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Given an elliptic curve $E/\mathbb{Q}$, is it possible to determine whether or not $E$ has torsion points just by looking at it's Hasse-Weil L function $L(E,s)$? In general, what effects does an elliptic curve having torsion points have on its L function?

The effect of having free points is clearly seen in the Birch and Swinnerton Dyer Conjecture which relates the order of the zero at $s=1$ of $L(E,s)$ to the rank of $E$. The BSD conjecture also relates the value of $L(E,1)$ to the order of the torsion group, but looking at $L(E,1)$ is not enough to determine the order $E(\mathbb{Q})_{\mathrm{tor}}$ because of all of the other invariants involved in the expression.

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    $\begingroup$ The $L$-function does not change under an isogeny, but the torsion point can. So there is no way one can detect the torsion points from the $L$-function alone. $\endgroup$
    – Chris Wuthrich
    Feb 6, 2021 at 23:49
  • $\begingroup$ @ChrisWuthrich The fact that the elliptic curve has torsion point is still preserved though, since isogenies are surjective, so my question still stands $\endgroup$
    – Milo Moses
    Feb 7, 2021 at 0:00
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    $\begingroup$ I am afraid that is incorrect. Typically there may well be a curve in the isogeny class without any torsion points. Isogenies are not surjective on $K$-rational points, only over algebarically closed fields. $\endgroup$
    – Chris Wuthrich
    Feb 7, 2021 at 0:04
  • $\begingroup$ @ChrisWuthrich Thank you for correcting me. I will make sure to learn more about isogenies. $\endgroup$
    – Milo Moses
    Feb 7, 2021 at 0:06
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    $\begingroup$ Although you can't completely determine the torsion, it may still be possible to find out something about it. From the L-function you can determine all its coefficients, and hence all sizes $|E(\mathbb F_p)|$. If $E$ has good reduction at $p$, then size of torsion divides this size of reduction, so you can determine a bound on torsion by finding the gcd of the reduction sizes. I doubt there is much more than this that you can do. $\endgroup$
    – Wojowu
    Feb 7, 2021 at 0:26

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This has been alluded to in one of the comments, but if $E(\mathbb Q)$ has an $\ell$-torsion point, then at every prime $p$ of good reduction we have $$ p+1-a_p = \#E(\mathbb F_p) \equiv 0 \pmod \ell, $$ so the local factor of the $L$-function at $p$ satisfies $$ L_p(T) = 1 - a_p T + p T^2 \equiv 1 - (p+1) T + p T^2= (1-T)(1-pT) \pmod\ell. $$ Of course, for $L(E/\mathbb Q,s)$, we need to evaluate this at $T=p^{-s}$, and then the meaning of this congruence gets a bit dicey, especially when we multiply them over all $p$ to get $L(E/\mathbb Q,s)$. On the other hand, if you just want to think about the reduction curve $\tilde E_p/\mathbb F_p$, then its zeta function is $$ Z(\tilde E_p/\mathbb F_p) = \frac{1-a_pT+pT^2}{(1-T)(1-pT)} \equiv 1 \pmod\ell, $$ i.e., the $\ell$-torsion point means that the local zeta function is congruent to 1 modulo $\ell$.

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    $\begingroup$ One can go a bit further, I believe. From the $L$-series, one gets the $\ell$-adic representations, i.e. the $\mathbb{Q}_{\ell}$-vector space with the action of the Galois group on them. That in turn allows us to list all $\mathbb{Z}_{\ell}$-lattices stable under the Galois group. Therefore, we would find all torsion subgroups as the curve varies in its isogeny class over $\mathbb{Q}$. $\endgroup$
    – Chris Wuthrich
    Feb 8, 2021 at 19:47
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    $\begingroup$ @ChrisWuthrich Nice point. Although my recollection is that it gets trickier for higher dimensional abelian varieties. $\endgroup$
    – Joe Silverman
    Feb 8, 2021 at 20:14

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