Lifting problem for universal quadratic forms

On Kitaoka's conjecture and lifting problem for universal quadratic forms

“…This theorem includes, as a significant special case, a new result on the sum of squares, for one can take 𝐹 = ℚ and 𝐿 = ℤ 𝑟 equipped with the quadratic form 𝑄 = 𝑥 2 1 + ⋯ + 𝑥 2 𝑟 . It also extends the previous results on the lifting problem (that is, whether a form can be universal over a larger number field, see [16] and potentially universal quadratic forms in [33]), and partly resolves an open question formulated in [16]: As Corollary 9, we show that a given quadratic form is universal only over finitely many totally real number fields of degree 𝑑.…”

“…Proof. The Pythagoras number of the ring of integers is bounded in terms of degree of the field extension [16,Corollary 3.3], that is, there is a function g(𝑑) such that ( 𝐾 ) ⩽ g(𝑑) whenever [𝐾 ∶ ℚ] = 𝑑. Thus, it suffices to consider the sum of a bounded number of squares 𝑄 = 𝑥 2 1 + ⋯ + 𝑥 2 g(𝑑) .…”

“…(For the required background in analytic number theory, see, for example, [24, chapter 7 and appendix II]). As for the Pythagoras number $$P$$, in [16, Corollary 3.3] we proved that $$P$$ is bounded from above by a bound that depends only on the degree $$d=[K:\mathbb {Q}]$$. Specifically, the proof of [16, Corollary 3.3] established that $$P\leqslant g(d)$$, where $$g(d)$$ is the $$g$$ ‐invariant , that is, the smallest rational integer such that any quadratic form with $$\mathbb {Z}$$‐coefficients of rank $$d$$ that is represented by the sum of any number of squares is represented by the sum of $$g(d)$$ squares.…”

“…As for the Pythagoras number $$P$$, in [16, Corollary 3.3] we proved that $$P$$ is bounded from above by a bound that depends only on the degree $$d=[K:\mathbb {Q}]$$. Specifically, the proof of [16, Corollary 3.3] established that $$P\leqslant g(d)$$, where $$g(d)$$ is the $$g$$ ‐invariant , that is, the smallest rational integer such that any quadratic form with $$\mathbb {Z}$$‐coefficients of rank $$d$$ that is represented by the sum of any number of squares is represented by the sum of $$g(d)$$ squares. Recently, Beli, Chan, Icaza, and Liu [1, Theorem 1.1] showed an upper bound for the $$g$$‐invariant that is exponential in $$\sqrt d$$, that is, $$g(d)\leqslant c_1 \exp (c_2\sqrt {d})$$.…”

“…This leads to the Pythagoras number, a constant well‐studied also in other settings (see, for example, [23]). Yet, in the case of the ring of integers of a totally real number field, all that is known in general is that the Pythagoras number is finite [28] and bounded by the degree of the field [16], but can grow arbitrarily large [28] (in the non‐totally real case, the Pythagoras number $$\leqslant 5$$ [27]; for some small degree cases see [22, 26, 31]). Furthermore, Siegel [29] proved that for each number field $$F$$ there exists $$m\in \mathbb {N}$$ such that all totally positive integers divisible by $$m$$ can be represented as the sum of squares.…”