Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

By Manjul Bhargava, A. Shankar, Takashi Taniguchi, F. Thorne, Jacob Tsimerman, Yongqiang Zhao. Journal of The American Mathematical Society, 2017

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<p>We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the trivial bound being <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O Subscript epsilon comma n Baseline left-parenthesis StartAbsoluteValue normal upper D normal i normal s normal c left-parenthesis upper K right-parenthesis EndAbsoluteValue Superscript 1 slash 2 plus epsilon Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ϵ<!-- ϵ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">D</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ϵ<!-- ϵ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O_{\epsilon ,n}(|\mathrm {Disc}(K)|^{1/2+\epsilon })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coming from the bound on the entire class group). This yields corresponding improvements to: (1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves, (2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves, (3) bounds on the sizes of 2-Selmer groups and ranks of Jacobians of hyperelliptic curves, and (4) bounds of Baily and Wong on the number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A 4"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">A_4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-quartic fields of bounded discriminant.</p>

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/409b44095b0615052a06c058aa9ac3f10597ce57