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Register a new userExplicit descent on elliptic curves and splitting Brauer classes
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We prove new results on splitting Brauer classes by genus 1 curves, settling in particular the case of degree 7 classes over global fields. Though our method is cohomological in nature, and proceeds by considering the more difficult problem of splitting $mu_N$-gerbes, we use crucial input from the arithmetic of modular curves and explicit $N$-descent on elliptic curves.
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Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $text{Br}, Y/ text{Br}_1, Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Neron-Severi lattices. Over a field of characteristic $0$, we prove that the existence of a strong uniform bound on the size of the odd-torsion of $text{Br}, Y / text{Br}_1, Y$ is equivalent to the existence of a strong uniform bound on integers $n$ for which there exist non-CM elliptic curves with abelian $n$-division fields. Using the same methods we show that, for a fixed prime $p$, a number field $k$ of fixed degree $r$, and a fixed discriminant of the geometric Neron-Severi lattice, $(text{Br}, Y / text{Br}_1, Y)[p^infty]$ is bounded by a constant that depends only on $p$, $r$, and the discriminant.
We discuss a non-computational elementary approach to a well-known criterion of divisibility by 2 in the group of rational points on an elliptic curve.
A cycle of elliptic curves is a list of elliptic curves over finite fields such that the number of points on one curve is equal to the size of the field of definition of the next, in a cyclic way. We study cycles of elliptic curves in which every curve is pairing-friendly. These have recently found notable applications in pairing-based cryptography, for instance in improving the scalability of distributed ledger technologies. We construct a new cycle of length 4 consisting of MNT curves, and characterize all the possibilities for cycles consisting of MNT curves. We rule out cycles of length 2 for particular choices of small embedding degrees. We show that long cycles cannot be constructed from families of curves with the same complex multiplication discriminant, and that cycles of composite order elliptic curves cannot exist. We show that there are no cycles consisting of curves from only the Freeman or Barreto--Naehrig families.
Let $k$ be a number field. We give an explicit bound, depending only on $[k:mathbf{Q}]$ and the discriminant of the N{e}ron--Severi lattice, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of isogenous CM elliptic curves. As an application, we show that the Brauer--Manin set for such a variety is effectively computable. Conditional on the Generalised Riemann Hypothesis, we also give an explicit bound, depending only on $[k:mathbf{Q}]$, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of CM elliptic curves. In addition, we show how to obtain a bound, depending only on $[k:mathbf{Q}]$, on the number of $mathbf{C}$-isomorphism classes of singular K3 surfaces defined over $k$, thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.
Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with $operatorname{char} k mid N$. For $P in C$, let $s_P$ be a rational function with divisor $N cdot P - N cdot O$. We ask whether the $N$ functions $s_P$ are linearly independent. For generic $(E,C)$, we prove that the answer is yes. We bound the number of exceptional $(E,C)$ when $N$ is a prime by using the geometry of the universal generalized elliptic curve over $X_1(N)$. The problem can be recast in terms of sections of an arbitrary degree $N$ line bundle on $E$.
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