We prove that all elliptic curves defined over real quadratic fields are modular.

Elliptic Curves over Real Quadratic Fields are Modular

https://hobbes.la.asu.edu/lmfdb-14/Siksek.pdf

Elliptic curves over real quadratic fields are modular

Memoria de investigación 2016. Facultad de Ciencias, UAM

We classify the possible torsion structures of rational elliptic curves over cubic fields. Along the way we find a previously unknown torsion structure over a cubic field, Z/21Z, which corresponds to a sporadic point on X1(21) of degree 3, which is the lowest possible degree of a sporadic point on a modular curve X1(n).

/pdf/torsion-of-rational-elliptic-curves-over-cubic-fields-and-2vhr4jn3f2.pdf

Torsion of rational elliptic curves over cubic fields and sporadic points on X_1(n)

We classify the possible torsion structures of rational elliptic curves over cubic fields. Along the way we find a previously unknown torsion structure over a cubic field, $\Z /21 \Z$, which corresponds to a sporadic point on $X_1(21)$ of degree 3, which is the lowest possible degree of a sporadic point on a modular curve $X_1(n)$.

Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1(n)$

We study how the torsion of elliptic curves over number fields grows upon base change, and in particular prove various necessary conditions for torsion growth. For a number field $F$, we show that for a large set of number fields $L$, whose Galois group of their normal closure over $F$ has certain properties, it will hold that $E(L)_{tors}=E(F)_{tors}$ for all elliptic curves $E$ defined over $F$. 
Our methods turn out to be particularly useful in studying the possible torsion groups $E(K)_{tors}$, where $K$ is a number field and $E$ is a base change of an elliptic curve defined over $\mathbb Q$. Suppose that $E$ is a base change of an elliptic curve over $\mathbb Q$ for the remainder of the abstract. We prove that $E(K)_{tors}=E(\mathbb Q)_{tors}$ for all elliptic curves $E$ defined over $\mathbb Q$ and all number fields $K$ of degree $d$, where $d$ is not divisible by a prime $\leq 7$. Using this fact, we determine all the possible torsion groups $E(K)_{tors}$ over number fields $K$ of prime degree $p\geq 7$. We determine all the possible degrees of $[\mathbb Q(P):\mathbb Q]$, where $P$ is a point of prime order $p$ for all $p$ such that $p\not\equiv 8 \pmod 9$ or $\left( \frac{-D}{p}\right)=1$ for any $D\in \{1,2,7,11,19,43,67,163\}$; this is true for a set of density $\frac{1535}{1536}$ of all primes and in particular for all $p<3167$. Using this result, we determine all the possible prime orders of a point $P\in E(K)_{tors}$, where $[K:\mathbb Q]=d$, for all $d\leq 3342296$. Finally, we determine all the possible groups $E(K)_{tors}$, where $K$ is a quartic number field and $E$ is an elliptic curve defined over $\mathbb Q$ and show that no quartic sporadic point on a modular curves $X_1(m,n)$ comes from an elliptic curve defined over $\mathbb Q$.

Growth of torsion groups of elliptic curves upon base change

We study elliptic curves over quadratic fields with isogenies of certain degrees. Let be a positive integer such that the modular curve is hyperelliptic of genus and such that its Jacobian has rank over . We determine all points of defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to -isomorphism, every elliptic curve over a quadratic field admitting an -isogeny is -isogenous, for some , to the twist of its Galois conjugate by a quadratic extension of . We determine and explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to -isomorphism, all elliptic curves with -isogenies over quadratic fields are in fact -curves.

/pdf/hyperelliptic-modular-curves-and-isogenies-of-elliptic-4bf65o6sus.pdf

Hyperelliptic modular curves and isogenies of elliptic curves over quadratic fields

Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to $\overline{\mathbb Q}$-isomorphism, all but finitely many elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb Q$-curves, and we list all exceptions. We also show that, again with finitely many exceptions up to $\overline{\mathbb Q}$-isomorphism, every $\mathbb Q$-curve $E$ over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by some quadratic extension $L$ of $K$; we determine $d$ and $L$ explicitly.

/pdf/hyperelliptic-modular-curves-x-0-n-and-isogenies-of-elliptic-4cjgc1x4xf.pdf

Filip Najman

Papers

Torsion of rational elliptic curves over cubic fields and sporadic points on X_1(n)

Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1(n)$

Growth of torsion groups of elliptic curves upon base change

Hyperelliptic modular curves and isogenies of elliptic curves over quadratic fields

Hyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fields