For elliptic curves ordered by height, the first moment of |Sel2(E)| contains a secondary term of order X^{3/4}, with a power-saving error of size O(X^{3/4-1/3804+epsilon}).
Improved bounds on number fields of small degree
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We study the number of degree $n$ number fields with discriminant bounded by $X$. In this article, we improve an upper bound due to Schmidt on the number of such fields that was previously the best known upper bound for $6 \leq n \leq 94$.
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Secondary terms in the first moment of $|{\rm Sel}_2(E)|$
For elliptic curves ordered by height, the first moment of |Sel2(E)| contains a secondary term of order X^{3/4}, with a power-saving error of size O(X^{3/4-1/3804+epsilon}).