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We state a conjecture concerning modular absolutely irreducible odd 2-dimensional
representations of the absolute Galois group over finite fields which is purely
combinatorial (without using modular forms) and proof that it is equivalent to Serre’s
strong conjecture. The main idea is to replace modular forms with coefficients in a
finite field of characteristic , by their counterparts in the theory of modular
symbols.
Let 𝓐₂(n) = Γ₂(n)∖𝔖₂ be the quotient of Siegel's space of degree 2 by the principal congruence subgroup of level n in Sp(4,ℤ). This is the moduli space of principally polarized abelian surfaces with a level n structure. Let 𝓐₂(n)* denote the Igusa compactification of this space, and ∂𝓐₂(n)* = 𝓐₂(n)* - 𝓐₂(n) its "boundary". This is a divisor with normal crossings. The main result of this paper is the determination of H(∂𝓐₂(n)*) as a module over the finite group Γ₂(1)/Γ₂(n). As an application...
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