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Let $E_1$ and $E_2$ be two Elliptic curve defined over $\mathbb Q$ . Let $p$ be an fixed given odd prime of $\mathbb Q$ at which both the curves have good ordinary reduction. Moreover p-adic $L$-function of $E_1$ and $E_2$ are same . Does it mean that the complex $L$-function of $E_1$ and $E_2$ are also same ? Is there some sufficient criteria on p-adic $L$-functions such that such that the $L$ function is determined?

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An interesting comment is that the arithmetic $p$-adic $L$-function (i.e. that defined as the characteristic power series of the Selmer group) is not invariant under $p$-isogeny: the $\mu$-invariant can change (see the appendix to Perrin-Riou's Fonctions $L$ $p$-adiques, théorie d'Iwasawa et points de Heegner (ams.org/mathscinet-getitem?mr=928018)). This doesn't necessarily imply anything either way regarding your question, but I thought I'd mention it. –  Rob Harron Apr 11 '10 at 23:59
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Though this is more an indication that the arithmetic $p$-adic $L$-function as defined in the article you quote is not the correct object than anything else. Add a few years and B.Perrin-Riou (among others) had an arithmetic $p$-adic function invariant under $p$-isogeny –  Olivier Apr 12 '10 at 11:48
    
Nono, the $p$-adic $L$-function must depend on the curve in the isogeny class. It is defined to interpolate values of the form $$ \frac{L(E,\chi,1)}{\Omega_E} = \frac{L(f,\chi,1)}{\Omega_E}$$ where $f$ is the modular form attached to the isogeny class of $E$ and $\Omega_E$ is the Néron period of $E$. So the $p$-adic $L$-function varies as the period varies. –  Chris Wuthrich Apr 12 '10 at 15:35
    
That is the analytic side, on the algebraic it has to vary as well. The $\mu$-invariant changes. For instance the $\mu$-invariant for 11a1 at $p=5$ is 1, while for 11a2, the $X_1$-optimal curve, it is zero. This means that the Tate-Shafarevich group of 11a1 grows exponentially in size in the cyclotomic $\mathbb{Z}_5$-tower while the one of 11a2 stays the same. –  Chris Wuthrich Apr 12 '10 at 15:36
    
Well, ditto for the choice of period then (and definition of the $p$-adic $L$-function): that is to say, in my opinion, this is more an indication that the choice of Neron period is not the correct one (replace it by the canonical period) than anything else. I want my conjectures on special values of $L$-functions to depend only on the motive, but I realize that this may be purely a question of taste. –  Olivier Apr 13 '10 at 8:53

4 Answers 4

up vote 10 down vote accepted

Hmmm. I am not so sure that the answer is "No". In fact I would rather bet on "Yes".

Of course, I totally agree that the characteristic ideal, i.e. the ideal generated by the $p$-adic $L$-function in $\Lambda$, is not enough to determine the elliptic curve. In particular there are plenty of curves for which the $p$-adic $L$-function is a unit in $\Lambda^{\times}$.

The $p$-adic $L$-function can be viewed as a measure $\mu$ on the Galois group of $F_\infty/\mathbb{Q}$. It is build up from modular symbols of the form $\bigl[\frac{a}{p^k}\bigr]$ as $a$ and $k$ varies over all positive integers. Knowing the measure $\mu$ it is easy to extract the unit root $\alpha$ of the Frobenius at $p$ and hence the value of $a_p$. Then it is not difficult to see from the definition of $\mu$ that one can compute all the modular symbols $\bigl[\frac{a}{p^k}\bigr]$. It is true that these values do not seem to carry the value of $a_{\ell}$ for primes $\ell\neq p$ with them needed to reconstruct the complex $L$-function; we would need modular symbols with $\ell$ in the denominator and I can not see immediately how to get them from $\mu$.

Nevertheless, there are plenty of values of $a$ and $k$. And it would be a big surprise to me if there very by chance two elliptic curves such that all the values of the modular symbols $\bigl[\frac{a}{p^k}\bigr]$ would be equal. But I have no clue of how to prove this intuition.

So I ran through some examples. Let $F_{\infty}$ be the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. I picked a few elliptic curves of small conductor such that

  • $E$ has good ordinary reduction at $p$.

  • There are no torsion points in $E(F_{\infty})$, simply by making sure that the $\ell$-adic Galois representation is surjective for all $\ell$.

  • The Tamagawa numbers are all 1 for $E/F_\infty$, by imposing that the Kodaira type at all bad places is $I_1$.

  • The curve is not anomalous at $p$, i.e. $a_p \neq 1$. Actually, I fix $a_p$.

  • The Tate-Shafarevich group of $E/\mathbb{Q}$ is trivial.

  • The rank of $E(F_{\infty})$ is $0$. This will follow from the previous conditions if the rank of $E(\mathbb{Q})$ is $0$, since the $p$-adic $L$-function will be a unit.

