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Adjoint modular Galois representations and their Selmer groups. (English) Zbl 0909.11025
Summary: In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation $$\text{ad}(\varphi)$$ of a two-dimensional modular Galois representation $$\varphi$$. We start with the $$p$$-adic Galois representation $$\varphi_0$$ of a modular elliptic curve $$E$$ and present a formula expressing in terms of $$L(1,\text{ad} (\varphi_0))$$ the intersection number of the elliptic curve $$E$$ and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group $$\text{Sel(ad} (\varphi_0))$$ from the proof of Wiles of the Shimura-Taniyama conjecture.
After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, $$T$$, is the weight variable of the universal $$p$$-ordinary Hecke algebra, and the second variable is the cyclotomic variable $$S$$. In the one-variable case, we let $$\varphi$$ denote the $$p$$-ordinary Galois representation with values in $$GL_2 (\mathbb{Z}_p [[T]])$$ lifting $$\varphi_0$$, and the characteristic power series of the Selmer group $$\text{Sel(ad} (\varphi))$$ is given by a $$p$$-adic $$L$$-function interpolating $$L(1,\text{ad} (\varphi_k))$$ for weight $$k+2$$ specialization $$\varphi_k$$ of $$\varphi$$. In the two-variable case, we state a main conjecture on the characteristic power series in $$\mathbb{Z}_p[[T,S]]$$ of $$\text{Sel(ad} (\varphi) \otimes \nu^{-1})$$, where $$\nu$$ is the universal cyclotomic character with values in $$\mathbb{Z}_p[[S]]$$. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using $$p$$-adic Siegel modular forms.

##### MSC:
 11F80 Galois representations 11F33 Congruences for modular and $$p$$-adic modular forms 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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##### References:
  SMINAIRE DE THORIE DES NOMBRES PARIS LMS LECTURE NOTES SERIES 235 pp 89– (1996)  INVENT MATH 109 pp 307– (1992) · Zbl 0781.14022  ANN MATH 142 pp 443– (1995)  INVENT MATH 44 pp 129– (1978) · Zbl 0386.14009  INVENT MATH 63 pp 225– (1981) · Zbl 0459.10018  AM J MATH 110 pp 323– (1988) · Zbl 0645.10029  COMPOSITIO MATH 99 pp 283– (1995)  ANN MATH 142 pp 553– (1995)  PUBL IHES 71 pp 65– (1990) · Zbl 0744.11053  COMPOSITIO MATH 65 pp 265– (1988)  DUKE MATH J 59 pp 629– (1989) · Zbl 0707.11079  INVENT MATH 117 pp 89– (1994) · Zbl 0819.11047  PERSPECT MATH 11 pp 93– (1990)  INVENT MATH 124 pp 1– (1996) · Zbl 0853.11059  C R ACAD SCI SER I 321 pp 5– (1995)  ANN MATH 131 pp 493– (1990) · Zbl 0719.11071
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