Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCharacteristic numbers of elliptic fibrations with non-trivial Mordell-Weil groups
118
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-trivial Mordell-Weil groups. We first consider the models of type E$_{9-d}$ with $d=1,2,3,4$ whose generic fibers are normal elliptic curves of degree $d$. We then analyze the characteristic numbers of the $Q_7$-model, which provides a smooth model for elliptic fibrations of rank one and generalizes the E$_5$, E$_6$, and E$_7$-models. Finally, we examine the characteristic numbers of $G$-models with $G=text{SO}(n)$ with $n=3,4,5,6$ and $G=text{PSU}(3)$ whose Mordell-Weil groups are respectively $mathbb{Z}/2mathbb{Z}$ and $mathbb{Z}/3 mathbb{Z}$. In each case, we compute the Chern and Pontryagin numbers, the Euler characteristic, the holomorphic genera, the Todd-genus, the L-genus, the A-genus, and the eight-form curvature invariant from M-theory.
rate research
Read More
We compute characteristic numbers of crepant resolutions of Weierstrass models corresponding to elliptically fibered fourfolds $Y$ dual in F-theory to a gauge theory with gauge group $G$. In contrast to the case of fivefolds, Chern and Pontryagin numbers of fourfolds are invariant under crepant birational maps. It follows that Chern and Pontryagin numbers are independent on a choice of a crepant resolution. We present the results for the Euler characteristic, the holomorphic genera, the Todd-genus, the $L$-genus, the $hat{A}$-genus, and the curvature invariant $X_8$ that appears in M-theory. We also show that certain characteristic classes are independent on the choice of the Kodaria fiber characterizing the group $G$. That is the case of $int_Y c_1^2 c_2$, the arithmetic genus, and the $hat{A}$-genus. Thus, it is enough to know $int_Y c_2^2$ and the Euler characteristic $chi(Y)$ to determine all the Chern numbers of an elliptically fibered fourfold. We consider the cases of $G=$ SU($n$) for ($n=2,3,4,5,6,7$), USp($4$), Spin($7$), Spin($8$), Spin($10$), G$_2$, F$_4$, E$_6$, E$_7$, or E$_8$.
In this note we study the constraints on F-theory GUTs with extra $U(1)$s in the context of elliptic fibrations with rational sections. We consider the simplest case of one abelian factor (Mordell-Weil rank one) and investigate the conditions that are induced on the coefficients of its Tate form. Converting the equation representing the generic hypersurface $P_{112}$ to this Tates form we find that the presence of a U(1), already in this local description, is consistent with the exceptional ${cal E}_6$ and ${cal E}_7$ non-abelian singularities. We briefly comment on a viable ${cal E}_6times U(1)$ effective F-theory model.
We use methods for computing Picard numbers of reductions of K3 surfaces in order to study the decomposability of Jacobians over number fields and the variance of Mordell-Weil ranks of families of Jacobians over different ground fields. For example, we give examples of surfaces whose Picard numbers jump in rank at all primes of good reduction using Mordell-Weil groups of Jacobians and show that the genus of curves over number fields whose Jacobians are isomorphic to a product of elliptic curves satisfying certain reduction conditions is bounded. The isomorphism result addresses the number field analogue of some questions of Ekedahl and Serre on decomposability of Jacobians of curves into elliptic curves.
We describe two constructions of elliptic K3 surfaces starting from the Kummer surface of the Jacobian of a genus 2 curve. These parallel the base-change constructions of Kuwata for the Kummer surface of a product of two elliptic curves. One of these also involves the analogue of an Inose fibration. We use these methods to provide explicit examples of elliptic K3 surfaces over the rationals of geometric Mordell-Weil rank 15.
Let $E$ be an elliptic curve defined over a number field $F$ with good ordinary reduction at all primes above $p$, and let $F_infty$ be a finitely ramified uniform pro-$p$ extension of $F$ containing the cyclotomic $mathbb{Z}_p$-extension $F_{cyc}$. Set $F^{(n)}$ be the $n$-th layer of the tower, and $F^{(n)}_{cyc}$ the cyclotomic $mathbb{Z}_p$-extension of $F^{(n)}$. We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $lambda$-invariant of the Selmer group over $F^{(n)}_{cyc}$ as $nrightarrow infty$. This method has its origins in work of A.Cuoco, who studied $mathbb{Z}_p^2$-extensions. Refined estimates for growth are proved that are close to conjectured estimates. The results are illustrated in special cases.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
