What effect do Torsion points have on an Elliptic Curve's L function?

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.

• 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. Feb 6 '21 at 23:49
• @ChrisWuthrich The fact that the elliptic curve has torsion point is still preserved though, since isogenies are surjective, so my question still stands Feb 7 '21 at 0:00
• 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. Feb 7 '21 at 0:04
• @ChrisWuthrich Thank you for correcting me. I will make sure to learn more about isogenies. Feb 7 '21 at 0:06
• 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. Feb 7 '21 at 0:26

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 $$\ell$$.
• 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 $\mathbb{Q}$. Feb 8 '21 at 19:47