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Hi!

This is my first question here.

In studying automorphic form, I am wondering the relation of critical L-values of some representation and its twisted representation by a character.

For example, let $F$ be a global number field and $G$ be an algebraic group over $F$. And $\pi$ is an irreducible cuspidal automorphic representation of $G(A_F)$ and $\chi$ is a unitary character of $G(A_F)$.

Then I am very curious whether there is some relationship between $L(\frac{1}{2},\pi \otimes \chi)$ and $L(\frac{1}{2},\pi)$. For example, their nonvanishing is equivalent especially when $\pi$ is character?

Since I am very beginner just started in this area and it might be a stupid question to many experts here. But, any help or remark will be greatly appreciated!

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The answer is no. Consider G=Gl(n). We have $L(s,\pi)=L(s-s',\pi\otimes|det|^{s'})$. The absolute value is the adelic norm. In general,eg in Tate's thesis, the parameter s is sometimes avoided for this reason.

So your conjecture violates GRH. For n=1 and $\pi$ trivial over the rational numbers gives you a concrete counter example, because there exist zeros on the critical line.

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    $\begingroup$ Just to give you another example so you can see that you can't hope to find a relationship -- let $G$ be $GL(2)$ and let $\pi$ correspond to a modular form, normalised so that BSD talks about what's going on at $1/2$. Then, under BSD, you're saying things like "say $y^2=x^3+T$ has infinitely many solutions in rationals. What can I say about the number of solutions to $2y^2=x^3+T$ in rationals?" and the answer is "pretty much nothing". $\endgroup$
    – user30035
    Commented Mar 29, 2013 at 17:35
  • $\begingroup$ gaargh -- for "modular form" read "elliptic curve". $\endgroup$
    – user30035
    Commented Mar 29, 2013 at 17:44
  • $\begingroup$ Palm, Thanks for kind reply. It helped me very much. I also thanks to wccanard. $\endgroup$
    – James
    Commented Mar 30, 2013 at 12:13
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I see this was posted awhile ago, but I would like to offer a counterpoint to Marc's answer: namely, yes there is a relation, at least in certain situations if you twist by quadratic characters, but it is not as clear as you were perhaps hoping. Here is a fundamental case.

Let $G$ be GL(2) and $\chi_1$ and $\chi_2$ be quadratic characters attached to quadratic extensions $E_1$ and $E_2$ of $F$. (You can view $\chi_i$ as a character of $G$ by composing with the determinant.) Then a famous result of Waldspurger (1985, "Sur les valeurs...") gives a formula for $L(1/2,\pi)L(1/2,\pi \otimes \chi_i)$ in terms of the absolute square of a period integral (over a compact torus associated to $E_i$). As a consequence (and one of Waldspurger's motivations) of this, Waldspurger shows that the ratio

$$L(1/2,\pi \otimes \chi_1) / L(1/2,\pi \otimes \chi_2)$$

is the square of an algebraic number (in a suitable field of rationality). If $\pi$ comes from a classical modular form $f$, this ratio is essentially a ratio of Fourier coefficients of the associated half-integral weight form (these ratios of Fourier coefficients were originally studied by Vigneras, and they are related to the $L$-values by a different formula of Waldspurger).

Understanding how central $L$-values vary under twists has various applications, such as the construction of $p$-adic $L$-functions. There are various conjectural generalizations of Waldspurger's period formula to other groups (e.g., Boecherer for GSp(4) and Ichino-Ikeda for SO(2n+1), and (Gan)-Gross-Prasad for less refined nonvanishing conjectures for various classical groups) but not too much is known yet. These conjectures should also give rationality results for ratios of different twists of central values.

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