Let $J(C)$ be the jacobian of a hyperelliptic curve $C$ of genus 2 defined over finite field $\mathbb{F}_q$. Let $\Theta$ be the image of the curve on the Jacobian under the embedding $P \mapsto P - \mathcal{O}$, which is also known as the theta divisor.

Do we know something about the structure of the following set: $$ J(C)_{\Theta}[n] := \{D \in J(C) : nD \in \Theta \}$$ Does this set have a name? Like "Theta n-torsion points"?

Clearly, the n-torsion points defined as $$ J(C)[n] := \{D \in J(C) : nD = \mathcal O \}$$ is a subset of $J(C)_{\Theta}[n]$.

I could not find many papers explicitly studying this set, the structure and cardinality. I found one paper Division polynomials and multiplication formulae of Jacobian varieties of dimension 2 by N. Kanayama where he defines this set in Section 3.2.2.

Is it correct/wrong to call it generalization of torsion points?

I am interested in the cardinality of the set $$ J(C)_{\Theta}[n] \cap J(C)_{\Theta}[m], m, n \in \mathbb{Z}, m \neq n$$.

  • 1
    $\begingroup$ Have your tried searching on "uniform Manin-Mumford conjecture" or "uniform Raynaud's theorem"? $\endgroup$ May 4 at 19:29
  • 2
    $\begingroup$ $J(C)_{\Theta}[n]$ is a curve. The map $D \mapsto nD$ describes it as an étale cover of $\Theta$ of degree $n^2$. To compute the intersection we have to know the algebraic equivalence class of the two curves with $m$ and $n$. My guess is that this curve is algebraically equivalent to $n\Theta$ but I haven't checked that. If that's the case, the intersection will have $2mn$ points. $\endgroup$ May 5 at 2:52

The set $J(C)_{\Theta}[n]$ has the structure of a smooth irreducible algebraic curve, and the restriction of $J(C)\xrightarrow{\times n } J(C)$ to $C$ defines a morphism $J(C)_{\Theta}[n]\rightarrow C$ which is a finite etale cover with Galois group $J(C)[n]$. I don't think it is good to think of $J(C)_{\Theta}[n]$ as generalized torsion points. I think it's better to think of $J(C)_{\Theta}[n]$ as a curve which is an unramified cover of $C$. For example, while $J(C)[n]$ has $n^4$ points over $\bar{\mathbb{F}}_q$ (when $q$ is coprime to $n$), $J(C)_{\Theta}[n]$ has infinitely many $\bar{\mathbb{F}}_q$-points. Also by Riemann-Hurwitz the genus of $J(C)_{\Theta}[n]$ is $n^4+1$, so the Hasse-Weil bound tells you something about its $\mathbb{F}_q$-points.

For your second question about $J(C)_{\Theta}[m] \cap J(C)_{\Theta}[n]$, I will interpret this as a scheme theoretic intersection. (In other words, incorporating multiplicities.) We may calculate this using some facts about intersection theory on surfaces and line bundles on abelian varieties. As noted before, the curve $J(C)_{\Theta}[m]$ is the pullback of the divisor $C\subset J(C)$ (embedded using your implicitly chosen point $\mathcal{O}$). By the theorem of the square, we have the following linear equivalence of divisors on $J(C)$: $$J(C)_{\Theta}[n] \sim \frac{n^2+n}{2} C + \frac{n^2-n}{2} [-1]^*C.$$ Here $[-1]^*C$ is the image of $C$ under the $[-1]$ map. Since $[-1]^*C$ is algebraically equivalent to $C$ (even linearly equivalent if you choose $\mathcal{O}$ to be a Weierstrass point), we conclude that $$ (J_{\Theta}(C)[m],J_{\Theta}(C)[n]) = m^2n^2(C,C). $$ By the adjunction formula (and the fact that the canonical bundle of $J(C)$ is trivial): $(C,C) = 2p_a(C)-2 = 2$.

Conclusion: When counted with multiplicity, $J_{\Theta}(C)[m]$ and $J_{\Theta}(C)[n]$ intersect in $2m^2n^2$ points.

  • $\begingroup$ Thanks a lot for this answer, it already helps me a lot, I am not very well-versed in Algebraic geometry, but I am taking my time to understand your answer to the first question. For references for the answer to the second question, I found the Chapter in Diophantine Geometry by Hindry and Silverman useful, it has all the theorems used. $\endgroup$
    – AVP82000
    May 5 at 19:43
  • $\begingroup$ It is still unclear to me why the preimage of C under [n] would be smooth and irreducible. I tried looking it up but was unsuccessful. I have a feeling that Prop A.3.2.4 from Diophantine Geometry by Hindry and Silverman, which says if $f : X \rightarrow Y$ is a finite morphism between 2 projective varities, if $D$ is an ample divisor on $Y$, then $f^* D$ is an ample divisor on $X$ maybe useful, but I don't understand it very well, so maybe it is completly wrong? $\endgroup$
    – AVP82000
    May 15 at 13:48
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    $\begingroup$ It's not obvious, and is specific to the theory of curves and jacobian varieties. You can look at Milne's notes on abelian varieties, Chapter III, Section 9 ('Obtaining coverings of a curve from its Jacobian') $\endgroup$
    – Jef
    May 15 at 15:43

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