Then I computed the $p$-adic $L$-functions $L_p(T)$ for these with $T$ corresponding to $1+p$ under the cyclotomic character Gal$(F_{\infty}/\mathbb{Q})$. By what I have imposed the leading term will be equal. For each $p^n$-th root of unity $\zeta$, the value of $L_p(\zeta-1)$ is, up to a power of $\alpha$ which is the same for all my curves because I fixed $a_p$, equal to the order of the Tate-Shafarevich group at the $n$-th level; at least if one believes the ($p$-adic version of the) Birch and Swinnerton-Dyer conjecture. From what I imposed, it is clear that the $p$-primary part will be trivial, but there may be different primes appearing in Sha for various curves. So there is no reason to believe that it would be easy to find two curves that have the same $p$-adic $L$-function in these family that I have chosen.

Here are some examples with $p=5$ and $a_5=-1$.

139a1 $4 + 4 \cdot 5 + 4 \cdot 5^2 + O(5^5) + (1 + 4 \cdot 5 + O(5^2)) \cdot T + (3 + 5 + O(5^2)) \cdot T^2 + (3 + 2 \cdot 5 + O(5^2)) \cdot T^3 + (1 + 5 + O(5^2)) \cdot T^4 + O(T^5)$ 141e1 $4 + 4 \cdot 5 + 4 \cdot 5^2 + O(5^5) + (4 + 3 \cdot 5 + O(5^2)) \cdot T + (3 \cdot 5 + O(5^2)) \cdot T^2 + (5 + O(5^2)) \cdot T^3 + (2 + 4 \cdot 5 + O(5^2)) \cdot T^4 + O(T^5) $ 346a1 $ 4 + 4 \cdot 5 + 4 \cdot 5^2 + O(5^5) + (2 + 5 + O(5^2)) \cdot T + (4 \cdot 5 + O(5^2)) \cdot T^2 + O(5^2) \cdot T^3 + (1 + 2 \cdot 5 + O(5^2)) \cdot T^4 + O(T^5)$ 906i1 $4 + 4 \cdot 5 + 4 \cdot 5^2 + O(5^5) + (3 + O(5^2)) \cdot T + (2 + 5 + O(5^2)) \cdot T^2 + (3 + 5 + O(5^2)) \cdot T^3 + (3 + 2 \cdot 5 + O(5^2)) \cdot T^4 + O(T^5)$

Finally a word why I believe the answer should be Yes. A big dream in the direction of BSD is to hope that there is a link between the $p$-adic and the complex $L$-function. Of course, we should believe that the order of vanishing at $s=1$ should be equal for instance (and this is only known when it is at most a simple zero). I would hope that such a link will be bijective. But that is a mere intuition and hence possibly wrong. But it also explains that the answer to this question might well be very difficult.

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thank you for your answer! –  Somnath Jha Apr 12 '10 at 6:22
    
This is very interesting; thanks. –  Emerton Apr 12 '10 at 12:53

Yes, it is true that the $p$-adic $L$-function of an elliptic curve $E$ over $\mathbf{Q}$ determines the isogeny class of $E$. As mentioned in the Luo-Ramakrishnan article pointed out by Idoneal, this follows from analytical results of Rohrlich. I thought I would sketch the argument of Rohrlich because it's interesting.

So, let us assume $L_p(E_1,\cdot)=L_p(E_2,\cdot)$ for two elliptic curves $E_1,E_2$ with good ordinary reduction at $p$. By the interpolation property of the $p$-adic $L$-function, this means that

\begin{equation} \frac{L(E_1,\chi,1)}{\Omega_{E_1}^\pm} = \frac{L(E_2,\chi,1)}{\Omega_{E_2}^\pm} \end{equation} for all Dirichlet characters $\chi$ of $p$-power conductor. Here $\pm = \chi(-1)$ and $\Omega^+$ (resp. $\Omega^{-}$) is the real (resp. imaginary) period of a Néron differential.

Now there is this nice analytical result by Rohrlich (On $L$-functions of elliptic curves and cyclotomic towers, Invent. Mat. 75, 1984) : if $\chi$ is a Dirichlet character of conductor $p^m$, then the average value of $L(E,\chi^{\sigma},1)$, where $\chi^\sigma$ runs through the conjugate characters of $\chi$, tends to $1$ as $m \to +\infty$. This is proved by writing $L(E,\chi,1)$ as a quickly converging series and using clever analytical arguments.

From this we deduce that $\Omega_{E_1}^{\pm}=\Omega_{E_2}^{\pm}$. This implies that there is an isogeny $\varphi : E_1 \to E_2$ (which is a priori defined over $\mathbf{C}$) such that $\varphi^* \omega_{E_2}$ is a non-zero rational multiple of $\omega_{E_1}$. By taking the sums of the conjugates of $\varphi$, we deduce that $E_1$ and $E_2$ are isogenous over $\mathbf{Q}$ (and thus their complex $L$-functions coincide).

Remark : we cannot always deduce that $E_1$ and $E_2$ are isomorphic over $\mathbf{Q}$. For example $E_1=15a1$ and $E_2=15a4$ are $2$-isogenous and have the same Néron periods $\Omega_E^{\pm}$, so they have the same $p$-adic $L$-function, but they are not isomorphic. To me this rather indicates that the definition of $p$-adic $L$-function is not the right one (as pointed out by Olivier in his comment). If we want a theory of $p$-adic $L$-functions which looks like as much as possible as the theory of complex $L$-functions, then the $p$-adic $L$-function should be invariant under isogeny.

