Elliptic curves over totally real quartic fields not containing √5 are modular

2021/09/15 by Josha Box · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Analytic Number Theory Research

paper · doi:10.1090/tran/8557

Abstract

We prove that every elliptic curve defined over a totally real number field of degree 4 not containing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartRoot 5 EndRoot"> <mml:semantics> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> <mml:annotation encoding="application/x-tex">√ 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is modular. To this end, we study the quartic points on four modular curves.

Citations

• Modular Elliptic Curves and Fermat's Last Theorem

Cited by

• Explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields
• A survey on the generalized Fermat equation of various signatures over totally real fields

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