Twisted L-functions and monodromy - download pdf or read online
By Nicholas M. Katz
For centuries, the learn of elliptic curves has performed a important function in arithmetic. The previous century specifically has noticeable large growth during this research, from Mordell's theorem in 1922 to the paintings of Wiles and Taylor-Wiles in 1994. still, there stay many primary questions the place we don't even understand what kind of solutions to anticipate. This e-book explores of them: what's the common rank of elliptic curves, and the way does the rank fluctuate in several types of households of elliptic curves?
Nicholas Katz solutions those questions for households of ''big'' twists of elliptic curves within the functionality box case (with a transforming into consistent field). The monodromy-theoretic equipment he develops prove to use, nonetheless within the functionality box case, both good to households of huge twists of items of every type, not only to elliptic curves.
The leisurely, lucid creation offers the reader a transparent photograph of what's recognized and what's unknown at the moment, and situates the issues solved during this ebook in the broader context of the general research of elliptic curves. The book's technical center uses, and explains, a variety of complicated subject matters starting from fresh leads to finite staff idea to the equipment of l-adic cohomology and monodromy. Twisted L-Functions and Monodromy is vital analyzing for an individual drawn to quantity thought and algebraic geometry.
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Additional resources for Twisted L-functions and monodromy
1) It will be convenient to introduce two generalizations of the notion of pseudoreflection. Suppose we are given a finite-dimensional vector space V over a field K. We write GL(V) for AutK(V), so long as there is no ambiguity about the field K. Recall that an element A in GL(V) is called a pseudoreflection if its space of fixed points, Ker(A-1), has codimension one in V, or, equivalently, if the quotient spaceV/Ker(A-1) has dimension one. 2) Given an integer r ≥ 0, and an element A in GL(V), we say that A has drop r if Ker(A-1) has codimension r in V.
Then we have: 1) If ç is trivial, then Ggeom contains a unipotent element A which is a quadratic of drop r, and Lie(Ggeom)der contains a nilpotent element n which, as endomorphism of V, has rank r. Moreover, n2 = 0 in End(V). 2) If ç is nontrivial, then ((Ggeom)0)der contains a semisimple element A such that for some scalar ¬ in $ ä …≠, ¬A is quadratic of drop r, and Lie(Ggeom)der contains a semisimple endomorphism f of V with precisely two distinct eigenvalues, ¬1 and ¬2, such that f-¬1 as endomorphism of V has rank r.
To prove 5), we argue as follows. We are given that Ï is Lie-irreducible, so Lie(Ggeom)der is an irreducible semisimple Lie subalgebra of End(V). Since R = 1, Lie(Ggeom) and hence its intrinsic subalgebra Lie(Ggeom)der is normalized by a pseudoreflection which is not a reflection. 5], Lie(Ggeom)der is either Lie(SL(V)) or, if dim(V) is even, Lie(Sp(V)). Now repeat the arguments given above for 3) and 4), which used only the inequality rank(Ï) > 2r2. The proof of 6) is similar to that of 5). 5] tells us that Lie(Ggeom)der is either Lie(SL(V)) or Lie(SO(V)).
Twisted L-functions and monodromy by Nicholas M. Katz