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Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. (English) Zbl 1307.11071
In this paper it is proved that the average rank of elliptic curves over $$\mathbb{Q}$$ is bounded. To be more precise, consider curves $E_{A,B}:\,y^2=x^3+Ax+B$ with integer coefficients $$A,B$$ normalized such that $$A^3$$ and $$B^2$$ have no nontrivial common 12-th power divisor. Set $$H(E_{A,B}):=\max(4|A|^3\,,\,27B^2)$$. Then $\limsup_{X\rightarrow\infty}\frac{\sum_{H(E_{A,B})\leq X} \mathrm{Rank}(E_{A,B})}{\#\{E_{A,B}:H(E_{A,B})\leq X\}}\leq \frac{3}{2}.$ This result follows immediately from an assertion about the average size of the 2-Selmer group: $\sum_{H(E_{A,B})\leq X}\# S_2(E_{A,B})\sim 3\#\{E_{A,B}:H(E_{A,B})\}\leq X$ as $$X\rightarrow\infty$$. Indeed the same is true when one restricts the coefficients $$A$$ and $$B$$ by any finite set of congruence conditions, or even a suitable natural set of infinitely many congruence conditions.
The proof handles the 2-Selmer group by counting binary quartic forms $$f(x,y)=ax^4+bx^3y+cx^2y^2+dxy^3+ey^4$$ such that the curve $$f(x,y)=z^2$$ has points everywhere locally. In order to relate such forms $$f$$ to elliptic curves with $$H(E_{A,B})\leq X$$, one counts forms for which the invariants $$I(f),J(f)$$ satisfy $$H(I,J):=\max(|I|^3,J^2/4)\leq Y$$, where again one may impose additional congruence conditions on $$f$$. Such congruence conditions allow one to account for the local solvability constraints. Write $$h^{(j)}(I,J)$$ for the number of equivalence classes under $$\mathrm{GL}_2(\mathbb{Z})$$ of forms $$f$$ having invariants $$I$$ and $$J$$, and having exactly $$4-2j$$ real roots. Then it is shown for example that $\sum_{H(I,J)\leq X}h^{(0)}(I,J)=\frac{4}{135}\zeta(2)X^{5/6} +O(X^{3/4+\varepsilon}),$ for any fixed $$\varepsilon>0$$. There are similar results for the other counting functions $$h^{(j)}(I,J)$$, and for the situation when there are congruence restrictions on the forms $$f$$.
The equivalence classes of binary quartic forms are counted by producing a suitable fundamental domain for the action of $$\mathrm{GL}_2(\mathbb{Z})$$. However this fundamental domain is unbounded, so that it is not straightforward to count integer points in its dilates. This difficulty is addressed using ideas developed from the first author’s earlier work [Ann. Math. (2) 162, No. 2, 1031–1063 (2005; Zbl 1159.11045); ibid. 172, No. 3, 1559–1591 (2010; Zbl 1220.11139)].

##### MSC:
 11G05 Elliptic curves over global fields 11E76 Forms of degree higher than two 11P21 Lattice points in specified regions
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