By David Mumford, C. P. Ramanujam, Yuri Manin

Now again in print, the revised version of this renowned research supplies a scientific account of the elemental effects approximately abelian types. Mumford describes the analytic equipment and effects acceptable while the floor box okay is the complicated box C and discusses the scheme-theoretic equipment and effects used to house inseparable isogenies whilst the floor box okay has attribute p. the writer additionally offers a self-contained facts of the lifestyles of a twin abeilan kind, stories the constitution of the hoop of endormorphisms, and comprises in appendices "The Theorem of Tate" and the "Mordell-Weil Thorem." this can be a longtime paintings through an eminent mathematician and the single ebook in this topic.

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**Extra resources for Abelian varieties**

**Example text**

We then have for a primitive Dirichlet character χ : (Z/ln Z)× → B × χdφ(t) = (la)−n χ(x)a(m, φ)(ζ x t)m x m ⎛ ⎞ ⎝ = (la)−n m χ(x)ζ mx ⎠ a(m, φ)tm x∈Z/ln Z = G(χ)(la)−n χ−1 (m)a(m, φ)tm , m where G(χ) is the Gauss sum: G(χ) = x∈Z/ln Z χ(x)ζ x = 0. 13 We may deﬁne φ|U (l) for any function φ : μl∞ (K) → B by the same formula (U). 3). 4 Evaluation Formula Take B = Zp [μN ] ⊂ Qp . Applying the computation in Sect. 1) (applied to χ−1 λ in place of λ) χdΦ = (lλ(l))−n G(χ)Φχ−1 λ (1) = (lλ(l))−n G(χ)L(0, χ−1 λ).

L We call this operator the Hecke operator of the prime l. (U) 14 1 Nontriviality of Arithmetic Invariants The Hecke operator has the following geometric interpretation: We have the l-power group homomorphism [l] : Gm → Gm given by x → xl . The map [l] is actually an endomorphism of the functor Gm . In this sense, it is essentially surjective (Gm is a divisible group under fppf topology over B in the geometric √ terminology in Sect. 10), as s ∈ Gm (A) is in the image of [l] from Gm (A[ l a]).

Waiting until Chaps. 2 and 6 for a more thorough treatment of the algebraization, here we give an outline of how to deﬁne modular forms algebraically. Pick two linearly independent numbers w = (w1 , w2 ) ∈ C2 . Writing u for the variable on C, the quotient C/Lw of C by a lattice Lw = Zw1 + Zw2 gives rise to a pair (E, ω) of elliptic curves and the diﬀerential ω = du of ﬁrst kind (nowhere vanishing diﬀerential). First of all, we note E(C) ∼ = C/Lw . Then we embed E into P2 by u → (x(u), y(u), 1) ∈ P2 (C) for Weierstrass ℘-functions (Sect.