On the extension of even families of non-congruent numbers
Rendiconti del Seminario Matematico della Università di Padova, Tome 148 (2022), pp. 23-49.
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A method that extends existing families of even non-congruent numbers to produce new families of non-congruent numbers with arbitrarily many distinct prime factors is presented. We show that infinitely many new non-congruent numbers can be generated by appending a suitable collection of primes onto any even non-congruent number whose corresponding congruent number elliptic curve has 2-Selmer rank of zero. Our method relies upon Monsky’s formula for computing the 2-Selmer rank of the congruent number elliptic curve. Even non-congruent numbers constructed according to our result have an unlimited number of prime factors in each odd congruence class modulo eight, and have congruent number elliptic curves with 2-Selmer rank equal to zero.

Accepté le :
Publié le :
DOI : 10.4171/rsmup/105
Classification : 11, 00
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     author = {Lindsey Reinholz and Qiduan Yang},
     title = {On the extension of even families of non-congruent numbers},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {23--49},
     volume = {148},
     year = {2022},
     doi = {10.4171/rsmup/105},
     zbl = {07673820},
     mrnumber = {4542371},
     language = {en},
     url = {http://www.numdam.org/articles/10.4171/rsmup/105/}
}
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Lindsey Reinholz; Qiduan Yang. On the extension of even families of non-congruent numbers. Rendiconti del Seminario Matematico della Università di Padova, Tome 148 (2022), pp. 23-49. doi : 10.4171/rsmup/105. http://www.numdam.org/articles/10.4171/rsmup/105/

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