Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTranscendental Brauer elements via descent on elliptic surfaces
128
0
0.0
(
0
)
Authors
Bianca Viray
Ask ChatGPT about the research
No Arabic abstract
Transcendental Brauer elements are notoriously difficult to compute. Work of Wittenberg, and later, Ieronymou, gives a method for computing 2-torsion transcendental classes on surfaces that have a genus 1 fibration with rational 2-torsion in the Jacobian fibration. We use ideas from a descent paper of Poonen and Schaefer to remove this assumption on the rational 2-torsion.
rate research
Read More
We recast elliptic surfaces over the projective line in terms of the non-commutative tori with real multiplication. The correspondence is used to study the Picard numbers, the ranks and the minimal models of such surfaces. As an example, we calculate the Picard numbers of elliptic surfaces with complex multiplication.
Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{e}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $Bto C$ arises from an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic. We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes showing that the Picard-Lefschetz transformations corresponding to an irreducible component of a singular fibre, can be extended as universal isometries. In the process, we get a family of homomorphisms of the affine Weyl group associated to $tilde{A}_{n-1}$ to that of $tilde{A}_{dn-1}$, indexed by natural numbers $d$, which are closed under composition.
Let $k$ be a field finitely generated over the finite field $mathbb F_p$ of odd characteristic $p$. For any K3 surface $X$ over $k$ we prove that the prime to $p$ component of the cokernel of the natural map $Br(k)to Br(X)$ is finite.
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.
Let ${P_i}_{1 leq i leq r}$ and ${Q_i}_{1 leq i leq r}$ be two collections of Brauer Severi surfaces (resp. conics) over a field $k$. We show that the subgroup generated by the $P_is$ in $Br(k)$ is the same as the subgroup generated by the $Q_is$ iff $Pi P_i $ is birational to $Pi Q_i$. Moreover in this case $Pi P_i$ and $Pi Q_i$ represent the same class in $M(k)$, the Grothendieck ring of $k$-varieties. The converse holds if $char(k)=0$. Some of the above implications also hold over a general noetherian base scheme.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
