By Michael Harris (auth.), Yuri Tschinkel, Yuri Zarhin (eds.)
Algebra, mathematics, and Geometry: In Honor of Yu. I. Manin involves invited expository and study articles on new advancements bobbing up from Manin’s awesome contributions to arithmetic.
Contributors within the moment quantity: M. Harris, D. Kaledin, M. Kapranov, N.M. Katz, R.M. Kaufmann, J. Kollár, M. Kontsevich, M. Larsen, M. Markl, L. Merel, S.A. Merkulov, M.V. Movshev, E. Mukhin, J. Nekovár, V.V. Nikulin, O. Ogievetsky, F. Oort, D. Orlov, A. Panchishkin, I. Penkov, A. Polishchuk, P. Sarnak, V. Schechtman, V. Tarasov, A.S. Tikhomirov, J. Tsimerman, okay. Vankov, A. Varchenko, A. Vishik, A.A. Voronov, Yu. Zarhin, Th. Zink.
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Additional info for Algebra, Arithmetic, and Geometry: Volume II: In Honor of Yu. I. Manin
Thus ρE , and ρE, become isomorphic over a ﬁnite extension of F + , hence E and E are isogenous by Faltings’ theorem. 3. Assume the Expected Theorems of Section 1. Let F + be a totally real field, let E and E be elliptic curves over F + , and assume E and E do not become isogenous over an abelian extension of F + . Let m and m m be positive integers. Then the L-function L s, ρm is invertible and E, ⊗ ρE , satisfies the expected functional equation. Proof. 2 of [HST]. It suﬃces to mention that the nonvanishing of the Rankin-Selberg L-function along the line Re(s) = 1 of a pair of cuspidal automorphic representations (with unitary central characters) is due in general to Shahidi [Shi63].
Consider now the case that R is not smooth but, on the contrary, local Artin. Moreover, assume that m2 = 0, so that R is itself a (commutative) square-zero extension of k. Then a deformation AR of A over R is also a squarezero extension of A, by the bimodule A ⊗ m (m here is taken as a k-vector space). But this square-zero extension is special: for a general square-zero extension A of A by some M ∈ A-bimod, there does not exist any analogue of the relative cyclic R-module AR# ∈ Fun(Λ, R). We observe the following: the data needed to deﬁne such an analogue is precisely a cyclic bimodule structure on the bimodule M .
HST] M. Harris, N. Shepherd-Barron, R. Taylor, A family of Calabi-Yau varieties and potential automorphy, Annals of Math. (in press). Potential Automorphy of Odd-Dimensional Symmetric Powers 21 [HT] M. Harris, R. Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Math. Studies, 151 (2001). [Ko] R. Kottwitz, On the λ-adic representations associated to some simple Shimura varieties, Invent. , 108, 653–665 (1992). -P. Labesse, Cohomologie, stabilisation et changement de base, Ast´erisque, 257 (1999).