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            <title xml:lang="en">ON MODIFIED DIAGONAL CYCLES AND THE BEAUVILLE DECOMPOSITION OF THE CERESA CYCLE</title>
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                <title xml:lang="en">ON MODIFIED DIAGONAL CYCLES AND THE BEAUVILLE DECOMPOSITION OF THE CERESA CYCLE</title>
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              <p>&lt;div&gt;&lt;p&gt;Let C be a curve of genus g ≥ 2, and let J be its Jacobian. The choice of a degree 1 divisor e on C gives an embedding of C into J; we denote by [C] e ∈ CH (J; Q) the class in the Chow group of J defined by the image of this embedding. It is known from the work of S.-W. Zhang that the vanishing of the Ceresa cycle Cer(C, e) := [C] e -[-1] * [C] e is equivalent to both the vanishing of the 1st Beauville component [C] e&lt;/p&gt;&lt;p&gt;(1) and the vanishing of the 3rd Gross-Kudla-Schoen modified diagonal cycle Γ 3 (C, e) ∈ CH(C 3 ; Q). In this paper, we extend this result to show that the vanishing of the s-th Beauville component [C] e (s) for s ≥ 1 is equivalent to the vanishing of the (s + 2)-nd modified diagonal cycle Γ s+2 (C, e) ∈ CH(C s+2 ; Q). We also establish "successive vanishing" results for these cycles: for instance, if Γ n (C, e) = Γ n+1 (C, e) = 0, then Γ k (C, e) = 0 for all k ≥ n. In the s = 1 case, we show an integral refinement to the original statement, relating the order of torsion of Cer(C, e) ∈ CH(J; Z) to that of Γ 3 (C, e) ∈ CH(C 3 ; Z).&lt;/p&gt;&lt;/div&gt;</p>
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