Download e-book for kindle: Abelian l-adic representations and elliptic curves by Jean-Pierre Serre

By Jean-Pierre Serre

ISBN-10: 0201093847

ISBN-13: 9780201093841

This vintage e-book comprises an creation to platforms of l-adic representations, an issue of serious value in quantity concept and algebraic geometry, as mirrored through the magnificent 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 unearths a pleasant correspondence among the l-adic representations and the linear representations of a few algebraic teams (now known as Taniyama groups). The final bankruptcy handles the case of elliptic curves with out complicated multiplication, the most results of that's that a dead ringer for the Galois staff (in the corresponding l-adic illustration) is "large."

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Qil ! Pi + ^T - ? ^i "• ^T } ^i=l ^ i=l ^« i=l ^ ^1 i=l ^ ^2-* Then F,_ -, is Zariski closed and projects isomorphically to ^l'^2 Zariski open subsets of each factor. Proof. Rewrite the definition of ] I^i' iQi I ! Pi + ! ^Qi + I L ^ 1 I i=l 1 i=l 1 T^ ^ as ^l'^2 B. + I i€T^ ^ B. = (2g+#T +#T )-ooJ. ,fj^ a basis of V, where M = N-g+1. Among these functions, those which have zeroes at i=l ^ i=l ^ Jl B. + J] B. ieT^ ^ ieT2 for fixed J^P. , I Q . 14. ,5:Q. I pP.. Q. i=l ^ _^(u^2)^^(2)^^(2)j Note tUtif h € V n Ij;p^2^Q then ( J^P^,, IQ^) € F^ ^ since h has exactly N zeroes, and poles only at «> .

N T (e + e ) = . 00) = i=l ^ 5;Q + J R -goo e Jac C-Gj 1 1 L J (because it's the image of a point in Z ) . If g-r is even, then D € (Jac C-0) + er„ D € (Jac C-0) + er „ \ ; if ^"^ is odd .. QED IR^,.. ,Kg__^,oo^ So, we take one copy of Z for each T, and we glue them together according to their identification as subsets of the Jacobian; we have to see that this glueing satisfies the conditions to give the atlas of a variety. _ Here - in Jac C-0 is a difference of sets, but + in (Jac C-0)+e_ means translation of a set by a point using the group law on Jac C.

S(P2) implies P2 = i(P,). ,P^) 1 z has no poles on (C )o , hence is in Thus rf(C ^) . For > )o , O f,2' n >_ 3, by induction and the expression for s (P. , • • ,Pj^) , s (P^ , • • ,P ) has poles only if t(P^) = t(P ) . But by symmetry, it has poles only if t(P^) ^ = t(P n ) too. ) 1 = t(P^) 2 = t(P n ) has codimension 2 in (C^) , so s(P-,»«,P ) has no poles at all in (C^) . s(P^,--,P^) . Thus the coefficients V. ) and s (P^ , • • • ,P, ) , hence are functions in that (C^) > V(a ) is a morphism.

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

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Categories: Algebraic Geometry