Additive combinatorics and descent

Abstract: We shall discuss the problem of constructing elliptic curves over number fields with positive but "controlled" rank. An example of this problem is: given a quadratic extension $L/K$ of number fields, construct an elliptic curve $E/K$ such that $0<\text{rk}(E(K))=\text{rk}(E(L))$. Another example is: for a number field $K$, find an elliptic curve $E/K$ such that $\text{rk}(E(K))=1$. With Peter Koymans we introduced a method to tackle this type of problems, combining additive combinatorics with 2-descent. I will explain our past work on the former problem, where we showed that Hilbert 10th problem has negative answer on ring of integers of general number fields. Next, I will explain our joint work in progress, where we settle the latter question, showing the following stronger result: if $E/K$ has full rational $2$-torsion and no cyclic degree $4$ isogeny defined over $K$, and it has at least one quadratic twist with odd root number, then it has infinitely many quadratic twists $d$ in $K^{\ast}/K^{\ast 2}$ such that $\text{rk}(E^d(K))=1$.

algebraic geometrynumber theory

Audience: researchers in the topic

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MIT number theory seminar

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