Abstract We make the polynomial dependence on the fixed representation π in our previous subconvex bound of L ⁢ ( 1 2 , π ⊗ χ ) {L(\frac{1}{2},\pi\otimes\chi)} for GL 2 × GL 1 {\mathrm{GL}_{2}\times\mathrm{GL}_{1}} explicit, especially in terms of the usual conductor 𝐂 ⁢ ( π fin ) {\mathbf{C}(\pi_{\mathrm{fin}})} . There is no clue that the original choice, due to Michel and Venkatesh, of the test function at the infinite places should be the optimal one. Hence we also investigate a possible variant of such local choices in some special situations.

中文翻译：

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GL2 × GL1 的显式类 Burgess 子凸边界

摘要 我们使多项式依赖于先前 L ⁢ ( 1 2 , π ⊗ χ ) {L(\frac{1}{2},\pi\otimes\chi)} 的子凸界中的固定表示 π，用于 GL 2 × GL 1 {\mathrm{GL}_{2}\times\mathrm{GL}_{1}} 显式，特别是对于通常的导体 𝐂 ⁢ ( π fin ) {\mathbf{C}(\pi_{ \mathrm{fin}})} 。由于 Michel 和 Venkatesh 的原因，没有任何线索表明在无限地方的测试函数的原始选择应该是最优的。因此，我们还研究了在某些特殊情况下这种局部选择的可能变体。