Rational points on modular curves

1977 ◽  
pp. 107-148 ◽  
Author(s):  
B. Mazur
Keyword(s):  
Rational Points ◽  
Modular Curves

Towers of GL($r$)-type of modular curves

2019 ◽  
Vol 2019 (754) ◽  
pp. 87-141 ◽  
Author(s):  
Ernst-Ulrich Gekeler
Keyword(s):  
Rational Points ◽  
Galois Groups ◽  
Modular Curves ◽  
Galois Covers

Abstract We construct Galois covers {X^{r,k}(N)} over {{\mathbb{P}}^{1}/{\mathbb{F}}_{q}(T)} with Galois groups close to {{\rm GL}(r,{\mathbb{F}}_{q}[T]/(N))} ( {r\geq 3} ) and rationality and ramification properties similar to those of classical modular curves {X(N)} over {{\mathbb{P}}^{1}/{\mathbb{Q}}} . As application we find plenty of good towers (with \limsup{\frac{\text{number~{}of~{}rational~{}points}}{{\rm genus}}>0} ) of curves over the field {{\mathbb{F}}_{q^{r}}} with {q^{r}} elements.

scholarly journals Quadratic Chabauty for modular curves and modular forms of rank one

2020 ◽  
Author(s):  
Netan Dogra ◽  
Samuel Le Fourn
Keyword(s):  
Modular Forms ◽  
Sufficient Conditions ◽  
Rational Points ◽  
Modular Curves ◽  
Type Result ◽  
Heegner Points ◽  
Modular Curve ◽  
Rank One ◽  
Finite Set ◽  
Set Of Points

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.

scholarly journals Modular curves of prime-power level with infinitely many rational points

2017 ◽  
Vol 11 (5) ◽  
pp. 1199-1229 ◽  
Author(s):  
Andrew Sutherland ◽  
David Zywina
Keyword(s):  
Prime Power ◽  
Power Level ◽  
Rational Points ◽  
Modular Curves

scholarly journals Quadratic points on non-split Cartan modular curves

2021 ◽  
pp. 1-23
Author(s):  
Philippe Michaud-Rodgers
Keyword(s):  
Elliptic Curves ◽  
Rational Points ◽  
Quadratic Fields ◽  
Modular Curves ◽  
Symmetric Powers

In this paper, we study quadratic points on the non-split Cartan modular curves [Formula: see text], for [Formula: see text] and [Formula: see text]. Recently, Siksek proved that all quadratic points on [Formula: see text] arise as pullbacks of rational points on [Formula: see text]. Using similar techniques for [Formula: see text], and employing a version of Chabauty for symmetric powers of curves for [Formula: see text], we show that the same holds for [Formula: see text] and [Formula: see text]. As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular.

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