Elliptic curves of rank two and generalized Kato classes

Heegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell–Weil generators whose heights encode first derivatives of the associated Hasse–Weil L-series. Yet the fruitful connection between Heegner points and L-series also accounts for their main limitation, namely that they are torsion in (analytic) rank >1. This partly expository article discusses the generalised Kato classes introduced in Bertolini et al. (J Algebr Geom 24:569–604, 2015) and Darmon and Rotger (J AMS 2016), stressing their analogy with Heegner points but explaining why they are expected to give non-trivial, canonical elements of the idoneous Selmer group in settings where the classical L-function (of Hasse–Weil–Artin type) that governs their behaviour has a double zero at the centre. The generalised Kato class denoted ¿(f,g,h) is associated to a triple (f, g, h) consisting of an eigenform f of weight two and classical p-stabilised eigenforms g and h of weight one, corresponding to odd two-dimensional Artin representations Vg and Vh of Gal(H/Q) with p-adic coefficients for a suitable number field H. This class is germane to the Birch and Swinnerton-Dyer conjecture over H for the modular abelian variety E over Q attached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that ¿(f,g,h) lies in the pro-p Selmer group of E over H precisely when L(E,Vgh,1)=0, where L(E,Vgh,s) is the L-function of E twisted by Vgh:=Vg¿Vh. In the setting of interest, parity considerations imply that L(E,Vgh,s) vanishes to even order at s=1, and the Selmer class ¿(f,g,h) is expected to be trivial when ords=1L(E,Vgh,s)>2. The main new contribution of this article is a conjecture expressing ¿(f,g,h) as a canonical point in (E(H)¿Vgh)GQ when ords=1L(E,Vgh,s)=2. This conjecture strengthens and refines the main conjecture of Darmon et al. (Forum Math Pi 3:e8, 2015) and supplies a framework for understanding the results of Darmon et al. (2015), Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016).Peer ReviewedPostprint (published version

Explicit Chabauty--Kim for the Split Cartan Modular Curve of Level 13

We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in an algorithm to compute the rational points on a curve of genus g ≥ 2 over the rationals whose Jacobian has Mordell-Weil rank g and Picard number greater than one, and which satisfies some additional conditions. This algorithm is then applied to the modular curve X_s(13), completing the classification of non-CM elliptic curves over Q with split Cartan level structure due to Bilu–Parent and Bilu–Parent–Rebolledo

Stark's points and units

El objetivo de este trabajo es el estudio de un análogo de la conjetura elíptica de Stark para unidades en cuerpos de números. Para ello, utilizamos el sistema de Euler de elementos de Beilinson-Flach y la posibilidad de interpolar p-ádicamente sus imágenes a través de reguladores étale. Los resultados de Kings, Loeffler y Zerbes nos permiten relacionar las clases de cohomología así construidas con ciertos valores de una función L p-ádica de Hida-Rankin de tres variables, donde también tenemos el concepto de variación p-ádica. Del mismo modo que el sistema de Euler de los ciclos diagonales permite estudiar la conjetura elíptica de Stark, el de Beilinson-Flach nos permitirá ofrecer evidencia teórica a la conjetura de Stark para unidades que formulan Darmon, Lauder y Rotger, que quedaría probada si fuésemos capaces de establecer ciertas relaciones conjeturales que involucran también a periodos p-ádicos. En este trabajo, además, hay una parte en la que se recuerdan ciertos resultados teóricos que ofrecen soporte a los que nosotros derivamos, centrándonos en el estudio de funciones L p-ádicas, formas modulares p-ádicas y sobreconvergentes y familias de Hida

Geometric quadratic Chabauty over number fields

This article generalizes the geometric quadratic Chabauty method, initiated over $\mathbb{Q}$ by Edixhoven and Lido, to curves defined over arbitrary number fields. The main result is a conditional bound on the number of rational points on curves that satisfy an additional Chabauty type condition on the Mordell-Weil rank of the Jacobian. The method gives a more direct approach to the generalization by Dogra of the quadratic Chabauty method to arbitrary number fields.Comment: Accepted Manuscript, to appear in Transactions of the American Mathematical Society. Minor changes in Section

Geometric and analytic methods for quadratic Chabauty

Let X be an Atkin-Lehner quotient of the modular curve X_0(N) whose Jacobian J_f is a simple quotient of J_0(N)^{new} over Q. We give analytic methods for determining the rational points of X using quadratic Chabauty by explicitly computing two p-adic Gross--Zagier formulas for the newform f of level N and weight 2 associated with J_f when f has analytic rank 1. Combining results of Gross-Zagier and Waldspurger, one knows that for certain imaginary quadratic fields K, there exists a Heegner divisor in J_0(N)(K) whose image is finite index in J_f(Q) under the action of Hecke. We give an algorithm to compute the special value of the anticyclotomic p-adic L-function of f constructed by Bertolini, Darmon, and Prasanna, assuming some hypotheses on the prime p and on K. This value is proportional to the logarithm of the Heegner divisor on J_f with respect to the differential form f dq/q. We also compute the p-adic height of the Heegner divisor on J_f using a p-adic Gross-Zagier formula of Perrin-Riou. Additionally, we give algorithms for the geometric quadratic Chabauty method of Edixhoven and Lido. Our algorithms describe how to translate their algebro-geometric method into calculations involving Coleman-Gross heights, logarithms, and divisor arithmetic. We achieve this by leveraging a map from the Poincaré biextension to the trivial biextension

Computational tools for quadratic Chabauty

http://math.bu.edu/people/jbala/2020BalakrishnanMuellerNotes.pdfhttp://math.bu.edu/people/jbala/2020BalakrishnanMuellerNotes.pdfFirst author draf

Computational tools for quadratic Chabauty

Lecture note