论文标题
模块化函数的乘法独立性
Multiplicative independence of modular functions
论文作者
论文摘要
我们提供了一个新的基本证明，证明了成对$ \ mathrm {gl} _2^+（\ mathbb {q}）$的乘法独立性 - 最初是pila and tsimerman的结果。因此，我们能够将此结果推广到更广泛的模块化功能。我们表明，此类包括一个包含模块化函数的集合，当Borcherds举起某些弱塑形模块化形式时，该集合自然会出现。对于$ f $，属于此类的模块化函数，我们将每个$ n \ geq 1 $，$ n $ tuplass的有限性推导，这些$ n $ tubles的有限$ f $ f $ - 特定点，这些点是多上依赖性且最小的。这概括了pila和tsimerman在单一模量上的定理。然后，我们展示这些结果与混合shimura品种$ y（1）^n \ times \ mathbb {g} _ {\ mathrm {m}}^n $的Zilber-粉碎构构相关。
We provide a new, elementary proof of the multiplicative independence of pairwise distinct $\mathrm{GL}_2^+(\mathbb{Q})$-translates of the modular $j$-function, a result due originally to Pila and Tsimerman. We are thereby able to generalise this result to a wider class of modular functions. We show that this class includes a set comprising modular functions which arise naturally as Borcherds lifts of certain weakly holomorphic modular forms. For $f$ a modular function belonging to this class, we deduce, for each $n \geq 1$, the finiteness of $n$-tuples of distinct $f$-special points that are multiplicatively dependent and minimal for this property. This generalises a theorem of Pila and Tsimerman on singular moduli. We then show how these results relate to the Zilber--Pink conjecture for subvarieties of the mixed Shimura variety $Y(1)^n \times \mathbb{G}_{\mathrm{m}}^n$ and prove some special cases of this conjecture.
