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edited Oct 27, 2023 at 6:41

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Let me upgrade my comment to an answer:

There exist (simple) abelian varieties over $\mathbb{Q}_{p}$ of every dimension $g\geq 2$ with supersingular good reduction and non-semisimple Tate module. Abelian surface examples are constructed in M. Volkov, Abelian surfaces with supersingular good reduction and non-semisimple Tate module, Int. J. Number Theory (2010), no.4, 811–818. Examples are constructed where the special fibre is a product of supersingular elliptic curves, and where the special fibre is a simple supersingular abelian surface.

The strategy is to write down an admissible filtered $\phi$-module $(D,\mathrm{Fil},\varphi)$ over $\mathbb{Q}_{p}$ which is not semisimple, has Hodge-Tate weights $(0,1)$ and totally isotropic $\mathrm{Fil}$ under a non-degenerate pairing, and such that the characteristic polynomial of $\varphi$ is a supersingular $p$-Weil polynomial. The conditions guarantee that there exists an abelian variety $A$ over $\mathbb{Q}_{p}$ with good reduction and $\mathbb{D}_{\mathrm{cris}}(V_{p}(A))=(D,\mathrm{Fil},\varphi)$ by M. Volkov, A class of $p$-adic Galois representations arising from abelian varieties over $\mathbb{Q}_{p}$, Math. Ann. 331 (2005), no.4, 889–923, and one knows that the special fibre is supersingular by Honda-Tate theory.

Let me upgrade my comment to an answer:

There exist (simple) abelian varieties over $\mathbb{Q}_{p}$ of dimension $g\geq 2$ with supersingular good reduction and non-semisimple Tate module. Abelian surface examples are constructed in M. Volkov, Abelian surfaces with supersingular good reduction and non-semisimple Tate module, Int. J. Number Theory (2010), no.4, 811–818. Examples are constructed where the special fibre is a product of supersingular elliptic curves, and where the special fibre is a simple supersingular abelian surface.

The strategy is to write down an admissible filtered $\phi$-module $(D,\mathrm{Fil},\varphi)$ over $\mathbb{Q}_{p}$ which is not semisimple, has Hodge-Tate weights $(0,1)$ and totally isotropic $\mathrm{Fil}$ under a non-degenerate pairing, and such that the characteristic polynomial of $\varphi$ is a supersingular $p$-Weil polynomial. The conditions guarantee that there exists an abelian variety $A$ over $\mathbb{Q}_{p}$ with good reduction and $\mathbb{D}_{\mathrm{cris}}(V_{p}(A))=(D,\mathrm{Fil},\varphi)$ by M. Volkov, A class of $p$-adic Galois representations arising from abelian varieties over $\mathbb{Q}_{p}$, Math. Ann. 331 (2005), no.4, 889–923, and one knows that the special fibre is supersingular by Honda-Tate theory.

Let me upgrade my comment to an answer:

There exist (simple) abelian varieties over $\mathbb{Q}_{p}$ of every dimension $g\geq 2$ with supersingular good reduction and non-semisimple Tate module. Abelian surface examples are constructed in M. Volkov, Abelian surfaces with supersingular good reduction and non-semisimple Tate module, Int. J. Number Theory (2010), no.4, 811–818. Examples are constructed where the special fibre is a product of supersingular elliptic curves, and where the special fibre is a simple supersingular abelian surface.

The strategy is to write down an admissible filtered $\phi$-module $(D,\mathrm{Fil},\varphi)$ over $\mathbb{Q}_{p}$ which is not semisimple, has Hodge-Tate weights $(0,1)$ and totally isotropic $\mathrm{Fil}$ under a non-degenerate pairing, and such that the characteristic polynomial of $\varphi$ is a supersingular $p$-Weil polynomial. The conditions guarantee that there exists an abelian variety $A$ over $\mathbb{Q}_{p}$ with good reduction and $\mathbb{D}_{\mathrm{cris}}(V_{p}(A))=(D,\mathrm{Fil},\varphi)$ by M. Volkov, A class of $p$-adic Galois representations arising from abelian varieties over $\mathbb{Q}_{p}$, Math. Ann. 331 (2005), no.4, 889–923, and one knows that the special fibre is supersingular by Honda-Tate theory.

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created Oct 26, 2023 at 21:09

1.4k
1
11
21

Let me upgrade my comment to an answer:

There exist (simple) abelian varieties over $\mathbb{Q}_{p}$ of dimension $g\geq 2$ with supersingular good reduction and non-semisimple Tate module. Abelian surface examples are constructed in M. Volkov, Abelian surfaces with supersingular good reduction and non-semisimple Tate module, Int. J. Number Theory (2010), no.4, 811–818. Examples are constructed where the special fibre is a product of supersingular elliptic curves, and where the special fibre is a simple supersingular abelian surface.

The strategy is to write down an admissible filtered $\phi$-module $(D,\mathrm{Fil},\varphi)$ over $\mathbb{Q}_{p}$ which is not semisimple, has Hodge-Tate weights $(0,1)$ and totally isotropic $\mathrm{Fil}$ under a non-degenerate pairing, and such that the characteristic polynomial of $\varphi$ is a supersingular $p$-Weil polynomial. The conditions guarantee that there exists an abelian variety $A$ over $\mathbb{Q}_{p}$ with good reduction and $\mathbb{D}_{\mathrm{cris}}(V_{p}(A))=(D,\mathrm{Fil},\varphi)$ by M. Volkov, A class of $p$-adic Galois representations arising from abelian varieties over $\mathbb{Q}_{p}$, Math. Ann. 331 (2005), no.4, 889–923, and one knows that the special fibre is supersingular by Honda-Tate theory.