A Diophantine Criterion for the Shafarevich-Tate Groups of Elliptic Curves from Heron Triangles
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866918519574102016 |
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| author | Ghale, Vinodkumar |
| author_facet | Ghale, Vinodkumar |
| contents | The solvability of Diophantine quartic equations is a contemporary area of interest due to its connection with generalized Fermat's equation. In this work, we are interested in the integer solutions of a similar Diophantine equation p u^2 = v^2 + w^2. For a particular form of u, v, and w, we prove that the elliptic curves E_p: y^2 = x(x-1)(x+p^2), which arise from Heron triangles, for primes p = 1 (mod 8) where q = (p^2+1)/2 is also prime, exhibit a sharp dichotomy based on the solution of the aforementioned Diophantine equation: either rank(E_p(Q)) = 2 with trivial Shafarevich-Tate group or rank = 0 with III(E_p/Q)[2] = (Z/2Z)^2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_03486 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Diophantine Criterion for the Shafarevich-Tate Groups of Elliptic Curves from Heron Triangles Ghale, Vinodkumar Number Theory 11G05, 11G07, 11G40, 11D25, 11G10, 14H52 The solvability of Diophantine quartic equations is a contemporary area of interest due to its connection with generalized Fermat's equation. In this work, we are interested in the integer solutions of a similar Diophantine equation p u^2 = v^2 + w^2. For a particular form of u, v, and w, we prove that the elliptic curves E_p: y^2 = x(x-1)(x+p^2), which arise from Heron triangles, for primes p = 1 (mod 8) where q = (p^2+1)/2 is also prime, exhibit a sharp dichotomy based on the solution of the aforementioned Diophantine equation: either rank(E_p(Q)) = 2 with trivial Shafarevich-Tate group or rank = 0 with III(E_p/Q)[2] = (Z/2Z)^2. |
| title | A Diophantine Criterion for the Shafarevich-Tate Groups of Elliptic Curves from Heron Triangles |
| topic | Number Theory 11G05, 11G07, 11G40, 11D25, 11G10, 14H52 |
| url | https://arxiv.org/abs/2301.03486 |