We show that if \(E/\mathbb {Q}\) is an elliptic curve with a rational p-torsion for \(p=2\) or 3, then there is a congruence relation between Ramanujan’s tau function and E modulo p. We make use of such congruences to compute the Iwasawa invariants of 2-adic and 3-adic Mazur–Tate elements attached to Ramanujan’s tau function. We also investigate numerically the Iwasawa invariants of the Mazur–Tate elements attached to an elliptic curve with additive reduction at a fixed prime number.

中文翻译：

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Ramanujan的tau函数与椭圆曲线以及加性素数的Mazur-Tate元素之间的同余

我们证明，如果\（E / \ mathbb {Q} \）是一个椭圆曲线，对于\（p = 2 \）或3具有合理的p扭转，那么Ramanujan的tau函数与E模p有一个全等关系。。我们利用这些等式来计算与Ramanujan的tau函数相关的2adic和3adic Mazur-Tate元素的Iwasawa不变量。我们还用数字方法研究了固定在椭圆曲线上的Mazur-Tate元素的Iwasawa不变量，并且在固定质数下具有加性折减。