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divisble --> divisible, deformations theoretic arguments --> deformation-theoretic argument
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Matthieu Romagny
Matthieu Romagny
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Concerning the first question, the answer is yes.

Given an abelian variety $A$ over $k$ of genus $g$, we have that the first cohomology of its tangent space is of dimension $g^2$. Moreover, by a theorem of Grothendieck, we know that the deformations of $A$ are unobstructed. This yields that the base ring of the universal deformation is isomorphic to $k[t_1, \dots, t_{g^2}]$.

Using the theory of p-divisbledivisible groups, one can show that in the case where $A$ is ordinary, the formal scheme representing the deformation functor can be equipped with the structure of a formal group scheme. This corresponds to a more canonical choice of parameters compared to the above one. In any case, the dimensions are the same.

In this deformations theoretic argumentsdeformation-theoretic argument, polarizations are not involved. If you add them as additional data to the moduli functor, dimensions can change.

Concerning the first question, the answer is yes.

Given an abelian variety $A$ over $k$ of genus $g$, we have that the first cohomology of its tangent space is of dimension $g^2$. Moreover, by a theorem of Grothendieck, we know that the deformations of $A$ are unobstructed. This yields that the base ring of the universal deformation is isomorphic to $k[t_1, \dots, t_{g^2}]$.

Using the theory of p-divisble groups, one can show that in the case where $A$ is ordinary, the formal scheme representing the deformation functor can be equipped with the structure of a formal group scheme. This corresponds to a more canonical choice of parameters compared to the above one. In any case, the dimensions are the same.

In this deformations theoretic arguments, polarizations are not involved. If you add them as additional data to the moduli functor, dimensions can change.

Concerning the first question, the answer is yes.

Given an abelian variety $A$ over $k$ of genus $g$, we have that the first cohomology of its tangent space is of dimension $g^2$. Moreover, by a theorem of Grothendieck, we know that the deformations of $A$ are unobstructed. This yields that the base ring of the universal deformation is isomorphic to $k[t_1, \dots, t_{g^2}]$.

Using the theory of p-divisible groups, one can show that in the case where $A$ is ordinary, the formal scheme representing the deformation functor can be equipped with the structure of a formal group scheme. This corresponds to a more canonical choice of parameters compared to the above one. In any case, the dimensions are the same.

In this deformation-theoretic argument, polarizations are not involved. If you add them as additional data to the moduli functor, dimensions can change.

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Holger Partsch
Holger Partsch
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Concerning the first question, the answer is yes.

Given an abelian variety $A$ over $k$ of genus $g$, we have that the first cohomology of its tangent space is of dimension $g^2$. Moreover, by a theorem of Grothendieck, we know that the deformations of $A$ are unobstructed. This yields that the base ring of the universal deformation is isomorphic to $k[t_1, \dots, t_{g^2}]$.

Using the theory of p-divisble groups, one can show that in the case where $A$ is ordinary, the formal scheme representing the deformation functor can be equipped with the structure of a formal group scheme. This corresponds to a more canonical choice of parameters compared to the above one. In any case, the dimensions are the same.

In this deformations theoretic arguments, polarizations are not involved. If you add them as additional data to the moduli functor, dimensions can change.