8

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We have a $L$-function $$L(E,s)$$ built from the local parameters $a_p(E)$.

If two elliptic curves are isomorphic, they clearly have the same $L$-function.

What about the converse ? If two elliptic curves $E,E'$ over $\mathbb{Q}$ have the same $L$-function, what can be said about them ? Would they be isomorphic ? Isogenous ? Or one of these properties would hold for "most" curves ? In other words, does this "local" analysis of the curve characterizes it ?

Klaus
  • 4,285

1 Answers1

7

Isogenous.

A theorem of Faltings says that two elliptic curves $E_1$ and $E_2$ over a number field $F$ are isogenous if and only if they have the same $L$-factors at almost all places. See, for example, this article, Theorem 3.1, or find the source:

G. Faltings, Endlichkeitssatze fur abelsche Varietaten uber Zahlkorpern, Invent. Math. 73 (1983), 349-366.

There are indeed cases where two elliptic curves are isogenous, not isomorphic, and all the $L$-factors are identical. For instance, take $E_1$ and $E_2$ be the curves 11a1 and 11a2, respectively, as in Cremona's tables. As the notation indicates, the two curves share an isogeny. Then, $$j(E_1)=-122023936/161051 \neq -52893159101157376/11 = j(E_2),$$ so the curves are not isomorphic. One can verify that all the $L$-factors coincide in this case (in general, they may differ at primes of bad reduction, but here both have bad reduction at 11, and both have bad split multiplicative reduction, so they get the same $L$-factor at 11). For instance, you can verify this using Sage or MAGMA, and verify that $$L(E_1,1)=L(E_2,1)=0.253841860855910684337758923351\ldots$$ In particular, their Mordell-Weil rank is $0$.

Álvaro Lozano-Robledo
  • 15,910
  • Two isogenous curves always have the same $L$-function (so it is not just "in this case"; it is a general feature of isogenous curves that the $L$-factors coincide at every prime.) – user160609 Jul 01 '14 at 00:04
  • The link to Zhang's article is dead; fortunately there is an archived version here. – Viktor Vaughn Jun 08 '21 at 22:32
  • Thank you very much for your answer. Over $\mathbb{Q}$, is it equivalent for elliptic curves $E_1$ and $E_2$ to be isogenous if and only if their $L$-functions coincide (including the bad primes)? – Poitou-Tate May 14 '25 at 08:26