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Semi-stable models, local heights and quadratic Chabauty for $X_0(N)^*$
Quadratic Chabauty computations for $X_0(N)^*$ are complicated by local height contributions at primes of bad reduction. For squarefree $N$, we explain a strategy for constructing a global $p$-adic height for which all the contributions at the primes of bad reduction vanish. As our first main result
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