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Let $\pi$ be a cuspidal automorphic representation of GL(2) over a number field (if you want, assume it's $\mathbb Q$ and $\pi$ comes from a holomorphic modular form).

In the case $\pi$ has trivial central character, the epsilon factor determines the parity of order of vanishing of $L(1/2,\pi)$. If $\chi$ is a quadratic character, and $\pi$ in fact comes from an elliptic curve, then one expects $L(s,\pi \otimes \chi)$ to have rank 0 half the time and rank 1 half the time (Goldfeld's conjecture).

(1) Is there a precise generalization of Goldfeld's conjecture to more general $\pi$ (assume what you need)?

I know there are several nonvanishing results and bounds on proportions for rank 0 and rank 1 if $\pi$ has trivial central character. However if $\pi$ does not have trivial central character (and is not self-dual), then I know little more than that Friedberg-Hoffstein says $L(s,\pi \otimes \chi)$ has rank 0 infinitely often.

(2) Is anything else known/expected when $\pi$ is not self-dual?

I know nothing about Katz-Sarnak philosophies and Random Matrix Models, but do they apply for non-self-dual representations?

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I am no expert, but let me have a punt. If $\pi$ is self-dual and $\chi$ is quadratic, then $\pi\otimes\chi$ is also self-dual. So the functional equation relates $L(s)$ to $L(1-s)$ and if the sign is $-1$ this forces $L(1/2)$ to be zero. However, the moment you leave this cosy situation things are very different. For example isn't it a result of Rohrlich that if you twist an elliptic curve by $\chi$ as $\chi$ runs through all Dirichlet characters of $p$-power conductor, then $L(E,\chi,1/2)$ is zero only finitely often? This makes sense because the $p$-adic $L$-functionwillbe0onlyfinitelyoften –  Kevin Buzzard Feb 24 '11 at 21:59
    
Kevin, I don't know about p-adic L-functions, nor did I know about the result of Rohrlich. Thanks for pointing this out. I'll take a look at it. Do you know a good reference for a summary of what's known about non/vanishing for p-adic L-functions? –  Kimball Feb 25 '11 at 0:38
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@Kevin: I believe that in fact results of the same sort as Rohrlich's that you point out are essentially the only way one knows how to show the p-adic L-functions constructed by interpolation of twisted L-values are not identically zero. Beyond that, maybe one can also deduce such nontriviality as a consequence of the appropiate divisibility, i.e. the one provided by means of congruences, in an Iwasawa Main Conjecture. ("Beyond", provided one can establish it, and the cotorsioness of the relevant Iwasawa module, independently of those non-vanishing results.) –  monodromy Feb 25 '11 at 2:04
    
After rereading Kevin's comment, I realized I may have not make it clear the epsilon factor condition is what gives us vanishing (to odd order) roughly half the time. So semi-naively I would expect for $\pi$ not self-dual to get vanishing only for a small proportion of quadratic twists. But how small? Density 0? Finitely many? –  Kimball Feb 25 '11 at 3:49
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@Kimball: It may suggest a little bit, I suppose. If $\pi=E\otimes\chi$ for $\chi$ of prime order $l>2$, then it is not self-dual, and its varying quadratic twists can be viewed are a subset of order $2l$ twists of $E$. And David-Fearnley-Kisilevsky's approach may suggest that even among this larger set there are only finitely many vanishing twists. –  Tim Dokchitser Feb 26 '11 at 11:32
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3 Answers 3

