zbMATH — the first resource for mathematics

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.

11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Full Text: DOI
[2] INVENT MATH 109 pp 307– (1992) · Zbl 0781.14022
[3] ANN MATH 142 pp 443– (1995)
[4] INVENT MATH 44 pp 129– (1978) · Zbl 0386.14009
[5] INVENT MATH 63 pp 225– (1981) · Zbl 0459.10018
[6] AM J MATH 110 pp 323– (1988) · Zbl 0645.10029
[7] COMPOSITIO MATH 99 pp 283– (1995)
[8] ANN MATH 142 pp 553– (1995)
[9] PUBL IHES 71 pp 65– (1990) · Zbl 0744.11053
[10] COMPOSITIO MATH 65 pp 265– (1988)
[11] DUKE MATH J 59 pp 629– (1989) · Zbl 0707.11079
[12] INVENT MATH 117 pp 89– (1994) · Zbl 0819.11047
[13] PERSPECT MATH 11 pp 93– (1990)
[14] INVENT MATH 124 pp 1– (1996) · Zbl 0853.11059
[15] C R ACAD SCI SER I 321 pp 5– (1995)
[16] ANN MATH 131 pp 493– (1990) · Zbl 0719.11071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.