- A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold,
with , , , with an appendix by
Research in Number Theory, to appear
(pdf) (arXiv)
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In this paper we investigate the Q-rational points of a class of simply connected Calabi-Yau threefolds, originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural 2-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer-Manin obstruction to weak approximation.
Hosono and Takagi also showed that over C each of these Calabi-Yau threefolds Y is derived equivalent to a Reye congruence Calabi-Yau threefold X. We show that these derived equivalences may also be constructed over Q and give sufficient conditions for X to not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi-Yau variety over C.
- Global Brill--Noether theory over the Hurwitz Space,
with and
(pdf) (arXiv)
(show abstracthide abstract)
Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps C to P^r of specified degree d. When C is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. Despite much study over the past three decades, a similarly complete picture has proved elusive for curves of fixed gonality. Here we complete such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting. As a corollary, we prove a conjecture of Eisenbud and Schreyer regarding versal deformation spaces of vector bundles on P^1.
- Stability of normal bundles of space curves,
with and
Algebra and Number Theory, to appear
(pdf) (arXiv)
(show abstracthide abstract)
In this paper, we prove that the normal bundle of a general Brill-Noether space curve
of degree d and genus g at least 2 is stable if and only if (d, g) is not one of (5, 2) or (6, 4). When g is at most 1 and the
characteristic of the ground field is zero, it is classical that the normal bundle is strictly semistable.
We show that this fails in characteristic 2 for all rational curves of even degree.
- An enriched count of the bitangents to a smooth plane quartic curve,
with
Research in the Mathematical Sciences, 2021
(pdf) (arXiv)
(show abstracthide abstract) (video)
Recent work of Kass--Wickelgren gives an enriched count of the 27 lines on a smooth cubic surface over arbitrary fields, generalizing Segre's signed count count of elliptic and hyperbolic lines. Their approach using A^1-enumerative geometry suggests that other classical enumerative problems should have similar enrichments, when the answer is computed as the degree of the Euler class of a relatively orientable vector bundle. Here, we consider the closely related problem of the 28 bitangents to a smooth plane quartic.
However, it turns out the relevant vector bundle is not relatively orientable and new ideas are needed to produce enriched counts.
We introduce a fixed ``line at infinity," which leads to enriched counts of bitangents that depend on their geometry relative to the quartic and this distinguished line.
- Low degree points on curves, with
International Mathematics Research Notices, 2020
(pdf)
(journal)
(arXiv)
(show abstracthide abstract) (video)
In this paper we investigate an arithmetic analogue of the gonality of a smooth projective curve C over a number field k: the minimal e such there are infinitely many points P with [k(P):k] at most e. Developing techniques that make use of an auxiliary smooth surface containing the curve, we show that this invariant can take any value subject to constraints imposed by the gonality. Building on work of Debarre--Klassen, we show that this invariant is equal to the gonality for all sufficiently ample curves on a surface S with trivial irregularity.
- A local-global principle for isogenies of composite degree,
Proceedings of the London Mathematical Society, 2020
(pdf) (arXiv)
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Let E be an elliptic curve over a number field K. If for almost all primes of K, the reduction of E modulo that prime has rational cyclic isogeny of fixed degree, we can ask if this forces E to have a cyclic isogeny of that degree over K. Building upon the work of Sutherland, Anni, and Banwait-Cremona in the case of prime degree, we consider this question for cyclic isogenies of arbitrary degree.
- Interpolation for Brill-Noether curves in P^4, with
European Journal of Mathematics, 2020
(pdf) (arXiv)
(journal)
(show abstracthide abstract)
In this paper, we compute the number of general points through which a general Brill-Noether curve in ℙ^4 passes. We also prove an analogous theorem when some points are constrained to lie in a transverse hyperplane.
- Abelian varieties isogenous to a power of an elliptic curve over a Galois extension,
Journal de Théorie des Nombres de Bordeaux, 31(1): 205-213, 2019
(pdf) (arXiv)
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Given an elliptic curve E/k and a Galois extension k′/k, we construct an exact functor from torsion-free modules over the endomorphism ring End(E/k′) with a semilinear Gal(k′/k) action to abelian varieties over k that are k′-isogenous to a power of E. As an application, we show that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.
- Constants in Titchmarsh divisor problems for elliptic curves,
with , Cliff Blakestad, , , , and
Research in Number Theory, 6(1): Art. 1, 24, 2020,
(pdf) (arXiv)
(show abstracthide abstract)
Inspired by the analogy between the group of units 𝔽_p^* of the finite field with p elements and the group of points E(𝔽_p) of an elliptic curve E/𝔽_p, E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum ∑_{p≤x} τ(p+a) ∼ Cx. In this paper, we present a comprehensive study of the constants C(E) emerging in the asymptotic study of these elliptic curve divisor sums. Specifically, by analyzing the division fields of an elliptic curve E/ℚ, we prove upper bounds for the constants C(E) and, in the generic case of a Serre curve, we prove explicit closed formulae for C(E) amenable to concrete computations. Moreover, we compute the moments of the constants C(E) over two-parameter families of elliptic curves E/ℚ. Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics.
- Elliptic fibrations on covers of the elliptic modular surface of level 5,
with , , , , and
WINE II: Contributions to Number Theory and Arithmetic Geometry, AWM Springer Series 11 (2018)
(pdf) (book) (arXiv)
(show abstracthide abstract)
We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, R_{5,5}. Such surfaces have a natural elliptic fibration induced by the fibration on R_{5,5}. Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on R_{5,5}. This has a major advantage over other methods of classification of elliptic fibrations, namely, a simple algorithm that has as input equations of linear systems of curves in the projective plane yields a Weierstrass equation for each elliptic fibration. We deal in detail with the cases for which the double cover is branched over the two reducible fibers of type I_5 and for which it is branched over two smooth fibers, giving a complete list of elliptic fibrations for these two scenarios.
- Interpolation for Brill-Noether space curves,
manuscripta mathematica 156(1), (2018) (pdf) (journal) (arXiv)
(show abstracthide abstract)
In this note we compute the number of general points through which a general Brill–Noether space curve passes.
- Powers in Lucas sequences via Galois representations, with
Proc. Amer. Math. Soc., 143 (2015) (pdf) (journal) (arXiv)
(show abstracthide abstract)
Let u_n be a nondegenerate Lucas sequence. We generalize the results of Bugeaud, Mignotte, and Siksek, 2006 to give a systematic approach towards the problem of determining all perfect powers in any particular Lucas sequence. We then prove a general bound on admissible prime powers in a Lucas sequence assuming the Frey-Mazur conjecture on isomorphic mod p Galois representations of elliptic curves.
