I've recently been learning about the special values of symmetric square $L$-functions of modular forms.

If $f$ is a cuspidal modular eigenform (of some weight $k \ge 2$) then its symmetric square $L$-function is the $L$-function given by $\prod_{\text{$\ell$ prime}} D_\ell(\ell^{-s})^{-1}$, where for almost all $\ell$ we have $D_\ell(X) = (1 - \alpha^2 X)(1 - \alpha \beta X)(1 - \beta^2 X)$, $\alpha, \beta$ being as usual the roots of the Hecke polynomial $X^2 - a_\ell(f) X + \ell^{k-1} \varepsilon_f(\ell)$.

Now, it seems to me that the critical values of $L(\operatorname{Sym}^2 f, s)$ (in the sense of Deligne) should be the odd integers in the range $1 \le s \le k-1$, and the even integers in the range $k \le s \le 2k-2$. If you twist the symmetric square by a Dirichlet character $\chi$ with $\chi(-1) = -1$, then the parity flips and you see even integers in $\{1, \dots, k-1\}$ and odds in $\{k, \dots, 2k-2\}$.

What's puzzling me is an assertion in Hida's paper "P-adic L-functions for base-change lifts from GL(2) to GL(3)" (see link, page 93 onwards). Hida works with a different $L$-function, a shifted version of the adjoint $L$-function; so his $L$-function is $L(\operatorname{Sym}^2(f) \otimes \varepsilon_f^{-1}, s)$ in my notation. Thus the critical values of Hida's $L$-function, by my computation, should be the integers in $\{1, \dots, k-1\}$ with the opposite parity to $k$, and the integers in $\{k, \dots, 2k-2\}$ with the same parity as $k$ (so the "near-central" values $k-1$ and $k$ are always critical).

However, Hida states on p96 of his paper that the critical values of his $L$-function are the odd integers in $\{1, \dots, k-1\}$ and the even ones in $\{k, \dots, 2k-2\}$, regardless of the parity of $k$. Have I got this computation wrong, or is this a mistake in Hida's paper?

  • $\begingroup$ What is the answer for $k=2$? Here $s=2$ is critical. $\endgroup$ – kantelope Sep 8 '15 at 9:33
  • $\begingroup$ For $k$ even there is no difference between Hida's statement and my computations. The issue is only for $k$ odd. $\endgroup$ – David Loeffler Sep 8 '15 at 9:34
  • $\begingroup$ Magma claims (I know not how) that for $k=5$, the critical points are 6,8, and when I twist it by some character $\epsilon_f^{-1}$ who has $\epsilon(-1)=-1$, then they become 5,7. $\endgroup$ – kantelope Sep 8 '15 at 9:47
  • $\begingroup$ Magma agrees with me and disagrees with Hida, then. (Magma only seems to give critical values in the right half of the $s$-plane, which is weird; I can't find any description of the CriticalPoints function in the Magma docs, so it's not clear if this is intended behaviour or a bug.) $\endgroup$ – David Loeffler Sep 8 '15 at 9:50
  • $\begingroup$ Reading Hida, is there any link between $\chi$ at the top of page 96 and anything previous? You seem to interpret it as the character of the form, but I do not see this explicitly. $\endgroup$ – kantelope Sep 8 '15 at 10:15

I have not yet checked Hida's normalization, but I think you're right about the critical range.

The critical range of the symmetric square L-function is studied in C.G. Schmidt's paper "p-adic measures attached to automorphic representations of GL(3)".

Using the notation of this paper (see bottom of page 603), for any Dirichlet character $\chi$, we have $L(\mathrm{Sym}^2 f \otimes \chi, s) = L(s-k+1, \Sigma \otimes (\chi \varepsilon_f)^{-1})$. (I'm simplifying notation a bit, since the $\chi$ in the RHS should be a Grössencharakter.)

Now the critical range is given in Lemma 2.1 of this paper (up to a shift). This gives the integers in $\{1,2,\ldots,k-1\}$ with opposite parity to $k$, together with the integers in $\{k,k+1,\ldots,2k-2\}$ with same parity as $k$.

This is consistent with what we assert about $s=k-1$ in the new version of my preprint with Chida (see Remark 5.4; the weight is $k+2$ in our article).


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