Split abelian surfaces over finite fields and reductions of genus-2 curves

Achter, Jeffrey D.; Howe, Everett W.

doi:10.2140/ant.2017.11.39

Cited by 9 publications

(11 citation statements)

References 36 publications

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“…Genesis of this project. We noticed this discrepancy while attempting to obtain numerical data in support of some earlier work [1]. Moreover, we found that one of us invoked an erroneous formula in a separate project [63] (see Section 5.2).…”

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confidence: 65%

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Hasse–Witt and Cartier–Manin matrices: A warning and a request

Achter¹,

Howe

2019

Contemporary Mathematics

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Let X be a curve in positive characteristic. A Hasse-Witt matrix for X is a matrix that represents the action of the Frobenius operator on the cohomology group H 1 (X, O X ) with respect to some basis. A Cartier-Manin matrix for X is a matrix that represents the action of the Cartier operator on the space of holomorphic differentials of X with respect to some basis. The operators that these matrices represent are adjoint to one another, so Hasse-Witt matrices and the Cartier-Manin matrices are related to one another, but there seems to be a fair amount of confusion in the literature about the exact nature of this relationship. This confusion arises from differences in terminology, from differing conventions about whether matrices act on the left or on the right, and from misunderstandings about the proper formulae for iterating semilinear operators. Unfortunately, this confusion has led to the publication of incorrect results. In this paper we present the issues involved as clearly as we can, and we look through the literature to see where there may be problems. We encourage future authors to clearly distinguish between Hasse-Witt and Cartier-Manin matrices, in the hope that further errors can be avoided. PrologueAn example. Consider the genus-2 hyperelliptic curve X over F 125 with affine modelwhere α ∈ F 125 satisfies α 3 +3α+3 = 0. Let us compute the 5-rank of (the Jacobian of) X.On one hand, we can follow Yui [14] and compute the effect of the Cartier operator on the space of regular one-forms. Let c m be the coefficient of x m in f (x) (5−1)/2 . Yui [14, p. 381] constructs a matrix (denoted A in her paper, but

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confidence: 65%

“…One is given a further opportunity to make a "sign error" when one chooses bases for these vector spaces and then decides whether the semilinear operator acts on the right or on the left. 1 Given these multiple opportunities for mistake, it is hardly surprising that there are occasional misstatements in the literature.…”

Section: Prologuementioning

confidence: 99%

Hasse–Witt and Cartier–Manin matrices: A warning and a request

Achter¹,

Howe

2019

Contemporary Mathematics

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“…One can give a heuristic justification for our theorem as follows. Following results of Achter and Howe in [AH17], the number of non-simple principally polarized abelian surfaces over F q n is roughly q n(5/2+o(1)) , and the number of non-simple isogeny classes is roughly q n(1+o(1)) . Similarly, the total number of isogeny classes in A 2 is roughly q n(3/2+o(1)) .…”

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confidence: 61%

Reductions of abelian surfaces over global function fields

Maulik¹,

Shankar²,

Tang³

2018

Preprint

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Let A be a non-isotrivial ordinary abelian surface over a global function field with good reduction everywhere. Suppose that A does not have real multiplication by any real quadratic field with discriminant a multiple of p. We prove that there are infinitely many places modulo which A is isogenous to the product of two elliptic curves. 3 4 I(m) by studying special endomorphisms. 1(3) We prove that the local contribution from a non-supersingular point is o(I(m)) as m → ∞. This allows us to conclude that, as m → ∞, more and more points of C contribute to the intersection C.Z(m). In order to prove Theorem 1(1), the sequence will consist only of squares, and in order to prove Theorem 2, the sequence will consist only of primes. Note that in A 2 , the Heegner divisor Z(m) for square m parametrizes abelian surfaces which are not geometrically simple, thereby allowing us to deduce Theorem 1(1). Similar arguments allow us to deduce part Theorem 1(2), and also Theorem 2.Compared to the number field situation, the main difficulty of the positive characteristic function field case is that the local contributions at supersingular points are of the same magnitude as the global contribution. More precisely, taking the Hilbert case as an example, Borcherds theory implies that the generating series of Z(m) is a non-cuspidal modular form of weight 2; on the other hand, the theta series attached to the special endomorphism lattice at a supersingular point is also a noncuspidal weight 2 modular form since the lattice is of rank 4. Therefore, even without considering 1 The ratio 3/4 is not a sharp upper bound.

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“…Proof of Proposition 6.14. From Lemma 6.13 (1), the curves in Orbit 1 for Z 1 and Z 2 are isogenous to one another. From Lemma 6.13(3), the curves in Orbits 2A and 2B for Z 1 are isogenous to the first pairs of Orbits 4A and 4B for Z 2 , respectively, and from Lemma 6.13 (5) applied to (4), the curves in Orbits 2A and 2B for Z 2 are isogenous to the second pairs of Orbits 4A and 4B for Z 1 , respectively.…”

Section: Description Of the Familiesmentioning

confidence: 94%

Doubly isogenous genus-2 curves with $D_4$-action

Arul¹,

Booher²,

Groen³

et al. 2021

Preprint

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We study the zeta functions of curves over finite fields. Suppose C and C are curves over a finite field K, with K-rational base points P and P , and let D and D be the pullbacks (via the Abel-Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C, P) and (C , P ) are doubly isogenous if Jac(C) and Jac(C ) are isogenous over K and Jac(D) and Jac(D ) are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naïve heuristics predict, and we provide an explanation for this phenomenon.

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Split abelian surfaces over finite fields and reductions of genus-2 curves

Cited by 9 publications

References 36 publications

Hasse–Witt and Cartier–Manin matrices: A warning and a request

Hasse–Witt and Cartier–Manin matrices: A warning and a request

Reductions of abelian surfaces over global function fields

Doubly isogenous genus-2 curves with $D_4$-action

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