WATKINS'S CONJECTURE FOR ELLIPTIC CURVES WITH NON-SPLIT MULTIPLICATIVE REDUCTION

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2022
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Abstract
Let E be an elliptic curve over the rational numbers. Watkins [Experiment. Math. 11 (2002), pp. 487-502 (2003)] conjectured that the rank of E is bounded by the 2-adic valuation of the modular degree of E. We prove this conjecture for semistable elliptic curves having exactly one rational point of order 2, provided that they have an odd number of primes of non-split multiplicative reduction or no primes of split multiplicative reduction.
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Watkins's conjecture, modular degree, rank, parity conjecture
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https://doi.org/10.1090/proc/15942
 https://repositorio.uc.cl/handle/11534/92903
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