D. Zywina

activity

20242026

collaborators

6 papers

math.NT2026

On the infinitude of elliptic curves over a number field with prescribed small rank

David Zywina

For any number field $K$ and integer $0 \leq r \leq 4$ , we prove that there are infinitely many elliptic curves over $K$ of rank $r$ . Our elliptic curves are obtained by specializin…

math.NT2025

Rank one elliptic curves and rank stability

David Zywina

For any quadratic extension $L / K$ of number fields, we prove that there are infinitely many elliptic curves $E$ over $K$ so that the abelian groups $E (K)$ and $E (L)$ both have rank…

math.NT2025

There are infinitely many elliptic curves over the rationals of rank 2

David Zywina

We show that there are infinitely many elliptic curves $E / Q$ , up to isomorphism over $Q$ , for which the finitely generated group $E (Q)$ has ra…

math.NT2025

Drinfeld modules with maximal Galois action

David Zywina

With a fixed prime power $q > 1$ , define the ring of polynomials $A = F_{q} [t]$ and its fraction field $F = F_{q} (t)$ . For each pair $a = (a_{1}, a_{2}) \in A^{2}$ with $a_{2}$ nonzer…

math.NT2025

An elliptic surface with infinitely many fibers for which the rank does not jump

David Zywina

Let $E$ be a nonisotrivial elliptic curve over $Q (T)$ and denote the rank of the abelian group $E (Q (T))$ by $r$ . For all but finitely many $t \in Q$ , spec…

math.NT2024

Open image computations for elliptic curves over number fields

David Zywina

For a non-CM elliptic curve $E$ defined over a number field $K$ , the Galois action on its torsion points gives rise to a Galois representation $Ï_E: Gal(\overline{K}/K)\to GL_2(\w…