F-theory models with 3 to 8 U(1) factors on K3 surfaces

In this study, we construct four-dimensional F-theory models with 3 to 8 U(1) factors on products of K3 surfaces. We provide explicit Weierstrass equations of elliptic K3 surfaces with Mordell–Weil ranks of 3 to 8. We utilize the method of quadratic base change to glue pairs of rational elliptic surfaces together to yield the aforementioned types of K3 surfaces. The moduli of elliptic K3 surfaces constructed in the study include Kummer surfaces of specific complex structures. We show that the tadpole cancels in F-theory compactifications with flux when these Kummer surfaces are paired with appropriately selected attractive K3 surfaces. We determine the matter spectra on F-theory on the pairs.

Isogenies on twisted Hessian curves

Elliptic curves are typically defined by Weierstrass equations. Given a kernel, the well-known Vélu's formula shows how to explicitly write down an isogeny between Weierstrass curves. However, it is not clear how to do the same on other forms of elliptic curves without isomorphisms mapping to and from the Weierstrass form. Previous papers have shown some isogeny formulas for (twisted) Edwards, Huff, and Montgomery forms of elliptic curves. Continuing this line of work, this paper derives explicit formulas for isogenies between elliptic curves in (twisted) Hessian form. In addition, we examine the numbers of operations in the base field to compute the formulas. In comparison with other isogeny formulas, we note that our formulas for twisted Hessian curves have the lowest costs for processing the kernel and our X-affine formula has the lowest cost for processing an input point in affine coordinates.

FORMAL GROUPS AND INVARIANT DIFFERENTIALS OF ELLIPTIC CURVES

In this paper, we find a power series expansion of the invariant differential ${\it\omega}_{E}$ of an elliptic curve $E$ defined over $\mathbb{Q}$, where $E$ is described by certain families of Weierstrass equations. In addition, we derive several congruence relations satisfied by the trace of the Frobenius endomorphism of $E$.