We give an explicit description of the category of central extensions of a group scheme by a sheaf of Abelian groups. Based on this, we describe a framework for computing with central extensions of finite locally free commutative group schemes, torsors under such group schemes and groups of isomorphism classes of these objects.
We present a simple and efficient algorithm to compute the sum of the algebraic conjugates of a point on an elliptic curve.
Nous proposons une “Conjecture d’André-Oort en pinceau arithmétique”.
C’est une extension de la conjecture d’André-Oort, disons “classique”, formulée à l’origine par Y. André et F. Oort. La conjecture fait intervenir les modèles entiers des variétés de Shimura.
Using our recent results on the algebraicity of the Hodge locus for variations of Hodge structures of level at least 3, we improve the results of Lawrence–Venkatesh in direction of the refined Bombieri–Lang conjecture.
We define and study a natural system of tautological rings on the moduli spaces of marked curves at the level of differential forms. We show that certain 2-forms obtained from the natural normal functions on these moduli spaces are tautological. Also we show that rings of tautological forms are always finite dimensional. Finally we characterize the Kawazumi–Zhang invariant as essentially the only smooth function on the moduli space of curves whose Levi form is a tautological form.
Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves . In this paper we compute the -rational points on the Atkin–Lehner quotients of these curves using a variety of techniques. We also determine which rational points are CM for many of these curves.
We compute the rational points on the Atkin–Lehner quotient using the quadratic Chabauty method. Our work completes the study of exceptional rational points on the curves of genus between 2 and 6. Together with the work of several authors, this completes the proof of a conjecture of Galbraith.
Two continued fractions and of order twenty-four are obtained from a general continued fraction identity of Ramanujan. Some theta-function and modular identities for and are established to prove general theorems for the explicit evaluations of and . From the theta-function identities of and , three colour partition identities are derived as application to partition theory of integer. Further, -, - and -dissection formulas are established for the continued fractions and , and their reciprocals.
In this paper, the composition of Bhargava’s cubes is generalized to the ring of integers of a number field of narrow class number one, excluding the case of totally imaginary number fields. The exclusion of the latter case arises from the nonexistence of a bijection between (classes of) binary quadratic forms and an ideal class group. This problem, together with a related mistake in another paper of the author, is addressed in the appendix.
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