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Stark's conjecture applied to the associated Galois representation of a newform f(z)=anqnf(z)=\sum a_n q^n of weight one [MR:0782485] states the following. Let E=Q((an)nN)E=\mathbb{Q}((a_n)_{n \in \mathbb{N}}), Δ=Gal(E/Q)\Delta=\text{Gal}(E/\mathbb{Q}) and fα(z)=α(an)qnf^\alpha(z)=\sum \alpha(a_n) q^n for αΔ\alpha \in \Delta. Let L(s,f)L(s, f) be the L-function of ff. Then, for all bEb \in E^* there exists an integer m1m \geq 1 and a unit ε\varepsilon in the Artin field of ff, called the Stark unit, such that

emαΔα(b)L(0,fα)=ε e^{m \sum_{\alpha \in \Delta} \alpha(b)L'(0, f^\alpha)} = \varepsilon

In the case where the coefficients of Tr(bf)\text{Tr}(bf) are in Z\mathbb{Z}, Chinburg further conjectured that there exists a Stark unit for m=1m=1 [10.1016/0001-8708(83)90006-3]. Notice that if we choose b=1b = 1, the preceding condition always holds. Here, we compute the Stark unit of the newform for b=1b=1 and m=1m=1.

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  • Last edited by David Roe on 2024-10-16 17:44:28
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