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I was reading this paper by de Weger and in conjecture 7 he mentions "the Riemann hypothesis for the Rankin-Selberg zeta function associated to the weight 3/2 modular form associated to E (an elliptic curve over $\mathbb{Q}$) by the Shintani-Shimura lift is true."

I didn't really understand this statement so I was hoping if someone could explain it to me. So far I've found this paper on the Rankin-Selberg zeta function. Is this just a statement about the zeros of the Rankin-Selberg zeta function? Also how is this zeta function associated to the elliptic curve? An expository reference would be appreciated.

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Since you asked for reference, have you looked at the Wikipedia article on Rankin-Selberg method? –  Sniper Clown May 22 '12 at 2:55

3 Answers 3

up vote 5 down vote accepted

There are various constructions known as Rankin--Selberg $\zeta$-functions (or Rankin--Selberg $L$-functions), and you should be able to find much more literature on this than the one paper you linked to (which doesn't address the particular $L$-function you are asking about, as far as I can tell).

You might want to begin by looking at de Weger's reference [10], since this is what he cites when he discusses Conjecture 7. I did look there, and there was no direct definition of the particular Rankin--Selberg $\zeta$-function in question, but following the references there ([5] and [9]; see p. 76), you might want to look at the Inventiones paper of Kohnen and Zagier, where various Rankin--Selberg-type computations take place. (This is reference [9] of Goldfeld--Szpiro.)

In any case, the Shintani--Shimura lift asociates a weight $k+{1/2}$ form to an eigenform $f$ of weight $2k$ (so a weight $3/2$-form gets associated to the weight two eigenform attached to a given elliptic curve $E$). The lifted form has the same system of Hecke eigenvalues as the original form, but because of the way Hecke operators work in the half-integral weight case, it has not just one leading coefficient (unlike in the integral weight case, where there is a single leading coefficient a_1 which, together with all the Hecke eigenvalues, determines the $q$-expansion) but a collection of "leading coefficients", indexed by fundamental discriminants $D$. The theorem of Waldspurger is that, up to an overall proportionality factor, the $D$th leading coefficient is equal to the $L$-value $L(f,\chi_D,1)$, where $\chi_D$ is the quadratic character of disc. $D$. It is reproved (under restricted hypotheses, I think) in the Kohnen--Zagier paper.

As BR indicates, the Rankin--Selberg $L$-function is determined from this half-integral weight modular form by multiplying by a certain Eisenstein series and integrating over a fundamental domain. It is (I'm pretty sure) one of the Rankin-type constructions given in the Kohnen--Zagier paper. The growth rates of its coefficients are related to growth rates of the values of $L(E,\chi_D,1)$, which, assuming BSD, are related to growth rates of Sha of quadratic twists of the elliptic curve $E$.

This is why it comes up in the kind of problems you are reading about.

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You can read the Szpiro-Goldfeld paper de Weger refers to here. See Theorem 2 and its proof.

I think the procedure is: start with an elliptic curve $E$. By modularity, we get a modular form of weight $2$. Use the Shintani-Shimura correspondence to get a modular form of weight $3/2$. To this, we have Shimura's Rankin-Selberg Zeta function for half-integral weight modular forms. This is not the same as the R-S zeta function in Ivic's paper (which is what I think of as the R-S zeta function), though it is not wildly different (unfortunately, I'm having trouble finding online descriptions of it, and even worse, Shimura has two Rankin-Selberg zeta functions attached to his name, one for even-integral weight and one for half-integral weight). It is supposed to be in Shimura's "On modular forms of half-integral weight", but the construction is not given a prominent place (at least, I'm having trouble finding it). Basically, you integrate (over a fundamental domain) a modular form of half-integral weight against a theta series and an Eisenstein series.

Finally, the R-S zeta function is an automorphic $L$-function (with a functional equation), so we have a Riemann Hypothesis for it (that it only vanishes on the center line of its functional equation).

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For more information on Shimura's R-S construction, there is a brief treatment in Gelbart-Shahidi "Analytic Properties of Automorphic $L$-functions", p. 59. They refer to three other papers by Gelbart/Piatetski-Shapiro: "On Shimura's Correspondence for modular forms of half-integral weight", "Distinguished representations and modular forms of half-integral weight", and "Some remarks on metaplectic cusp forms and the correspondence of Shimura and Waldspurger". –  B R May 22 '12 at 3:02

Apparently the Rankin-Selberg zeta function is explicitly defined in this paper by Roland Matthes.

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