(This is part of a series of occasional posts discussing results and observations in my joint paper with Boxer, Gee, and Pilloni mentioned here.)

Brumer and Kramer made a conjecture positing a bijection between isogeny classes of abelian surfaces over the rationals of conductor N with and paramodular Siegel newforms of level N with rational eigenvalues (up to scalar) that are not Gritsenko lifts (Gritsenko lifts are those of Saito-Kurokawa type). This conjecture is closely related to more general conjectures of Langlands, Clozel, etc., but its formulation was made more specifically with a view towards computability and falsifiability (particularly in relation to the striking computations of Poor and Yuen).

The recognition that the “optimal level” of the corresponding automorphic forms is paramodular is one that has proved very useful both computationally and theoretically. Moreover, it is almost certain that something very close to this conjecture is true. However, as literally stated, it turns out that the conjecture is false (though easily modifiable). There are a few possible ways in which things could go wrong. The first is that there are a zoo of cuspidal Siegel forms for GSp(4); it so happens that the forms of Yoshida, Soudry, and Howe–Piatetski-Shapiro type never have paramodular eigenforms (as follows from a result of Schmidt), although this depends on the accident that the field has odd degree and no unramified quadratic extensions (and so the conjecture would need to be modified for general totally real fields). Instead, something else goes wrong. The point is to understand the relationship between motives with -coefficients and motives with -coefficients which are invariant under the Galois group (i.e. Brauer obstructions and the motivic Galois group.)

It might be worth recalling the (proven) Shimura-Taniyama conjecture which says there is a bijection between cuspidal eigenforms of weight two with rational eigenvalues and elliptic curves over the rationals. Why might one expect this to be true from general principles? Let us imagine we are in a world in which the Fontaine-Mazur conjecture, the Hodge conjecture, and the standard conjectures are all true. Now start with a modular eigenform with rational coefficients and level Certainly, one can attach to this a compatible family of Galois representations:

with the property that the characteristic polynomials of Frobenius at any prime have *integer* coefficients, and the representations are all de Rham with Hodge-Tate weights [0,1]. But what next? Using the available conjectures, one can show that there must exist a corresponding simple abelian variety which gives rise to The key to pinning down this abelian variety is to consider its endomorphism algebra over the rationals. Because it is simple, it follows that the endomorphism algebra is a central simple algebra for some number field F. From the fact that the coefficients of the characteristic polynomial are rational, one can then show that the number field F must be the rationals. But the Albert classification puts some strong restrictions on endomorphism rings of abelian varieties, and the conclusion is the following:

Either:

- is an elliptic curve.
- is a fake elliptic curve; that is, an abelian surface with endomorphisms over by a quaternion algebra

The point is now that the second case can never arise; the usual argument is to note that there will be an induced action of the quaternion algebra on the homology of the real points of A, which is impossible since the latter space has dimension two. (This is related to the non-existence of a general cohomology theory with rational coefficients.) In particular, we do expect that such modular forms will give elliptic curves, and the converse is also true by standard modularity conjectures (theorems in this case!). A similar argument also works for all totally real fields. On the other hand, this argument does *not* work over an imaginary quadratic field (more on this later). In the same way, starting with a Siegel modular form with rational eigenvalues whose transfer to GL(4) is cuspidal, one should obtain a compatible family of irreducible 4-dimensional symplectic representations with cyclotomic similitude character. And now one deduces (modulo the standard conjectures and Fontaine-Mazur conjecture and the Hodge conjecture) the existence of an abelian variety A such that:

Either:

- is an abelian surface.
- is a fake abelian surface; that is, an abelian fourfold with endomorphisms over by a quaternion algebra

There is now no reason to suspect that fake abelian surfaces cannot exist. Taking D to be indefinite, the corresponding Shimura varieties have dimension three, and they have an abundance of points — at least over totally real fields. But it turns out there is a very easy construction: take a fake elliptic curve over an imaginary quadratic field, and then take the restriction of scalars!

You have to be slightly careful here: one natural source of fake elliptic curves comes from the restriction of certain abelian surfaces of GL(2)-type over and one wants to end up with fourfolds which are simple over Hence one can do the following:

**Example:** Let be an abelian surface of GL(2)-type which acquires quaternion multiplication over an imaginary quadratic field K, but is not potentially CM. For example, the quotient of with coefficient field with Take the restriction to K, twist by a sufficiently generic quadratic character and then induce back to Then the result will be a (provably) modular fake abelian surface whose corresponding Siegel modular form has rational eigenvalues. Hence the paramodular conjecture is false.

Cremona (in his papers) has discussed a related conjectural correspondence between Bianchi modular forms with rational eigenvalues and elliptic curves over K. His original formulation of the conjecture predicted the existence of a corresponding elliptic curve over K, but one also has to allow for fake elliptic curves as well (as I think was pointed out in this context by Gross). The original modification of Cremona’s conjecture was to only include (twists of) base changes of abelian surfaces of GL(2)-type from Q which became fake elliptic curves over K, but there is no reason to suppose that there do not exist fake elliptic curves which are autochthonous to K, that is, do not arise after twist by base change. Indeed, autochthonous fake elliptic curves do exist! We wrote down a family of such surfaces over for example. (We hear through Cremona that Ciaran Schembri, a student of Haluk Sengun, has also found such curves.) On the other hand, the examples coming from base change forms from Q have been known in relation to this circle of problems for 30+ years, and already give (by twisting and base change) immediate counter-examples to the paramodular conjecture, thus the title.

It would still be nice to find fake abelian surfaces over (rather than totally real fields) which are geometrically simple. I’m guessing that (for D/Q ramified only at 2 and 3 and a nice choice of auxiliary structure) the corresponding 3-fold may be rational (one could plausibly prove this via an automorphic form computation), although that still leaves issues of fields of rationality versus fields of definition. But let me leave this problem as a challenge for computational number theorists! (The first place to look would be Jacobians of genus four curves [one might be lucky] even though the Torelli map is far from surjective in this case.)

Let me finish with one fake counter example. Take any elliptic curve (say of conductor 11). Let be any Galois extension with Galois group the quaternion group of order 8. The group has an irreducible representation of dimension 4 over the rationals, which preserves a lattice If you take

then is a simple abelian fourfold with an action of an order in (now the definite Hamilton quaternions) and so gives rise to compatible families of 4-dimensional representations which are self-dual up to twisting by the cyclotomic character. However, the four dimensional representations are only symplectic with respect to a similitude character which is the product of the cyclotomic character and a non-trivial quadratic character of and instead they are orthogonal with cyclotomic similitude character. So these do not give rise to counterexamples to the paramodular conjecture. A cursory analysis suggests that the quaternion algebra associated to a fake abelian surface which gives rise to a symplectic with cyclotomic similitude character should be indefinite.

In case you don’t get the reference in the title:

https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/HodgeConj.pdf

“a bijection between cuspidal eigenforms of weight two with rational eigenvalues and elliptic curves over the rationals” only if you restrict to strong Weil curves, otherwise it is elliptic curves up to isogeny.

Many things I say are also false for trivial reasons too!

I ran into the level 243 example a decade ago when I was trying to understand related stuff. Half-baked notes of mine at http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/inner_twists.pdf .

Another way to see why you always get case 1 (an elliptic curve) associated to an eigenform of weight 2 over Q with integral Fourier coefficients is that the associated \ell-adic representations are odd, so complex conjugation gives an involution which is not central in GL_2. But the only involutions in D* are +1 and -1. This argument also works over totally real fields, but not over an imaginary quadratic field.

Thank you Frank for this important correction! JV