We relate the study of Landau–Siegel zeros to the ranks of Jacobians $J_0(q)$ of modular curves for large primes $q$. By a conjecture of Brumer–Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level $q$ have analytic rank $⩽1$. We show that either Landau–Siegel zeros do not exist, or that, for wide ranges of $q$, almost all such newforms have analytic rank $⩽2$. In particular, in wide ranges, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes $q$ in a wide range, we show that the rank of $J_0(q)$ is asymptotically equal to the rank predicted by the Brumer–Murty conjecture.

中文翻译：

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自守 L 函数和 Landau-Siegel 零点的解析秩

我们将朗道-西格尔零点的研究与雅可比行列式联系起来$J_0（q）$大素数的模曲线$q$。根据布鲁默-穆蒂的猜想，等级应该等于维度的一半。同样，几乎所有重量级和级别的新形式$q$有分析等级$⩽1$。我们证明朗道-西格尔零点不存在，或者对于大范围的$q$，几乎所有此类新形式都具有解析等级$⩽2$。特别是，在大范围内，几乎所有奇怪的新形式的分析等级都等于 1。此外，对于稀疏素数集$q$在一个广泛的范围内，我们表明$J_0（q）$渐近等于布鲁默-穆蒂猜想预测的等级。