two copies of Y glued at the supersingular points, so the Euler characteristic is 2*(-1/12) - S (where S is the "number" of supersingular elliptic curves).

However, since Y_0(p) has semi-stable reduction (and is constant at infinity) over Z_p, the special fiber is gotten from the generic fiber by contracting a bunch of "circles" to points, one for each nodal point of the special fiber; thus our second (char p) Euler characteristic is equal to our first (char 0) Euler characteristic plus S. Comparing and solving for S gives the formula pretty quickly.
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You can also do this "topologically". The idea is to separately calculate the (l-adic etale) Euler characteristics of the stack Y_0(p) in char. 0 and in char. p, then see how they must relate. Here we go:

Over C, and hence over Q_p-bar as well, the Euler characteristic of Y_0(p) is (p+1)*(-1/12), since Y_0(p) is a (p+1)-fold cover of Y=M_{ell}.

On the other hand, over F_p-bar, up to "homeomorphism" Y_0(p) is two copies of Y glued at the supersingular points, so the Euler characteristic is 2*(-1/12) - S (where S is the "number" of supersingular elliptic curves).

However, since Y_0(p) has semi-stable reduction over Z_p, the generic special fiber is gotten ("topologially") from the special generic fiber by fattening up the nodal contracting a bunch of "circles" to pointsinto closed cylinders then removing , one for each nodal point of the open centers from thesespecial fiber; thus our first second (char 0p) Euler characteristic is equal to our second first (char p0) Euler characteristic minus plus S. Comparing and solving for S gives the formula pretty quickly.
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You can also do this "topologically". The idea is to separately calculate the (l-adic etale) Euler characteristics of the stack Y_0(p) in char. 0 and in char. p, then see how they must relate. Here we go:

Over C, and hence over Q_p-bar as well, the Euler characteristic of Y_0(p) is (p+1)*(-1/12), since Y_0(p) is a (p+1)-fold cover of Y=M_{ell}.

On the other hand, over F_p-bar, up to "homeomorphism" Y_0(p) is two copies of Y glued at the supersingular points, so the Euler characteristic is 2*(-1/12) - S (where S is the "number" of supersingular elliptic curves).

However, since Y_0(p) has semi-stable reduction over Z_p, the generic fiber is gotten ("topologially") from the special fiber by fattening up the nodal points into closed cylinders then removing the open centers from these; thus our first (char 0) Euler characteristic is equal to our second (char p) Euler characteristic minus S. Comparing and solving for S gives the formula pretty quickly.