March 7 (Thu), 2024, 16:00--17:00, Room 8101

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Title:
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Representation numbers of Bell-type quadratic forms
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Speaker:
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Yeongwook Kwon (UNIST)

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Abstract:
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In 1834, Jacobi proved his four-square theorem. The sum of four squares can be viewed as an example of the so-called Bell-type quadratic forms, and the representation number of Bell-type quadratic forms were studied by several authors. However, most preceding results only dealt with Bell-type forms of class number 1. In this talk, we derive a closed formula for the representation numbers of each Bell-type quadratic form of class number less than or equal to 2. This is joint work with Chang Heon Kim, Kyoungmin Kim and Soonhak Kwon.

May 30 (Thu), 2024, 16:00--17:00, Room 1424

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Title:
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Growth of torsion groups of elliptic curves over number fields, rational isogenies, and number fields without rationally defined CM
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Speaker:
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Hansol Kim (KAIST)

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Abstract:
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We find an equivalent condition that a number field $K$ has the following property: There is a prime $p_{K}$ depending only on $K$ such that if $d$ is a positive integer whose minimal prime divisor is greater than $p_{K}$, then for any extension $L/K$ of degree $d$ and any elliptic curve $E/K$, we have $E\left(L\right)_{\operatorname{tors}} = E\left(K\right)_{\operatorname{tors}}$. For the purpose, we study the relations among torsion groups of elliptic curves over number fields, rational isogenies, and number fields without rationally defined CM. As a collorary of our result, we prove that any quadratic number field which is not an imaginary number field whose class number is not $1$ has the above property. This is a joint work with Bo-Hae Im.