Abelian l-adic representations and elliptic curves by Jean-Pierre Serre

By Jean-Pierre Serre

This vintage e-book includes an advent to platforms of l-adic representations, a subject of significant value in quantity concept and algebraic geometry, as mirrored via the miraculous fresh advancements at the Taniyama-Weil conjecture and Fermat's final Theorem. The preliminary chapters are dedicated to the Abelian case (complex multiplication), the place one reveals a pleasant correspondence among the l-adic representations and the linear representations of a few algebraic teams (now referred to as Taniyama groups). The final bankruptcy handles the case of elliptic curves with out complicated multiplication, the most results of that's that clone of the Galois team (in the corresponding l-adic illustration) is "large."

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Example text

11. The map xp’ o q : T C* is a character of T;so it is of the form xp, and its exponent p = q*(p’) clearly lies in u”. It is not difficult t o see that in fact, the T’-orbit O(T’) is the G-orbit space O(T)/G. 6). Any (strongly convex) N-cone also is an “-cone; the face structure is of course independent of the lattice, so there is a one-toone correspondence between T-orbits and TI-orbits. In that case, the induced map X/G --H X//G is a homeomorphism, so the algebraic quotient is just the topological orbit space.

We may interpret U := X u , as an open subset of X := X u . Since X \ U is a closed T-invariant subvariety, its vanishing ideal in O ( X ) is generated by finitely many characters xi = x p ’ E Sx. As a consequence, U is the union of the (affine) principal open toric subvarieties (xi# 0) = X,, corresponding to the faces ~i = CJ n ( p i ) L (cf. 4). 8, this description U = UX,” yields an analogous description u’ = ri. Hence, there is a face ri 5 u satisfying dimTi = dimu’. Together with ~i u’ u,this implies that a’ is included in lin(ri) n u = ri, thus 0 proving u’= ri.

E, of curves as depicted in Figure 17: Each Ei Figure 17. System of exceptional curves only intersects its neighbours; the intersection is transverse and consists of one point (transversality means that the curves meet like coordinate axes). (2) To curves Ei and E j , one attaches their “intersection number” Ei . Ej. For i # j , this is just the number of intersection points since a non-empty intersection is transverse. Ei = E: expresses the “twisting” of the ambient smooth surface along the curve.

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