# Weight 3 modular form associated to singular abelian surfaces?

Given an extremal K3 surface $$S$$ over $$\mathbb{Q}$$ (i.e. a K3 surface with maximal Picard rank) there is a 2-dimensional Galois representation on the transcendental lattice $$T(S)$$, and an associated cusp modular form of weight three. This is of course the modularity of extremal K3 surfaces, generalizing the famous result on elliptic curves. See, for example, Theorem 2.2 of (https://arxiv.org/pdf/1212.4308.pdf).

So are there weight 3 modular forms similarly associated to singular abelian surfaces over $$\mathbb{Q}$$? Can anyone point me to any literature where these are computed?

A singular abelian surface is an abelian surface with maximal Picard rank 4, and any such is isogenous to a product of elliptic curves with CM. The transcendental lattice (middle cohomology modulo the algebraic cycles) is then 2-dimensional and should conceivably carry a Galois representation. Which should have an associated modular form. Is this out there in the literature somewhere?

• This paper discusses modularity of products of elliptic cuvers. It refers to some examples by Howe for products of elliptic curves with CM. Maybe check there? Mar 8 '20 at 22:43
• May I ask why it is of weight 3 ? Mar 8 '20 at 23:19
• @reuns Someone can probably give a better comment than me, but if a $d$-dimensional variety is modular, I think it is expected that the weight should be $d+1$. For example, elliptic curves correspond to weight 2 forms, and rigid Calabi-Yau threefolds give weight 4 forms. So K3 and abelian surfaces should be weight 3. Mar 9 '20 at 0:21
• For the intuition: for $f\in M_k(\Gamma)$ the period polynomials make $\Bbb{C}[X]_{\deg \le k-2}$ into a $\Gamma$ module, quotienting $\Bbb{C}[X]_{\deg \le k-2}$ by the action of $\Gamma$ gives a dimension $k-1$ complex torus which in general is not algebraic. If it is then it is a dimension dimension $k-1$ abelian variety. Mar 9 '20 at 1:09

This more of comment. If $$A$$ is an abelian surface with Picard rank $$4$$, then the associated Kummer surface $$X$$ has Picard rank $$20$$. So the answer to your question is presumably yes. To be a bit more explicit, the etale cohomology $$H^2(\bar X_{et}, \mathbb{Q}_\ell)= H^2(\bar A_{et}, \mathbb{Q}_\ell)\oplus \mathbb{Q}_\ell(-1)^{16}$$ So you'll see the same Galois representation $$S^2H^1(E)$$ in both surfaces, where $$E$$ is the underlying CM elliptic curve. This presumably is what gives rise to your modular form.
• Interesting, I didn't know about this nice connection to extremal K3s. Thanks! So you're expecting that the modular form for $A$ and it's Kummer K3 should be the same? Also, do you mind if I clarify what you mean by $S^{2}$? I was guessing symmetric product, but I thought we wanted specifically 2-dimensional Galois representations. Mar 9 '20 at 0:59
• There is $\epsilon$ more to this, because if $E$ is CM, then $S^2 H^1(E)$ is reducible. $H^1(E)$ is the induction of a 1-diml representation $\psi$ of $G_K$, where $K$ is the CM field, and $S^2 H^1(E)$ splits as (cyclo char) + (induction of $\psi^2$) and the induction of $\psi^2$ corresponds to a wt 3 mod form. Mar 9 '20 at 7:08
• Yes, sorry I was a bit hasty, I meant a subquotient of $S^2H^1(E)$ Mar 9 '20 at 10:03