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I have been reading elliptic curves and had a discussion with my professor regarding the torsion subgroup of the Mordell-Weil group. He advised me to read Loïc Merel's "Bornes pour la torsion des courbes elliptiques sur les corps de nombres" (Bounds for the torsion of elliptic curves on number fields) where Merel proves the uniform boundedness conjecture for number fields (that for any elliptic curve E over a number field K the torsion is bounded independent of E).

I have the paper but it's in French and translation ruins the math typesetting. Is there an English version that anyone is aware of? Or any other work that contains this but in English. Thanks.

Ref: Merel, L. (1996). "Bornes pour la torsion des courbes elliptiques sur les corps de nombres". Inventiones Mathematicae (in French). 124 (1–3): 437–449. doi:10.1007/s002220050059.

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Googling a little, I found this expository paper: https://lmbp.uca.fr/~rebolledo/page-fichiers/07-goettingen.pdf

I haven't read it and can't vouch for its completeness, but between reading this and looking at the original paper in French you should be able to piece together any missing steps.

If you go further into arithmetic geometry, you will probably run into this problem again. It is probably worth forcing yourself to read a few papers in French (with help from Google and people who already know the math) until you are reasonably comfortable with reading mathematical French.

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  • $\begingroup$ Thanks a lot for the paper, I think it is based on this paper of Merel. I do hope to learn French someday to atleast be able to read more in the subject. $\endgroup$ Dec 23, 2021 at 16:53

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