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> then the p-adic L-function should be invariant under isogeny. Would you then want the Selmer groups to be invariant as well ? Or do you want the main conjecture to look really complicated ? It is a question of taste and one of how far above the ground you are walking. The concrete description of the Selmer groups and the p-adic L-functions in terms of basic arithmetic information on the curve can be very valuable. E.g. Dokchitsers relie on the variation of the Selmer group by isogeny for many of their results. Of course for an "arbitary motive" you may be reluctant to choose a lattice. –  Chris Wuthrich Oct 22 '11 at 12:32
    
@Chris : I don't really know what would be the best definition of $p$-adic $L$-function. As you say, this depends much on the applications you're interested in. Thinking a bit more, you're right that having information like the $\mu$-invariant from the $p$-adic $L$-function is certainly useful (but then it's strange that the $p$-adic $L$-function doesn't always detect 2-isogenies). Also I think it's good to at least try to think purely $p$-adically e.g. try to prove things without using the complex $L$-function. –  François Brunault Oct 22 '11 at 19:35

Yes. See the Invent. Math. paper by Luo and Ramakrishnan titled "Determination of modular forms by twists of critical L-values". They mention this result on the third page.

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thanks idoneal ! –  Somnath Jha Apr 12 '10 at 15:49

This is not a definitive answer to the question. It's goal is simply to establish enough background and perspective on the question to explain why I think that the answer is likely to be ``No''.

The $p$-adic $L$-function does not interpolate infinitely many special values of the $L$-function of $E$. Rather, the $p$-adic $L$-function (and this is a feature of $p$-adic $L$-functions generally) interpolates critical values (in the sense of Deligne) of twisted $L$-functions of $E$. For an elliptic curve, the only critical value is $s = 1$; it is because one allows twisting that one ends up with a function, and not just a number.

A little more precisely: the $p$-adic $L$-function of an elliptic curve $E$ over $\mathbb Q$ interpolates the twisted central values, i.e. the values $L(f\_{\chi},1),$ where $f$ is the weight two modular form attached to $E$ by the modularity theorem for elliptic curves, $\chi$ denotes a character of $p$-power conductor, and $f\_{\chi}$ denotes the twist of $f$ by $\chi$. (There are some extra factors to do with interpolation and so on, which I will ignore here.)

More precisely again, supposing that $E$ has good ordinary reduction at $p$ (and perhaps some other technical conditions which I'll suppress) the $p$-adic $L$-function is an element of the completed group ring ${\mathbb Z}\_p[[{\mathbb Z}\_p^{\times}]]$, whose specialization under a finite order character $\chi:{\mathbb Z}\_p^{\times} \rightarrow \overline{\mathbb Q}^{\times}$ is (essentially) the algebraic part of the classical $L$-value $L(f\_{\chi},1)$.

Now the complex $L$-function of $E$ determines (and is determined by) the isogeny class of $E$. The reason for this is that a consideration of the Euler product allows one to determine the traces of all Frobenius elements on the Tate modules of $E$, and hence by Faltings' theorem (the Tate conjecture) determine $E$ up to isogeny. On the other hand, the $p$-adic $L$-function does not have a corresponding Euler product, and so it is not clear (to me, at least) that it determines the $p$-adic Tate module of $E$ (and hence $E$, up to isogeny, or equivalently, the complex $L$-function of $E$).

One complication (at least at a psychological level) in thinking about this question is that in Iwasawa theory one often just thinks about the ideal in ${\mathbb Z}\_p[[{\mathbb Z}\_p^{\times}]]$ that is generated by the $p$-adic $L$-function, since it is this ideal which is (according to the main conjecture) supposed to relate to the Selmer group of $E$ over the $p$-adic cyclotomic tower. And this ideal surely won't determine $E$ up to isogeny; it is much too coarse a piece of information (e.g. it could just be the unit ideal in many cases, say if $E$ has no points over ${\mathbb Q}(\zeta\_p)$ and the $p$-torsion part of Sha$(E)$ over this field is trivial (and maybe some condition on Tamagawa numbers).)

The $p$-adic $L$-function carries more information than this ideal, of course; it really does know about all those special values. But it's not clear to me exactly what to do with this information. So while I'm pretty sure that the answer to your question is ``No'', I'm not sure how to actually prove it either way.

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One question. From what is written above, does it follow that the $p$-adic L-function determines the (algebraic part of) complex L-values of the twist by all Dirichlet character of conductor $p^r$ for all $r$ sufficiently large. –  Idoneal Jan 4 '10 at 6:36
    
Yes, Idoneal, by definition the $p$-adic $L$-function interpolates excatly these. –  Chris Wuthrich Apr 11 '10 at 19:26
    
Excellent! If this is the case, then I suspect analytic arguments should give what is being sought. –  Idoneal Apr 12 '10 at 5:12

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