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A Prym variety with everywhere good reduction over $\mathbb{Q}(\sqrt{61})$

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Pith's one-line read The paper explicitly realizes a modular abelian surface over Q(√61) with everywhere good reduction and no principal polarization as the Prym of a genus 3 curve.

arxiv 1908.00421 v2 pith:PBHMJHEV submitted 2019-08-01 math.NT

Nicolas Mascot , Jeroen Sijsling , John Voight This is my paper · ORCID
classification math.NT
keywords everywheregoodmathbbreductionsqrtabelianadmitcompute
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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The reading

The paper works with the quadratic number field K obtained by adjoining the square root of 61 to the rational numbers. The authors write down an explicit polynomial F in three variables with coefficients in K, and the zero set of F is a smooth curve X of genus 3. This curve comes with a symmetry that flips the sign of the first coordinate. Taking the quotient by this symmetry gives an elliptic curve E. The Jacobian of X breaks up, up to isogeny, into E together with a two-dimensional piece A, called the Prym variety. The main result is that A has good reduction at every prime of K, meaning that the surface does not break or develop singularities modulo any prime. It is also shown that A does not admit a principal polarization over K, which is a strong symmetry condition on its cohomology.
Extended reading notes

Core claim

Theorem 1.1 (restated as Theorem 5.7): the curve X over K defined by F(x,y,z)=0 is smooth of genus 3 and has bad reduction only at the prime (2); its Prym variety A is an abelian surface over K with everywhere good reduction, End(A) = Z[√3], A is isogenous to A61 over K, A has a polarization of type (1,2), and no abelian surface isogenous to A over K admits a principal polarization over K.

Load-bearing premise

The Faltings-Serre argument in the proof of Theorem 5.7(c) relies on the computational class field theory statement that the primes of L above 3, 5, 61, 97 generate the elementary 2-abelian extension of L. This is stated as a computed fact in Section 5 ('Verification'). If this generation statement were false, checking traces only at those primes would not suffice to prove ρA,l ≃ ρfK,l, and hence would not establish that A has everywhere good reduction.

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Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities appear. The proof rests on standard theorems (Deligne bounds, Eichler-Shimura, Faltings-Serre, polarization theory) and on the correctness of computational assertions, most notably the Faltings-Serre prime generation statement in Section 5.

assumptions (6)
  • standard math Deligne's bound |a_n(f)| ≤ σ_0(n)√n for Fourier coefficients of weight-2 newforms.
    Invoked in Section 2 to recover exact coefficients from modular computations and to justify the truncation bound for the twisted winding series.
  • standard math The Eichler-Shimura construction associates an abelian variety A_f to a Galois orbit of newforms f, with End(A_f)⊗Q = K_f and the isogeny decomposition of J_H(N).
    Used in Section 2 to identify A61 as the factor of J_H(61) attached to the newform orbit 61.2.b.a.
  • standard math The Faltings-Serre method: two absolutely irreducible Galois representations are isomorphic if their Frobenius traces agree on a set of primes generating the relevant ray class field.
    Used in Section 5 to prove that the residual representation of A matches that of the base change of the newform fK, yielding modularity.
  • domain assumption The base change of the newform f of level 61 with quadratic nebentypus to K = Q(√61) has level (1).
    Used in Section 5 to conclude that fK has level (1), so the associated abelian variety has good reduction everywhere; this follows from standard quadratic base change, though the level computation is not shown in detail.
  • domain assumption The outputs of the computer algebra computations (LLL reduction, Groebner bases, point counting, class field theory, endomorphism ring certification) are correct.
    The paper relies on computations performed in Magma and with referenced packages; a bug in any of these could affect the construction, though the final equation is explicit and independently checkable.
  • ad hoc to paper The primes of L above 3, 5, 61, 97 generate the elementary 2-abelian extension of L, as computed by computational class field theory.
    This computed fact is load-bearing for the finite set of primes used in the Faltings-Serre step in Section 5; it is asserted but not proved in the paper.

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Pith. "Pith review of A Prym variety with everywhere good reduction over $\mathbb{Q}(\sqrt{61})$." pith.science (2026). https://pith.science/paper/PBHMJHEV

@misc{pith2026190800421,
  author       = {Pith},
  title        = {Pith review of: A Prym variety with everywhere good reduction over $\mathbbQ(\sqrt61)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PBHMJHEV}},
  note         = {Machine review of arXiv:1908.00421}
}
abstract

We compute an equation for a modular abelian surface $A$ that has everywhere good reduction over the quadratic field $K = \mathbb{Q}(\sqrt{61})$ and that does not admit a principal polarization over $K$.

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Works this paper leans on

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