• Arithmetic Geometry
• BSD Conjecture
• Elliptic Curves
• Modular Curves
• Modularity Theorem
• Sylvester's Conjecture

In this thesis we address the following old question: which positive integers can be written as the sum of two rational cubes?
That is to say, for which $n\in \Zz_{>0}$ the elliptic curve $E_n: x^3+y^3=nz^3$, with base point $\infty=[1:-1:0]$, has non-trivial Mordell$-$Weil group $E_n(\Qq)$?

When $n\ge 3$ is a cube-free integer, we have $E_n(\Qq)_{\tors}=\{\infty\}$, therefore we need to understand when $\rank E_n(\Qq)>0$. In the case $n=p$ is a prime number, Sylvester formulated the following conjecture (~1879), which can be seen as a particular case of the celebrated Birch and Swinnerton-Dyer conjecture: if $p\equiv 4, 7, 8 \pmod 9$, then $\rank E_p(\Qq)>0$.

In 1994, Noam Elkies announced a proof of Sylvester's conjecture in the cases $p\equiv 4, 7 \pmod 9$, but the aforementioned proof has never been published. In 2017, Samit Dasgupta and John Voight proved the following result: let $p\equiv 4, 7 \pmod 9$ be a prime such that 3 is not a cube modulo $p$, then $\rank E_p(\Qq)=\rank E_{p^2}(\Qq)=1$.

Our main goal will be to explain the proof of this theorem. The work is organized as follows.

In Chapter 1 we provide the background useful to understand the proof of Dasgupta-Voight's theorem. We review the theory of the classical modular curves $X_0(N)$; in particular, we explain how to construct an integral model of $X_0(N)$ by carefully defining a moduli problem of elliptic curves with Drinfeld level structure, and we analyze its reduction over $\Ff_p$. Then, we state the celebrated modularity theorem, which provides a bridge between the theory of the modular curves $X_0(N)$ and the elliptic curves over $\Qq$. Next, we introduce an important class of points on $X_0(N)$, called (mock) Heegner points, we study their Galois theory (aka the Shimura reciprocity law) and, thanks to the Gross-Zagier formula, we explain why and when such points can be useful to construct nontorsion algebraic points on elliptic curves defined over $\Qq$.

Chapter 2 is mainly devoted to the proof of Dasgupta-Voight's theorem. For a prime $p\equiv 4, 7 \pmod 9$, we first show that $\rank E_{p^i}(\Qq)\le 1$, for $i=1, 2$, then we find a nontorsion point in $E_{p^i}(\Qq)$ as follows. We construct an explicit modular parametrization $\Phi: X_0(N)\longrightarrow E_9$ in terms of $\eta-$quotients, where $N=243$ is the conductor of $E_9$. Then, we choose a particular mock Heegner point $P_0\in X_0(N)(H_{9p})$, where $H_{9p}$ denotes the ring class field of $K=\Qq(\sqrt{-3})$ of conductor $9p$. Using the Shimura reciprocity law and the twisting theory of elliptic curves, we are able to move $P:=\Phi(P_0)\in E_9(H_{9p})$ to a point $R\in E_1(L)$, where $L=K(\sqrt[3]{p})$. Next, we prove that $R$ is nontorsion, when $3$ is not a cube mod $p$, by reducing it modulo $p$; a key ingredient here is a general congruence relation between modular functions, which is a refinement of the classical Kronecker's congruence.
At this point, we will move $R$ to a point in $E_{p^i}(\Qq)$, for $i=1, 2$, and show that it is nontorsion (when $3$ is not a cube mod $p$).

We conclude the work outlining a possible strategy to attack Sylvester's conjecture in the case $p\equiv 8 \pmod 9$, which is still, to this day, untouched.


We will assume that the reader is acquainted with algebraic number theory, including the class field theory, with scheme theory, and the basic theory of elliptic curves. However, we will recall all the results we need. Moreover, we included two appendices for the reader's convenience: the first one on group schemes, and the other one on Selmer groups.