## Tokyo Journal of Mathematics

### On a Conjecture for Rubin-Stark Elements in a Special Case

Takamichi SANO

#### Abstract

A conjecture on a relation between two different Rubin-Stark elements was recently proposed by the author, and also by Mazur and Rubin. For a tower of finite extensions of global fields $K/L/k$ such that $K/k$ is abelian, this conjecture gives a relation between Rubin-Stark elements $\varepsilon_{K,S,T,V}$ and $\varepsilon_{L,S,T,V'}$, where $S, T, V$ and $V'$ are suitable sets of places of $k$. In this paper, we prove this conjecture under the following three assumptions: (i) $V$ contains all infinite places of $k$; (ii) all $v\in S$ split completely in $L$; (iii) $\mathrm{Gal}(K/L)$ is the direct product of the inertia groups at $v\in S\setminus V$.

#### Article information

Source
Tokyo J. Math., Volume 38, Number 2 (2015), 459-476.

Dates
First available in Project Euclid: 14 January 2016

https://projecteuclid.org/euclid.tjm/1452806050

Digital Object Identifier
doi:10.3836/tjm/1452806050

Mathematical Reviews number (MathSciNet)
MR3448867

Zentralblatt MATH identifier
1377.11117

#### Citation

SANO, Takamichi. On a Conjecture for Rubin-Stark Elements in a Special Case. Tokyo J. Math. 38 (2015), no. 2, 459--476. doi:10.3836/tjm/1452806050. https://projecteuclid.org/euclid.tjm/1452806050

#### References

• Burns, D., Congruences between derivatives of abelian $L$-functions at $s=0$, Invent. Math. 169 (2007), 451–499.
• Burns, D., Kurihara, M. and Sano, T., On arithmetic properties of zeta elements, I, preprint (2014), arXiv:1407.6409v1.
• Darmon, H., Thaine's method for circular units and a conjecture of Gross, Canad. J. Math. 47 (1995), 302–317.
• Gross, B., On the values of abelian L-functions at $s=0$, J. Fac. Sci. Univ. Tokyo 35 (1988), 177–197.
• Mazur, B. and Rubin, K. Refined class number formulas for $\mathbb{G}_{m}$, preprint (2013), arXiv:1312.4053v1.
• Rubin, K., A Stark conjecture “over $\bbZ$” for abelian $L$-functions with multiple zeros, Ann. Inst. Fourier (Grenoble) 46 (1996), 33–62.
• Sano, T., Refined abelian Stark conjectures and the equivariant leading term conjecture of Burns, Compositio Math. 150 (2014), 1809–1835.
• Serre, J.-P., Local Fields, Graduate Texts in Math. 67, Springer Verlag (1979).
• Tate, J., Les conjectures de Stark sur les fonctions $L$ d'Artin en $s=0$, vol. 47 of Progress in Mathematics, Boston, Birkh”auser (1984).