By Haruzo Hida

This publication offers a finished account of the idea of moduli areas of elliptic curves (over integer jewelry) and its program to modular varieties. the development of Galois representations, which play a primary function in Wiles' evidence of the Shimura-Taniyama conjecture, is given. furthermore, the publication offers an overview of the evidence of various modularity result of two-dimensional Galois representations (including that of Wiles), in addition to a few of the author's new ends up in that course.

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0, k is equivalent to a Note that Δ(kδ1 ) = Δ(kδ2 ), if and only if there exists a fourth root of unity ζ with ζ · δ1 = δ2 . 7. Show that, for any fourth root of unity ζ, there is a matrix g ∈ SL2 ( ) which carries the form kδ into the form kζ·δ . 8. Any invariant of binary cubic forms is a polynomial in the discriminant. Proof. Let I ∈ [a1 , a2 , a3 , a4 ] be an invariant polynomial, and deﬁne I ∈ [a], by setting a2 = −a3 = a and a1 = 0 = a4 . 7, I must be a polynomial in a4 , that is, there exists a polynomial I with I(a) = I(a4 ).

R( f, h) := b1 b2 b1 ··· b2 .. a1 be+1 ··· a2 be+1 ··· b1 ··· b2 · · · ad+1 , .. ··· be+1 the displayed matrix having e + d rows. As is well-known, the condition R( f, h) = 0 holds, if and only if f and h share a common factor. Furthermore, R(g· f, g·h) = R( f, h) for all g ∈ SL2 ( ). The discriminant of the form f is then declared to be a1 · Δ( f ) := R( f, f ). 3: C I T 38 The polynomial Δ is, by construction, an invariant polynomial which vanishes in the form f exactly when f has a multiple zero.

7, with the symbolic method. The reader is advised to check that the result is—under the above SL2 ( )-equivariant identiﬁcation ∨ ∨ 2 2∨ of Sym3 ( 2 ) —the same which we will state below. with Sym3 ( 2 ) We begin by constructing several invariants in A. 3: C I T 40 is clearly equivariant and surjective. For this reason, it yields an inclusion of the ring of invariants of binary quartics into A. 10, all invariants of binary quartics are polynomials in the invariants I and J.