Related papers: Explicit sections on Kuwata's elliptic surfaces

We enlarge a Coxeter group into a category, with one object for each finite parabolic subgroup, encoding the combinatorics of double cosets. This category, the singular Coxeter monoid, is connected to the geometry of partial flag varieties.…

In this paper, we use the conjugate surface construction to prove the existence of certain non-periodic symmetric immersed minimal surfaces. These surfaces have finite total curvature and embedded catenoid ends, and they have positive genus…

We compute the genus one family Gromov-Witten invariants of K3 surfaces for non-primitive classes. These calculations verify Gottsche-Yau-Zaslow formula for non-primitive classes with index two. Our approach is to use the genus two…

We study monodromy groups of elliptic fibrations over the projective line.

Explicit embeddings of the group $\mathbb{Q}$ into a finitely presented group $\mathcal{Q}$ and into a $2$-generator finitely presented group $T_{\mathcal{Q}}$ are suggested. The constructed embeddings reflect questions mentioned by…

Extending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato-Tate Law. We present two methods of…

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice…

We present an elementary introduction to the construction of integrable models on hyper-elliptic surfaces for non specialists; also, we present some of the details of the paper `solv-int/9504002' for the more interested readers. (Based on a…

In this paper, we construct a one-parameter family of minimal surfaces in the Euclidean $3$-space of arbitrarily high genus and with three ends. Each member of this family is immersed, complete and with finite total curvature. Another…

This note for the Proceedings of the International Congress of Mathematical Physics gives an account of a construction of an ``elliptic quantum group'' associated with each simple classical Lie algebra. It is closely related to elliptic…

In this article, we construct the first example of an elliptic surface with infinitely many smooth \((-1)\)-curves of genus \(g>1\), settling an open question of Bauer et al. [Duke Math. J. \textbf{162} (10) (2013), 1877-1894].

We give explicit formulas for the number of distinct elliptic curves over a finite field, up to isomorphism, in two families of curves introduced by C.~Doche, T.~Icart and D.~R. Kohel.

We explicitly compute family GW invariants of elliptic surfaces for primitive classes. That involves establishing a TRR formula and a symplectic sum formula for elliptic surfaces and then determining the GW invariants using an argument from…

We introduce a new elliptic quantum toroidal algebra $U_{q,t,p}(gl_{1,tor})$. Various representations in the quantum toroidal algebra $U_{q,t}(gl_{1,tor})$ are extended to the elliptic case including the level (0,0) representation realized…

We give a simple and explicit constructions of various semi-discrete surfaces and discrete $K$-surfaces in terms of the Jacobi elliptic functions using $\tau$-functions. Their periodicities are also determined.

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…

Let $k$ denote an algebraically closed field. We revisit a construction of the author of families of elliptic curves over the rational function field $k(t)$. Combining a combinatorial analysis with a rank formula of Ulmer we prove that, for…

We consider a rational elliptic surface with a relatively minimal fibration. We compute the number of integral sections in the above rational elliptic surface. As an application, we obtain an estimate of polynomial solutions of some…

Orbifold elliptic genus and elliptic genus of singular varieties are introduced and relation between them is studied. Elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of…

The first part surveys the push forward formula for elliptic class and various applications obtained in the papers by L.Borisov and the author. In the remaining part we discuss the ring of quasi-Jacobi forms which allow to characterize the…