The equivariant Tamagawa number conjecture and modular symbols
Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\), \(K/\mathbb{Q}\) a finite Galois extension with \(G:=\mathrm{Gal}(K/\mathbb{Q})\) and \(E_K\) the base change of \(E\) to \(K\). The paper deals with the \(\ell\)-part of the Equivariant Tamagawa Number Conjecture (ETNC\(_\ell\) from now on), as formulated in [\textit{D. Burns} and \textit{M. Flach}, Doc. Math., J. DMV 6, 501--570 (2001; Zbl 1052.11077)], for the motive \(h^1(E_K)(1)\) with the action of \(\mathbb{Q}[G]\) and for \(\ell\)-\textit{arithmetically trivial pairs} \((E,K)\). A pair \((E,K)\) is called \(\ell\)-arithmetically trivial if it verifies: \noindent a) \([K:\mathbb{Q}]\) is a power of \(\ell\); \noindent b) \((d_K,\ell)=(N_E,\ell)=(d_K,N_E)=1\) (\(d_K\) is the discriminant of \(K\), \(N_E\) is the conductor of \(E\)); \noindent c) \(\ell \nmid \# E(\mathbb{Q})_{\mathrm{tors}} \prod_{p\mid d_K} \# \overline{E}(\mathbb{F}_p)\) (\(\overline{E}\) is the reduced curve); \noindent d) \(\ell \nmid \prod_{p\mid N_E} c_p(E,\mathbb{Q})\) (\(c_p(E,\mathbb{Q})\) is the Tamagawa factor at \(p\)); \noindent e) rank\(_{\mathbb{Z}}\) \((E(K))=0\); \noindent f) \(\ell\) does not divide the order of the Tate-Shafarevich group of \(E\) over \(K\). \noindent In a previous paper [Math. Comput. 81, No. 279, 1681--1705 (2012; Zbl 1277.11071)] the author proved the equivalence between the ETNC\(_\ell\) and a congruence, modulo \(\mathbb{Z}_\ell[G]^*\,\), between the product on primes dividing \(d_K\) of local Euler factors (twisted by the absolutely irreducible characters \(\chi\) of \(G\)) for \(E/\mathbb{Q}\) at \(s=1\) and the leading coefficients for the (twisted) \(L\)-functions \((L(E/\mathbb{Q},\overline{\chi},s))_\chi\) at \(s=1\). The rationality and an explicit description of the last object are the main features of the ETNC and the ETNC\(_\ell\) simply considers the same problem modulo \(\ell\)-adic units. \noindent In the paper under review the author proves first the congruence mentioned above (hence the ETNC\(_\ell\)) for \(\ell\)-arithmetically trivial pairs \((E,K)\) such that \noindent 1) \(K/\mathbb{Q}\) is abelian; \noindent 2) \(\ell\) does not divide the product of the Manin constant for the modular parametrization of \(E\) and of the order of the Tate-Shafarevich group for \(E\) over \(\mathbb{Q}\); \noindent 3) the \(\ell\)-part of the Birch and Swinnerton-Dyer conjecture holds for \(E\) over \(\mathbb{Q}\). \noindent The proof uses the modular symbols of [\textit{B. Mazur} and \textit{J. Tate}, Duke Math. J. 54, 711--750 (1987; Zbl 0636.14004)] to compute the elements under investigation in terms of the quantities mentioned before (\(E(\mathbb{Q})_{\mathrm{tors}}\,\), Tamagawa factors, Tate-Shafarevich groups and so on), then the hypotheses and direct computations show that their quotient lies in \(\mathbb{Z}_\ell[G]^*\,\). In the last section the author produces explicit examples of extensions \(K/\mathbb{Q}\) of degree \(\ell\) satisfying all the hypotheses and shows that the composite of such extensions still verifies them. This procedure provides (for any elliptic curve \(E\) with \(L(E/\mathbb{Q},1)\neq 0\)) infinitely many \(\ell\) and (for each \(\ell\)) infinitely many abelian \(\ell\)-extensions \(K/\mathbb{Q}\) for which ETNC\(_\ell\) holds for the pair \((h^1(E_K)(1),\mathbb{Z}[G])\).
scientific article; zbMATH DE number 6168467
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