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1. Moduli Spaces of Quadratic Maps: Arithmetic and Geometry

   https://doi.org/10.1093/imrn/rnae126  

   Ramadas, Rohini (June 2024, International Mathematics Research Notices)

   Abstract We establish an implication between two long-standing open problems in complex dynamics. The roots of the $$n$$th Gleason polynomial $$G_{n}\in{\mathbb{Q}}[c]$$ comprise the $$0$$-dimensional moduli space of quadratic polynomials with an $$n$$-periodic critical point. $$\operatorname{Per}_{n}(0)$$ is the $$1$$-dimensional moduli space of quadratic rational maps on $${\mathbb{P}}^{1}$$ with an $$n$$-periodic critical point. We show that if $$G_{n}$$ is irreducible over $${\mathbb{Q}}$$, then $$\operatorname{Per}_{n}(0)$$ is irreducible over $${\mathbb{C}}$$. To do this, we exhibit a $${\mathbb{Q}}$$-rational smooth point on a projective completion of $$\operatorname{Per}_{n}(0)$$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $$n$$, $$\operatorname{Per}_{n}(0)$$ itself has no $${\mathbb{Q}}$$-rational points. 

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2. On Single-Distance Graphs on the Rational Points in Euclidean Spaces

   https://doi.org/10.4153/S0008439520000181  

   Bau, Sheng; Johnson, Peter; Noble, Matt (March 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract For positive integers n and d > 0, let $$G(\mathbb {Q}^n,\; d)$$ denote the graph whose vertices are the set of rational points $$\mathbb {Q}^n$$ , with $$u,v \in \mathbb {Q}^n$$ being adjacent if and only if the Euclidean distance between u and v is equal to d . Such a graph is deemed “non-trivial” if d is actually realized as a distance between points of $$\mathbb {Q}^n$$ . In this paper, we show that a space $$\mathbb {Q}^n$$ has the property that all pairs of non-trivial distance graphs $$G(\mathbb {Q}^n,\; d_1)$$ and $$G(\mathbb {Q}^n,\; d_2)$$ are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of $$G(\mathbb {Q}^n,\; d)$$ . 

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3. Tamagawa Products of Elliptic Curves Over ℚ

   https://doi.org/10.1093/qmath/haab042  

   Griffin, Michael; Onowei-Lun Tsai, Ken; Tsai, Wei-Lun (October 2021, The Quarterly Journal of Mathematics)

   Abstract We explicitly construct the Dirichlet series $$\begin{equation*}L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},\end{equation*}$$ where $$P_{\mathrm{Tam}}(m)$$ is the proportion of elliptic curves $$E/\mathbb{Q}$$ in short Weierstrass form with Tamagawa product m. Although there are no $$E/\mathbb{Q}$$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $$P_{\mathrm{Tam}}(1)={0.5053\dots}$$. As a corollary, we find that $$L_{\mathrm{Tam}}(-1)={1.8193\dots}$$ is the average Tamagawa product for elliptic curves over $$\mathbb{Q}$$. We give an application of these results to canonical and Weil heights. 

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4. LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER

   https://doi.org/10.1112/S0025579319000032  

   Bienvenu, Pierre-Yves; Lê, Thái Hoàng (January 2019, Mathematika)

   We examine correlations of the Möbius function over $$\mathbb{F}_{q}[t]$$ with linear or quadratic phases, that is, averages of the form 1 $$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f0$$ if $$Q$$ is linear and $$O(q^{-n^{c}})$$ for some absolute constant $c>0$ if $$Q$$ is quadratic. The latter bound may be reduced to $$O(q^{-c^{\prime }n})$$ for some $$c^{\prime }>0$$ when $Q(f)$ is a linear form in the coefficients of $$f^{2}$$ , that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem. 

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5. LARGE FIELDS IN DIFFERENTIAL GALOIS THEORY

   https://doi.org/10.1017/S1474748020000018  

   Bachmayr, Annette; Harbater, David; Hartmann, Julia; Pop, Florian (January 2020, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over  $$\mathbb{Q}$$ . More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over $$\mathbb{Q}_{p}(x)$$ . 

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