Though it is perhaps not an "answer" as such, let me try to explain some intuition. In certain settings, it is possible to formulate subtle analogues of Mazur's conjecture for nonvanishing of central values (think Cornut-Vatsal) from the refined conjecture of Birch and Swinnerton-Dyer via Iwasawa theory. A general method is explained in section 4 of Coates-Fukaya-Kato-Sujatha, " Root numbers, Selmer groups, and non-commutative Iwasawa theory" (available at http://www.math.tifr.res.in/~sujatha/root.pdf). CFKS consider the setting of the so-called False-Tate curve extension, but a similar (and simpler) set of arguments can be used to deduce an analogous conjecture for the setting of the ${\bf{Z}}_p^2$ of an imaginary quadratic field. To be slightly more precise, fix a rational prime $p$. Fix an eigenform $f \in S_2(\Gamma_0(N))$ with $(N,p)=1$. Fix an imaginary quadratic field $k$ with discriminant prime to $pN$. Let $k_{\infty}$ denote the ${\bf{Z}}_p^2$-extension of $k$, which is the compositum of the cyclotomic ${\bf{Z}}_p$-extension $k^c$ with the anticyclotomic ${\bf{Z}}_p$-extension $k^a$. Write $\lambda_f(k)$ to denote the cyclotomic $\lambda$-invariant associated to $f$, with $\mu_f(k)$ the cyclotomic $\mu$-invariant. Let $\mathcal{W}$ be any finite order character of $\operatorname{Gal}(k_{\infty}/k)$. Such a character can always be written as a product of characters $\rho \cdot \chi$, where $\rho$ is a finite order character of $\operatorname{Gal}(k^a/k)$, and $\psi$ is a finite order character of $\operatorname{Gal}(k^c/k)$. What the CFKS conjecture predicts, very roughly, is the following assertion. Assume that $\mu_f(k)=0$, and fix a finite order character $\rho$ of $\operatorname{Gal}(k^a/k)$. Let $\Psi$ denote the set of finite order character of $\operatorname{Gal}(k^c/k)$. Then, assuming the refined Birch and Swinnerton-Dyer conjecture, \begin{align*} \sum_{\psi \in \Psi} \operatorname{ord}_{s =1}L(f \times \rho \cdot \psi, s) &\leq \lambda_f(k). \end{align*} So, what does this tell us? Well for one, it tells us that Rohrlich nonvanishing (at least in this setting) should be a general phenomenon. One can make this intuition slightly more precise via the following heuristic argument. View any finite extension of $k^c$ over $k$ as a totally imaginary quadratic extension of its maximal totally real subfield. Suppose that the nonvanishing theorem of Cornut-Vatsal were uniformly effective (in the sense that their $n$ sufficiently large could be replaced by some absolute $n_k$ that does not grow as we ascend the cyclotomic tower). Then, invoking their result systematically and decomposing via Artin formalism, we should (I think) expect the following behaviour. Let $\epsilon(f/k, s)$ denote the root number of the Rankin-Selberg $L$-function $L(f/k, s)$. Given $\rho$ a finite order character of $\operatorname{Gal}(k^a/k)$ of conductor greater that $p^{n_k}$, we should have:

\begin{align*} \sum_{\psi \in \Psi} \operatorname{ord}_{s=1} L(f \times \rho \cdot \psi, s) &= 0 \text{ if $\epsilon(f/k, 1) = 1;$} \end{align*}

\begin{align*} \sum_{\psi \in \Psi} \operatorname{ord}_{s=1} L(f \times \rho \cdot \psi, s) &= 1 \text{ if $\epsilon(f/k, 1) = -1;$} \end{align*}

Sorry to have skipped steps, or if this is perhaps somewhat unclear in places. The main idea is that there are subtle generalizations of Mazur's conjecture to the types of settings that you will likely want to consider. These generalizations suggest that one should expect generic nonvanishing à la Rohrlich, even in the case where the root number at the bottom is $-1$. And though it is not a priori clear, it might be possible to use these sorts of ideas to obtain the general formulation of Goldfeld's conjecture that you ask for.

