# Heegner Points on Cartan Non-split Curves

@article{Kohen2016HeegnerPO, title={Heegner Points on Cartan Non-split Curves}, author={Daniela Kohen and Ariel Pacetti}, journal={Canadian Journal of Mathematics}, year={2016}, volume={68}, pages={422 - 444} }

Abstract Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ , and let $K$ be an imaginary quadratic field such that the root number of $E/K$ is −1. Let $O$ be an order in $K$ and assume that there exists an odd prime $p$ such that ${{p}^{2}}\,\parallel \,N$ , and $p$ is inert in $O$ . Although there are no Heegner points on ${{X}_{0}}(N)$ attached to $O$ , in this article we construct such points on Cartan non-split curves. In order to do that, we give a method to compute Fourier…

## 8 Citations

### An unexpected trace relation of CM points

- Mathematics
- 2018

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N=p^2M$ where $p$ is an odd prime not dividing $M$. Let $\mathcal{O}_f$ be the order of conductor $f$ (relatively prime to $N$) in an imaginary…

### A MODULI INTERPRETATION FOR THE NON-SPLIT CARTAN MODULAR CURVE

- MathematicsGlasgow Mathematical Journal
- 2017

Abstract Modular curves like X0(N) and X1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ),…

### Local Galois representations and Frobenius traces

- Mathematics
- 2022

Ramified l-adic local Galois representations tend to be unwelcome. We give an explicit character formula which reconstructs such a representation from its Frobenius traces over extensions where it is…

### Heegner points on Hijikata-Pizer-Shemanske curves

- Mathematics
- 2016

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rather general type of quaternionic or- ders…

### On Heegner points for primes of additive reduction ramifying in the base field

- Mathematics
- 2015

Let E be a rational elliptic curve, and K be an imaginary quadratic
field. In this article we give a method to construct Heegner points when E has a prime bigger than 3 of additive reduction…

### HEEGNER POINTS CONSTRUCTION FOR RAMIFIED ADDITIVE REDUCTION PRIMES

- Mathematics
- 2015

Let E be a rational elliptic curve, and K an imaginary quadratic field. In this article we study how to construct Heegner points when E has a prime of additive reduction ramifying in the field K.

### Quadratic Chabauty for modular curves and modular forms of rank one

- MathematicsMathematische Annalen
- 2020

<jats:p>In this paper, we provide refined sufficient conditions for the quadratic Chabauty method on a curve <jats:italic>X</jats:italic> to produce an effective finite set of points containing the…

## References

SHOWING 1-10 OF 23 REFERENCES

### A MODULI INTERPRETATION FOR THE NON-SPLIT CARTAN MODULAR CURVE

- MathematicsGlasgow Mathematical Journal
- 2017

Abstract Modular curves like X0(N) and X1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ),…

### Rational Points on Modular Elliptic Curves

- Mathematics
- 2003

Elliptic curves Modular forms Heegner points on $X_0(N)$ Heegner points on Shimura curves Rigid analytic modular forms Rigid analytic modular parametrisations Totally real fields ATR points…

### The Jacobians of Non‐Split Cartan Modular Curves

- Mathematics
- 1998

The mod p representation associated to an elliptic curve is called split or non‐split dihedral if its image lies in the normaliser of a split or non‐split Cartan subgroup of GL2(fp), respectively.…

### On a result of Imin Chen

- Mathematics
- 1996

We give another proof of Imin Chen's result that the jacobian of the modular curve X(p)_{non-split}, for p a prime number, is isogeneous to the new part of the jacobian of X_0(p^2), using only the…

### Heegner Points and Rankin L-Series

- Mathematics
- 2004

1. Preface Henri Darmon and Shour-Wu Zhang 2. Heegner points: the beginnings Bryan Birch 3. Correspondence Bryan Birch and Benedict Gross 4. The Gauss class number problem for imaginary quadratic…

### Euler factors determine local Weil representations

- Mathematics
- 2011

We show that a Frobenius-semisimple Weil representation over a local fieldK is determined by its Euler factors over the extensions ofK. The construction is explicit, and we illustrate it for l-adic…

### ELLIPTIC FUNCTIONS

- Mathematics

The first systematic account of the theory of elliptic functions and the state of the art around the turn of the century. Preceding general class field theory and therefore incomplete. Contains a…

### Sur un resultat d'Imin Chen

- Mathematics
- 1999

We generalize a result of Chen concerning an isogeny between products of jacobians of modular curves associated to subgroups of GL(2,F_p). This generalization concerns objects with an action by…

### Introduction to the arithmetic theory of automorphic functions

- Mathematics
- 1971

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary…

### Lectures On The Mordell-Weil Theorem

- Mathematics
- 1989

Contents: Heights - Nomalized heights - The Mordell-Weil theorem - Mordell's conjecture - Local calculation of normalized heights - Siegel's method - Baker's method - Hilbert's irreducibility theorem…