Related papers: Automorphic forms and cubic twists of elliptic cur…

In this paper, we study relations between automorphism groups of cubic fourfolds and Kuznetsov components. Firstly, we characterize automorphism groups of cubic fourfolds as subgroups of autoequivalence groups of Kuznetsov components using…

In this paper, we study non-hyperelliptic curves of genus $3$ with cyclic automorphism group of order $6$. Over an algebraically closed field $K$ of characteristic $\neq 2,3$, such curves are written as plane quartics $C_r: x^3 z + y^4 + r…

Given any cube-free integer $\lambda>0$, we study the $3$-adic valuation of the algebraic part of the central $L$-value of the elliptic curve $$X^3+Y^3=\lambda Z^3.$$ We give a lower bound in terms of the number of distinct prime factors of…

Let $ E $ be an elliptic curve defined over a number field, the conjecture of Birch and Swinnerton-Dyer (BSD, for short) asserts a deep relation between the group $ E(K) $ of rational points and the $ L-$function $ L(E/K, s)$ of $ E $ at $…

Let $G$ be the graph attached to the $\mathbb Q$-isogeny class of an elliptic curve defined over $\mathbb Q$: that is, a vertex for every elliptic curve defined over $\mathbb Q$ in the isogeny class, and edges in correspondence with the…

For each positive integer k, we investigate the L-function attached to the k-th symmetric power of the F-crystal associated to the family of cubic exponential sums of x^3 + \lambda x. We explore its rationality, field of definition, degree,…

We determine explicit generators for the ring of modular forms associated with the moduli spaces of K3 surfaces with automorphism group $(\mathbb{Z}/2\mathbb{Z})^2$ and of Picard rank 13 and higher. The K3 surfaces in question carry a…

The combinatorial type of an elliptic K3 surface with a zero section is the pair of the ADE -type of singular fibers and the torsion part of the Mordell-Weil group. We determine the set of connected components of the moduli of elliptic K3…

In this note we extend the theory of twists of elliptic curves as presented in various standard texts for characteristic not equal to two or three to the remaining characteristics. For this, we make explicit use of the correspondence…

We classify all possible charge-3 monopole spectral curves with non-trivial automorphism group and within these identify those with elliptic quotients. By focussing on elliptic quotients the transcendental constraints for a monopole…

We study an infinite family of Mordell curves (i.e. the elliptic curves in the form y^2=x^3+n, n \in Z) over Q with three explicit integral points. We show that the points are independent in certain cases. We describe how to compute bounds…

Global F-theory compactifications whose fibers are realized as complete intersections form a richer set of models than just hypersurfaces. The detailed study of the physics associated with such geometries depends crucially on being able to…

In this paper, we sharpen earlier work of the first author, Luca and Mulholland, showing that the Diophantine equation $$ A^3+B^3 = q^\alpha C^p, \, \, ABC \neq 0, \, \, \gcd (A,B) =1, $$ has, for "most" primes $q$ and suitably large prime…

The aim of this paper is to present elliptic curves defined over function fields of even characteristic having arbitrarily large Mordell-Weil rank. More precisely, we study elliptic curves arising as quartic twist of a supersingular…

Let $f$ be a positive definite integral quadratic form in $d$ variables. In the present paper, we establish a direct link between the genus representation number of $f$ and the order of higher even $K$-groups of the ring of integers of real…

Let l be a prime number and let E and E' be l-isogenous elliptic curves defined over Q. In this paper we determine the proportion of primes p for which E(F_p) is isomorphic to E'(F_p). Our techniques are based on those developed in…

We give several results concerning the connected component ${\rm Aut}_X^0$ of the automorphism scheme of a proper variety $X$ over a field, such as its behaviour with respect to birational modifications, normalization, restrictions to…

Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very difficult…

We use contact fiber sums of open book decompositions to define an infinite hierarchy of filling obstructions for contact 3-manifolds, called planar k-torsion for nonnegative integers k, all of which cause the contact invariant in Embedded…

Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices…