heegner points
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Author(s):  
Massimo Bertolini ◽  
Marco Adamo Seveso ◽  
Rodolfo Venerucci
Keyword(s):  

AbstractThis article proves a case of the p-adic Birch and Swinnerton–Dyer conjecture for Garrett p-adic L-functions of (Bertolini et al. in On p-adic analogues of the Birch and Swinnerton–Dyer conjecture for Garrett L-functions, 2021), in the imaginary dihedral exceptional zero setting of extended analytic rank 4.


Author(s):  
Jie Shu ◽  
Shuai Zhai

Abstract In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over ℚ {{\mathbb{Q}}} . We prove the existence of explicit infinite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank 1. In addition, we show that these families of quadratic twists satisfy the 2-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.


Author(s):  
Netan Dogra ◽  
Samuel Le Fourn

AbstractIn this paper, we provide refined sufficient conditions for the quadratic Chabauty method on a curve X to produce an effective finite set of points containing the rational points $$X({\mathbb {Q}})$$ X ( Q ) , with the condition on the rank of the Jacobian of X replaced by condition on the rank of a quotient of the Jacobian plus an associated space of Chow–Heegner points. We then apply this condition to prove the effective finiteness of $$X({\mathbb {Q}})$$ X ( Q ) for any modular curve $$X=X_0^+(N)$$ X = X 0 + ( N ) or $$X_\mathrm{{ns}}^+(N)$$ X ns + ( N ) of genus at least 2 with N prime. The proof relies on the existence of a quotient of their Jacobians whose Mordell–Weil rank is equal to its dimension (and at least 2), which is proven via analytic estimates for orders of vanishing of L-functions of modular forms, thanks to a Kolyvagin–Logachev type result.


2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Asbjørn Christian Nordentoft

AbstractIn this paper, we study hybrid subconvexity bounds for class group 𝐿-functions associated to quadratic extensions K/\mathbb{Q} (real or imaginary). Our proof relies on relating the class group 𝐿-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the uniform sup norm bound for Eisenstein series E(z,1/2+it)\ll_{\varepsilon}y^{1/2}(\lvert t\rvert+1)^{1/3+\varepsilon}, y\gg 1, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.


Author(s):  
Dimitar Jetchev ◽  
David Loeffler ◽  
Sarah Livia Zerbes
Keyword(s):  

2020 ◽  
Vol 362 ◽  
pp. 106938
Author(s):  
Ashay A. Burungale ◽  
Ye Tian
Keyword(s):  

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