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I remember to have read that the L-function of an elliptic curve, which a priori only converges for $\Re s > \frac{3}{2}$ also converges at $s=1$ provided that the $L$-function satisfies the functional equation.

I always thought that this is due to the fact that in this case the L-function is also the L-function of a modular form and in this case we have better convergence. However, the modular forms which correspond to these curves are cusp forms of weight 2 and so have a priori even worse convergence properties, namely convergence for $\Re s > 2$.

So I now wonder, whether I remember correctly and the the claim above is indeed correct? If so, I would like to see a reference for the proof. What is the reason for this fact. Does some non-trivial fact about elliptic curves play a role. It will (almost surely) not hold for arbitrary modular forms or cusp forms, since they always satisfy the functional equation. What do we need? An Euler product?

EDIT: The $L$-function extends to an entire function. But what I am interested in is the original series representation. Is it true that the series representation for the $L$-function of an elliptic curve which converges a priori for $\Re s > \frac{3}{2}$, is valid also for a bigger strip $\Re s > c$ with $c<\frac{3}{2}$.

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up vote 16 down vote accepted

Wood, see the article by Kumar Murty in Seminar on Fermat's Last Theorem. He shows how the L-series converges (conditionally!) on Re(s) > 5/6, thus in particular at s = 1. You can find the book on Google books and do a search on "5/6" to find the page. OK, I just did that and will tell you: it's on page 15. He proves the theorem for any Dirichlet series converging abs. for Re(s) > 3/2 and having an analytic continuation and suitable functional equation relating values at s and 2-s. (Thus in practice it is a theorem about L-functions of suitable weight 2 modular forms, certainly nothing directly about elliptic curves!) He also says that what he describes is a special case of a more general result, with citation.

Where the Dirichlet series converges, it still represents the L-function that may have been analytically continued to the wider region by some other method, since Dirichlet series are analytic on the half-plane to the right of any point where they converge and moreover at any point where they converge the value is the limit of the function taken along the line to the right of the point (Abel's theorem for power series on discs works for Dirichlet series on right half-planes). So we are assured that if you happen to know the series itself converges somewhere new it still equals the orthodox analytic continuation (whatever that means).

On the other hand, the Euler product has surprises. Goldfeld discovered that if the Euler product for an ell. curve over Q converges at s = 1, in the natural sense of partial Euler products over primes up to x as x goes to $\infty$, and the value of the Euler product is nonzero, then this value is not L(1) but rather is off from this by a factor of $\sqrt{2}$. Of course there was no real input about elliptic curves directly: Goldfeld was assuming the ell. curve was modular (he was writing in the 1980s) and used that right away. It turns out exactly the same thing happens for quadratic Dirichlet L-functions at s = 1/2: if the partial Euler product at s = 1/2 converges to a nonzero value then again you're off by a factor of $\sqrt{2}$ from L(1/2), but in one setting the factor is $\sqrt{2}$ and in the other it's $1/\sqrt{2}$. For non-quadratic Dirichlet L-functions there's no funny business: if the Euler product converges at s = 1/2 to a nonzero value (which, by the way, it always should since nobody expects Dirichlet L-functions vanish at 1/2) then the value will be L(1/2). I first heard about Goldfeld's result at a talk by Karl Rubin (he gave the usual heuristic for BSD conjecture by looking at the Euler product at s = 1 and some wiseguy in the audience asked if it really did converge at s = 1 and Rubin mentioned there was a paper of Goldfeld on that), but when I read Goldfeld's paper I was confused by part of it, so in trying to work it out in the simpler example of Dirichlet L-functions I wound up seeing I could prove the same kind of theorem for any Euler product over a global field having the properties everyone expects it should have. This turns out to be related to properties of symmetric and exterior square Euler products and is morally the same quadratic bias (called Chebyshev's bias) that Sarnak and Rubinstein found when they worked out comparative statistics on the number of primes up to x in different congruence class mod $m$: classes of squares or non-squares exhibit different fine growth rates compared to one another. For more details on the partial Euler products, including some numerical examples, see my paper

By the way, one can definitely observe this $\sqrt{2}$ business happening numerically but we'll never expect to prove it happens since it actually implies the Riemann hypothesis for the relevant L-function. If the product converges to a nonzero value at a point on the critical line it is no real surprise that you could prove the Dirichlet series for the log of the Euler product converges everywhere to the right of the critical line, which implies the Euler product itself converges to a nonzero value everywhere to the right of the critical line, hence Riemann hypothesis. In fact, as I show in that paper, this (suitable) Euler product convergence at a point on the critical line is actually equivalent to something which at present appears to lies deeper than the Riemann hypothesis but is still plausible. There's no lack of results which imply RH but are themselves false, you see. This probably isn't one of them.

In summary, the Dirichlet series for ell. curve L-functions (over $\mathbf Q$) can provably be shown to converge on a wider region than they are usually said to converge and still equal the L-function there, including s = 1, while the Euler product probably does converge at s = 1 too but if the value is not 0 then it's not going to converge to what you expect... unless you think the Riemann hypothesis is false.

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Thanks. But still surprising. – wood May 22 '10 at 11:45

The $L$-function attached to any cuspform is obtained by a Mellin transform, and converges on the whole complex plane. (See Ogg's book on $L$-functions and modular forms, or probably any other text.) If you just write down the Mellin transform integral, this is is not obvious: you have to use the transformation $\tau \mapsto -1/N\tau$ (where $N$ is the level) to get rid of the end-point $0$ in the Mellin transform integral (more precisely break the interval $(0,i\infty)$ into the interval $(0,i/{\sqrt{N}})$ and $(i/\sqrt{N}, i\infty)$, and then apply $\tau \mapsto -1/N\tau$ to move the first interval to $(i/\sqrt{N},i\infty)$, so that now there are no convergence issues, for any value of $s$), and use the fact that this transformation preserves the space of cuspforms.

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Maybe I did not explain good enough what I meant, or I am now really surprised. I made an edit to clarify this. My concern is with the series representation for the $L$-function. Of course we take the series representation and then make the Mellin transform and we see that we can extend everything to the whole plane and get the functional equation. But this does not imply that the series represenatation is valid for all $s$, does it. – wood May 20 '10 at 13:45
There are some surprises which happen when using the actual Dirichlet series at $s=1$, as KConrad will explain in a later posting, I am sure. – BCnrd May 20 '10 at 15:29
Dear Wood, Sorry, I misunderstood the question (obviously!). Let's wait for Keith's answer. – Emerton May 20 '10 at 15:44

The Dirichlet series representation of a GL2/$\mathbf{Q}$ L-function converges slightly to the left of the critical point. The key thing is the estimate $\sum_{n \leq X} a(n) \ll X^{1/3+\epsilon}$, from which the result follows by partial summation. See for example this paper.

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