B.H. Gross, B. Mazur's Arithmetic on Elliptic Curves with Complex Multiplication PDF

By B.H. Gross, B. Mazur

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I=1 Thus in order to prove the proposition it suffices to consider the case where (α, β) = (γ(0), γ(i∞)) for some element γ in SL(2, Z). 31) {α, β, p, q}ω,χ = γ · {0, i∞, p , q }ω,χ with (p , q ) = (p, q)χ(γ)−1 or (p, q) = (p , q )χ(γ); hence we have p = a1 p + c1 q, for χ(γ) = aχ bχ cχ d χ q = b1 p + d1 q ∈ SL(2, Z). Let p = (p1 , . . , pm ), q = (q1 , . . , qm ), and define ck (p , q ) by m m ck (p, q)X k Y m−k . 16, it can be shown that m {0, i∞, p , q }ω,χ = ck (p , q ){0, i∞, 1k , 1m − 1k }ω,χ .

Now for arbitrary α, β ∈ Q, we consider a finite sequence of points η1 , . . , ηl ∈ Q such that (α, η1 ) = (γ1 (0), γ1 (i∞)), . . , (ηi , ηi+1 ) = (γi+1 (0), γi+1 (i∞)), . . . , (ηl , β) = (γl+1 (0), γl+1 (i∞)) for γ1 , . . 6] for the proof of the existence of such a sequence). 15 provides us the existence of the modular symbol {α, β, p, q}ω,χ , and therefore the proof of the theorem is complete. 3. In particular, we establish relations among such periods, which generalize Eichler-Shimura relations for classical ˇ cusp forms studied by Sokuov [118].

Then we define the element {α, p, q}ω,χ ∈ H0 (Σ, (R1 π∗ Q)m ) to be trivial on the summands H0 (s, (R1 π∗ Q)m ) for s = s0 so that we may write {α, p, q}ω,χ ∈ H0 (s0 , (R1 π∗ Q)m ). Now we choose a small disk D of s0 , and let D be the corresponding neighborhood of α that covers D. If zD ∈ D ⊂ H∗ , p = (p1 , . . , pm ) ∈ Zm and q = (q1 , . . , qm ) ∈ Zm , then we define the element m {zD , p, q}D ω,χ ∈ H0 (D, (R1 π∗ Q) ) to be the homology class of the cycle m (pi η1 + qi η2 )vD , i=1 where vD ∈ X is the point corresponding to zD ∈ H∗ .

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Arithmetic on Elliptic Curves with Complex Multiplication by B.H. Gross, B. Mazur

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