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The Tate elliptic curve product formula for the algebraic intermediate Jacobian

Let X=E1×E2×E3X=E_1\times E_2\times E_3 be a product of three Tate elliptic curves with parameters qiq_i for i=1,2,3i=1,2,3. Let rr be the rank of the space of integer triples (n1,n2,n3)(n_1,n_2,n_3) satisfying

∏i=13qini=1.\prod_{i=1}^3q_i^{n_i}=1.

Let Ja2(X)J_a^2(X) denote the algebraic intermediate Jacobian and J2(X)J^2(X) the full intermediate Jacobian. The Tate elliptic curve product formula. One has

dim⁡Ja2(X)=6+r.\dim J_a^2(X)=6+r.

Thus for generic parameters with no multiplicative relations, the dimension is 66, while if all three parameters are equal, it is 88. Consequently, the restriction of the Abel–Jacobi map to CH2(X)algCH^2(X)_{\mathrm{alg}} is never surjective onto J2(X)J^2(X). The statement gives an explicit computation in a basic totally degenerate setting and shows that the algebraic intermediate Jacobian is always a proper target for the Abel–Jacobi image.

Sources & referencesView supporting material

Primary source

Wayne Raskind and Xavier Xarles, “On p-adic intermediate Jacobians”, arXiv:math/0601401 (2006).

abel-jacobi-mapintermediate-jacobiansp-adic-cohomology

Progress summary

Refreshed
Partially solved

The proposed dimension formula is known in several special cases but has not been proved for all choices of the three parameters.

The formula for three Tate elliptic curves appears as Conjecture 23 in a 2006 paper: it predicts dim⁡Ja2(X)=6+r\dim J_a^2(X)=6+r, where rr counts independent multiplicative relations among the parameters. It also predicts that the Abel–Jacobi image is never all of J2(X)J^2(X).

Known results

Proposition 24 proves the formula when there are two independent multiplicative relations; when there is one relation and two curves are isogenous; or when there are no nontrivial multiplicative relations. The generic dimension is therefore established as 66, and the case of three equal parameters is established as 88.

Current status (as of August 2026): The formula remains conjectural in the case of exactly one multiplicative relation not induced by an isogeny between two curves; the cited source gives no general proof, counterexample, or subsequent verification.

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Solutions 0

No solutions have been posted yet.