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Thanks, though I must confess I don't understand much of what you say. So the bottom two sums mean $L(f \times \rho \cdot \psi, 1)$ is nonzero for all $\psi$ in the first case, and all but 1 $\psi$ in the second? Am I misreading the epsilon factor condition? –  Kimball Feb 26 '11 at 1:25
    
Yes, that is exactly what the notation means, and there is no mistake with the root number. Sorry to have compressed so much information into such a short paragraph. The deduction in fact requires far more justification than I have given. I wrote a paper about this while I was a grad student, but never submitted it. (The result seemed to be well known to experts, and has also been subsumed by subsequent results/projects). Perhaps I'll write a note about this now, as the deduction is perhaps not as well understood as I had assumed ... in which case, I will post a link here :) –  jvo Feb 26 '11 at 9:48
    
I at least would appreciate it, but don't feel the need to write it just on my account. It may be well-understood by experts. I just never learned p-adic/Iwasawa theories (though I'd like to better understand it). –  Kimball Feb 26 '11 at 16:17
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(1) In a "true" Katz-Sarnak context, i.e., over finite fields, non self-dual situations definitely make sense, exist, and are among those studied in their book (chapters 9 and following). In fact the basic question of proportion of vanishing v.s. non-vanishing is really an application of Deligne's Equidistribution Theorem. Basically, when the size of the field goes to infinity, if you have a known (compactifo-complexified) monodromy group, the proportion of vanishing at the central point will converge to the Haar probability -- in the monodromy group -- of matrices with $1$ as an eigenvalue. In most cases this is computable once the monodromy is known. However, this story has the usual limitation (field goes to infinity can not be replaced, for the moment, with size of matrices goes to infinity).

(2) For L-functions over $\mathbf{Q}$, the analytic methods which give a "decent" proportion of non-vanishing for quadratic twists (not positive proportion, but about $1/(\log D)^A$ for twists by quadratic characters of size up to $D$, for some fixed $A$) work whether the form is self-dual or not. But one definitely expects that twists of a fixed form which is not self-dual should have "density" zero of order of vanishing at least $1$ (this is as precise as Goldfeld's Conjecture). In many cases, it might well be that no twist vanishes at $1/2$, or at most finitely many.

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Thanks. This is very helpful. Can you suggest any references for your comments in (2)? –  Kimball Feb 27 '11 at 15:39
    
Actually, I now realize that $1/(\log D)^A$ was too optimistic for analytic methods... Sorry for the misleading statement: the point is that one needs an upper bound for the second moment of the special values which is sharp (up to power of log), and this is not known. (See Soundararajan's paper arxiv.org/pdf/0907.4747 for what is probably the state of the art). But one can get $1/D^{\epsilon}$ along these lines for any $\epsilon>0$; a good reference for this is the paper of Perelli and Pomykala in Acta Arithmetica 1997. –  Denis Chaperon de Lauzières Feb 27 '11 at 16:17
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I suggest you try asking Dipendra Prasad. I have some vague memory that he once told me that (conjecturally) if pi is not self dual then the central value will never vanish.

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Hmmm.... maybe Dipendra has some ideas, but the work Dokchitser refers to in his comment says Random Matrix Theory predicts there are non self-dual examples vanishing at the center. –  Kimball Feb 26 '11 at 1:02
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He probably told you the conjecture that if $\pi$ on $GL(n)$ isn't twist-equivalent to its dual, then $L(1/2,\pi)\neq 0$. (See e.g. his paper with Gan and Gross.) This is vacuous for $GL(2)$ since everything is a twist of its dual. –  David Hansen Feb 26 '11 at 17:15
    
@David: This sounds like a very interesting conjecture, actually. Do you have a precise reference? Does this really mean that the only L-functions that may vanish at the central point come from representations that are twist-equivalent to their duals? –  Tim Dokchitser Feb 27 '11 at 16:45
    
math.ucsd.edu/~wgan/work8-3.pdf Do you mean page 96, conjecture 24.1 and following? –  Junkie Feb 28 '11 at 12:59
    
Tim, Junkie: I think it's been deleted from this more recent draft of GGP. I remember explicitly seeing the never-vanishing conjecture for non-self-dual things in this paper, but now for the life of me I cannot find it! –  David Hansen Mar 1 '11 at 19:11
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