# Self-dual automorphic forms on $GL(4)$

As is known among experts, all self-dual automorphic forms on $GL(3)$ come from symmetric square lifts from $GL(2)$. You can find this in Ramakrishnan (http://www.math.caltech.edu/~dinakar/papers/exercise-Selfdual-GL(3).pdf) or Flicker or probably Gelbart-Jacquet.

I would like to know what is known for self-dual forms on $GL(4)$. An easy construction shows that symmetric cube lifts from $GL(2)$ and Rankin-Selberg lifts from $GL(2)\times GL(2)$ will give us self-dual forms on $GL(4)$.

I raise the following question: Are there "many" automorphic forms on GL(4) which are NOT coming from the mentioned lifts (symmetric cube and Rankin-Selberg)?

• I think most Siegel modular forms should not come from the above methods. OTOH, many SMFs you see in the literature do come about from lifting (though not necessarily the lifts you list), as it is one way to exhibit examples. Writing down a generic SMF is harder (AFAIK). You can also look at the Kedlaya (et al) work on Sato-Tate groups in deg 4. Another non-SMF example that is not in your set of lifts is Hilbert modular forms (over GL2/K, but induced to be thought of as GL4/Q, similarly with Grossenchar over a quartic field). Jun 28, 2014 at 21:48
• The examples on the top half of the page at this link are not lifts: lmfdb.org/L/degree4/r0r0r0r0 Surely most self-dual degree 4 L-functions are not lifts, which probably follows from the previous comment about Sato-Tate groups. Nov 19, 2020 at 14:27

As I say in my comment, a possibly useful way to think about this is from $L$-functions, and the Sato-Tate distributions for degree 4 $L$-functions.

There is work of Fite, Kedlaya, Rotger, and Sutherland on this.

In weight 3: http://arxiv.org/abs/1212.0256

I will concentrate on the second paper (weight 3).

In brief: almost everything (from dimension counts) should have generic $USp_4$ Sato-Tate group, but neither ${\rm Sym}^3 A$ nor $A\otimes B$ has such a group. So the answer to the specific raised question is "no".

In particular, there are a few "standard" (functorial) operations to construct a degree 4 $L$-function (here symplectic, so indeed self-dual).

You listed ${\rm Sym}^3 A$ and $A\otimes B$, which are in Section 6 of the "weight 3" link above.

Another example is $A\oplus B$, which is Section 5 (the $L$-functions are imprimitive here).

They don't mention inductions from $GL_1$ (Grossencharacters) or $GL_2$ (Hilbert modular forms), but those are other "functorial" constructions.

Most of the "smaller" Sato-Tate groups they list come about from eg complex multiplication, or a tensor/sum of "correlated" $L$-functions.

But the main bulk (the "generic" case) should be Siegel modular forms.

My knowledge is not that deep, but I think there are still "lifts" in the SMF class which give "generic" $L$-functions (the full $Sp_4$ Sato-Tate group), and that these are often used in the literature to exhibit examples, but that the "generic" SMF is not a lift (correspondingly, eg some small level modular forms for $GL_2$ are eta-products, they have the "generic" Sato-Tate group, but perhaps are not truly "generic" in the sense one might want).

Examples of SMF lifts include the Saito-Kurokawa and (Miyawaki-)Ikeda lifts (I am no expert, there are likely more I think, I just know these words). EDIT: these are not really relevant, they give imprimitive deg 4 $L$-functions it seems. In fact, the Yoshida lift would be more relevant (it can give both the direct sum and tensor product from the spinor/standard $L$-functions).

From the standpoint of what is expected, the $d$-column on page 10 of the "weight 3" preprint gives the dimension of the associated space, so indeed almost everything is thought to be

• $USp_4$ Sato-Tate group (dimension 10)

• while the ${\rm Sym}^3$ construction lands you in $D$ (dimension 3),

• the tensor product is generically in the normalizer of $U(2)$ (dimension 2),

• the direct sum (dimension 6 or 4) is $G_{3,3}$ or $G_{1,3}$, depending on whether you direct sum the same weights (wt 4 modular forms), or first Tate twist (wt 2 plus wt 4)

• the induction from $GL_2$ (Hilbert modular forms) is dimension 6, giving the normalizer of $G_{3,3}$ (they do not mention this one, maybe it's the normalizer of $G_{1,3}$ in non-parallel weight $[2,4]$ and with $G_{3,3}$ in parallel weight $[4,4]$, I would have to think).

• the induction from $GL_1$ (Grossencharacters) is in the $F_\bullet$ classes (I think, again they don't do this explicitly, and get various $F$-classes as "degenerations" as mentioned next), so dimension 2 (I don't think a full dihedral example occurs over ${\bf Q}$ in the sense they demand, but $F_{ac}$, with its $C_4$ in the fourth column, should occur for cyclic quartic fields).

Everything else (the cyclic and J-cyclic data) is dimension 1 in the appropriate moduli space, as I say, coming about from having "complex multiplication" on the underlying data in one of the above constructions. For instance, tensor a CM elliptic curve with a CM modular form of weight 3, depending on how the CMs line up, you get different behavior, landing in $F$-classes, see 6.15 (the point being, you did the CM-induction before the tensor, but this can be unwound, so that's why induced $F$-classes appear).

Sorry for such a sprawling answer, and indeed it is certainly not from the automorphic perspective you phrased, but I hope it helps.

In brief: almost everything should have generic $USp_4$ Sato-Tate group, but neither ${\rm Sym}^3 A$ nor $A\otimes B$ has such a group. So the answer is no.

• It seems that Saito-Kurokawa lift $F$ of a classical modular form $f$ has $L(s,F)=\zeta(s-k+1)\zeta(s-k+2)L(s,f)$, so automorphically this is probably just taking the direct sum of two Tate twists of the trivial rep with the automorphic rep corresponding to $f$. Again I don't really know. The Ikeda lift generalizes this to higher degree, so is probably not relevant. Langlands has the S-K case as functorality, $PGL_2\times PGL_2\rightarrow PGSp_4$, coming from the $SL_2\times SL_2 \rightarrow Sp_4$ on $L$-groups, sect 3 sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/autoreps-ps.pdf Jun 28, 2014 at 23:15
• There is also the Yoshida lift on classical modular forms $f,g$, whose Andrianov/spinor $L$-function is $L(s-1,f)L(s,g)$ and whose standard $L$-function is $\zeta(s)L(s-2,f\otimes g)$, at least under certain circumstances. See the last paragraph (and indeed all of) Section 3 of dx.doi.org/10.1016/j.jnt.2006.11.005 Jun 28, 2014 at 23:38