Analogue of the Brauer-Siegel theorem for Legendre elliptic curves. (English) Zbl 1440.11093
Summary: We prove an analogue of the Brauer-Siegel theorem for the Legendre elliptic curves over \(K = \mathbb{F}_q(t)\). Namely, denoting by \(E_d\) the elliptic curve with model \(y^2 = x(x + 1)(x + t^d)\) over \(K\), we show that, for \(d \rightarrow \infty\) ranging over the integers, one has
\[
\log(| \mathrm{III}(E_d / K) | \cdot \mathrm{Reg}(E_d / K)) \sim \log H(E_d / K) \sim \frac{\log q}{2} \cdot d .
\]
Here, \(H(E_d / K)\) denotes the exponential differential height of \(E_d\), \(\mathrm{III}(E_d / K)\) its Tate-Shafarevich group (which is known to be finite), and \(\mathrm{Reg}(E_d / K)\) its Néron-Tate regulator.
MSC:
| 11G05 | Elliptic curves over global fields |
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 11M99 | Zeta and \(L\)-functions: analytic theory |
| 11R47 | Other analytic theory |
Keywords:
elliptic curves over function fields; \(L\)-functions and BSD conjecture; estimates on special valuesReferences:
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