We solve the lifting problem for Galois representations in every dimension and in every characteristic. That is, we determine all pairs (n, k), where n is a positive integer and k is a field of characteristic <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, such that for every field F, every continuous homomorphism <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Gamma _F\rightarrow \textrm{GL}_n(k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mi>F</mi> </msub> <mo stretchy="false">→</mo> <msub> <mtext>GL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> lifts to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{GL}_n(W_2(k))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>GL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>W</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma _F$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Γ</mi> <mi>F</mi> </msub> </math></EquationSource> </InlineEquation> is the absolute Galois group of F and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$W_2(k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the ring of p-typical length 2 Witt vectors of k.

The lifting problem for Galois representations

期刊论文
发表日期：

2026年2月13日

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Alexander Merkurjev,
Federico Scavia

摘要

We solve the lifting problem for Galois representations in every dimension and in every characteristic. That is, we determine all pairs (n, k), where n is a positive integer and k is a field of characteristic $p>0$ $p > 0$ , such that for every field F, every continuous homomorphism $\Gamma _F\rightarrow \textrm{GL}_n(k)$ $Γ_{F} \to {GL}_{n} (k)$ lifts to $\textrm{GL}_n(W_2(k))$ ${GL}_{n} (W_{2} (k))$ , where $\Gamma _F$ $Γ_{F}$ is the absolute Galois group of F and $W_2(k)$ $W_{2} (k)$ is the ring of p-typical length 2 Witt vectors of k.