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(This answer has been edited -- it used to say that a finite cover of $\overline M_{g,n}$ gives a counterexample, which no longer seems obvious.)

If you had written "Deligne-Mumford stack" instead of "scheme", then a counterexample would be given by the spaces $\overline M_{g,n}$, which are certainly smooth and proper but far from rational in the large $g$ limit (or, for $g > 0$, in the large $n$ limit). The original references here are, I guess, Deligne (for $\overline M_{1,11}$) and Harris-Mumford (for $\overline M_{25}$).

Kevin Buzzard's hint with the Ramanujan $\Delta$ function is relevant here; indeed, $H^{11,0}(\overline M_{1,11})$ is nonzero, and the $\ell$-adic Galois representation corresponding to $H^{11,0}\oplus H^{0,11}$ is the representation attached to $\Delta$.

My answer to the question Is the moduli space of curves defined over the field with one element? contains some more detailed information about these things.

The natural way to find a scheme instead of a stack would then be to consider finite or generically finite covers of $\overline M_{g,n}$ by smooth proper schemes. There are several closely related constructions of such covers in the literature by Looijenga, Boggi-Pikaart, Pikaart-de Jong, Abramovich-Corti-Vistoli, all using some kind of non-abelian level structure on curves, but as far as I can tell none of them work over the integers.

(This answer has been edited -- it used to say that a finite cover of $\overline M_{g,n}$ gives a counterexample, which no longer seems obvious.)

If you had written "Deligne-Mumford stack" instead of "scheme", then a counterexample would be given by the spaces $\overline M_{g,n}$, which are certainly smooth and proper but far from rational in the large $g$ limit (or, for $g > 0$, in the large $n$ limit). The original references here are, I guess, Deligne (for $\overline M_{1,11}$) and Harris-Mumford (for $\overline M_{25}$).

Kevin Buzzard's hint with the Ramanujan $\Delta$ function is relevant here; indeed, $H^{11,0}(\overline M_{1,11})$ is nonzero, and the $\ell$-adic Galois representation corresponding to $H^{11,0}\oplus H^{0,11}$ is the representation attached to $\Delta$.

My answer to the question Is the moduli space of curves defined over the field with one element? contains some more detailed information about these things.

The natural way to find a scheme instead of a stack would then be to consider finite or generically finite covers of $\overline M_{g,n}$ by smooth proper schemes. There are several closely related constructions of such covers in the literature by Looijenga, Boggi-Pikaart, Pikaart-de Jong, Abramovich-Corti-Vistoli, all using some kind of non-abelian level structure on curves, but as far as I can tell none of them work over the integers.

I think this is very false. A counterexample is given by the spaces $\overline M_{g,n}$, which are certainly smooth and proper but far from rational in the large $g$ limit (or, for $g > 0$, in the large $n$ limit). The original references here are, I guess, Deligne (for $\overline M_{1,11}$) and Harris-Mumford (for $\overline M_{25}$). If you are troubled by the fact that this is a stack and not a variety, then you are still fine since $\overline M_{g,n}$ is a finite cover of a smooth proper scheme.

Kevin Buzzard's hint with the Ramanujan $\Delta$ function is relevant here; indeed, $H^{11,0}(\overline M_{1,11})$ is nonzero, and the $\ell$-adic Galois representation corresponding to $H^{11,0}\oplus H^{0,11}$ is the representation attached to $\Delta$.

Edit: My answer to the question Is the moduli space of curves defined over the field with one element? contains some more detailed information about these things.

(This answer has been edited -- it used to say that a finite cover of $\overline M_{g,n}$ gives a counterexample, which no longer seems obvious.)

If you had written "Deligne-Mumford stack" instead of "scheme", then a counterexample would be given by the spaces $\overline M_{g,n}$, which are certainly smooth and proper but far from rational in the large $g$ limit (or, for $g > 0$, in the large $n$ limit). The original references here are, I guess, Deligne (for $\overline M_{1,11}$) and Harris-Mumford (for $\overline M_{25}$).

Kevin Buzzard's hint with the Ramanujan $\Delta$ function is relevant here; indeed, $H^{11,0}(\overline M_{1,11})$ is nonzero, and the $\ell$-adic Galois representation corresponding to $H^{11,0}\oplus H^{0,11}$ is the representation attached to $\Delta$.

My answer to the question Is the moduli space of curves defined over the field with one element? contains some more detailed information about these things.

The natural way to find a scheme instead of a stack would then be to consider finite or generically finite covers of $\overline M_{g,n}$ by smooth proper schemes. There are several closely related constructions of such covers in the literature by Looijenga, Boggi-Pikaart, Pikaart-de Jong, Abramovich-Corti-Vistoli, all using some kind of non-abelian level structure on curves, but as far as I can tell none of them work over the integers.

I think this is very false. A counterexample is given by the spaces $\overline M_{g,n}$, which are certainly smooth and proper but far from rational in the large $g$ limit (or, for $g > 0$, in the large $n$ limit). The original references here are, I guess, Deligne (for $\overline M_{1,11}$) and Harris-Mumford (for $\overline M_{25}$). If you are troubled by the fact that this is a stack and not a variety, then you are still fine since $\overline M_{g,n}$ is a finite cover of a smooth proper scheme.

Kevin Buzzard's hint with the Ramanujan $\Delta$ function is relevant here; indeed, $H^{11,0}(\overline M_{1,11})$ is nonzero, and the $\ell$-adic Galois representation corresponding to $H^{11,0}\oplus H^{0,11}$ is the representation attached to $\Delta$.

I think this is very false. A counterexample is given by the spaces $\overline M_{g,n}$, which are certainly smooth and proper but far from rational in the large $g$ limit (or, for $g > 0$, in the large $n$ limit). The original references here are, I guess, Deligne (for $\overline M_{1,11}$) and Harris-Mumford (for $\overline M_{25}$). If you are troubled by the fact that this is a stack and not a variety, then you are still fine since $\overline M_{g,n}$ is a finite cover of a smooth proper scheme.

Kevin Buzzard's hint with the Ramanujan $\Delta$ function is relevant here; indeed, $H^{11,0}(\overline M_{1,11})$ is nonzero, and the $\ell$-adic Galois representation corresponding to $H^{11,0}\oplus H^{0,11}$ is the representation attached to $\Delta$.

Edit: My answer to the question Is the moduli space of curves defined over the field with one element? contains some more detailed information about these things.