D. W. Anderson, E. H. Brown, J. , and F. P. Peterson, Pin cobordism and related topics, Comment. Math. Helv, vol.44, pp.462-468, 1969.

J. F. Adams, On the groups J(X). IV. Topology, vol.5, pp.21-71, 1966.

J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, vol.77, 1974.

T. Bauer, Computation of the homotopy of the spectrum tmf, Groups, homotopy and configuration spaces, vol.13

. Topol and . Monogr, Geom. Topol. Publ, vol.108, p.85, 2008.

T. Barthel, A. Beaudry, P. G. Goerss, and V. Stojanoska, Constructing the determinant sphere using a Tate twist, vol.37, p.124, 2018.

H. Edgar, J. Brown, and M. Comenetz, The Pontrjagin dual of a spectrum, New developments in topology (Proc. Sympos. Algebraic Topology, p.36, 1972.

A. Beaudry, The algebraic duality resolution at p = 2, Algebr. Geom. Topol, vol.15, issue.6, pp.3653-3705, 2015.

A. Beaudry, Towards the homotopy of the K(2)-local Moore spectrum at p = 2, Adv. Math, vol.306, pp.722-788, 2017.

M. Behrens, A modular description of the K(2)-local sphere at the prime 3, Topology, vol.45, issue.2, pp.343-402, 2006.

M. Behrens, The homotopy groups of S E(2) at p ? 5 revisited, Adv. Math, vol.230, issue.2, pp.458-492, 2012.

P. Bhattacharya, P. Egger, and M. Mahowald, On the periodic v 2 -self-map of A 1, Algebr. Geom. Topol, vol.17, issue.2, pp.657-692, 2017.

A. K. Bousfield and E. M. Friedlander, Homotopy theory of ?spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf, vol.II, p.17, 1977.

I. Bobkova and P. G. Goerss, Topological resolutions in K(2)-local homotopy theory at the prime 2, Topology, vol.11, issue.4, pp.918-957, 2018.

A. Beaudry, P. G. Goerss, and H. Henn, Chromatic splitting conjecture for the K(2)-local sphere at p = 2, vol.33, 2017.

M. Behrens and K. Ormsby, On the homotopy of Q(3) and Q(5) at the prime 2, Algebr. Geom. Topol, vol.16, issue.5, p.94, 2016.

A. K. Bousfield, The localization of spectra with respect to homology, Topology, vol.18, issue.4, pp.257-281, 1979.

K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol.87, 1982.

E. S. Devinatz, Morava's change of rings theorem, The?ech centennial, vol.181, pp.83-118, 1993.

C. L. Douglas, J. Francis, A. G. Henriques, and M. A. Hill, Topological modular forms, Mathematical Surveys and Monographs, vol.201, 2014.

E. S. Devinatz and M. J. Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology, vol.43, issue.1, pp.1-47, 2004.

M. Donald, M. Davis, and . Mahowald, v 1 -and v 2 -periodicity in stable homotopy theory, Amer. J. Math, vol.103, issue.4, pp.615-659, 1981.

M. Donald, M. Davis, and . Mahowald, Ext over the subalgebra A 2 of the Steenrod algebra for stunted projective spaces, Current trends in algebraic topology, Part 1, vol.2, pp.297-342, 1981.

M. Donald, M. Davis, and . Mahowald, Connective versions of T M F (3), Int. J. Mod. Math, vol.5, issue.3, pp.223-252, 2010.

P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, vol.349, p.27, 1972.

P. Gabriel, Des catégories abléliennes, Bull. Soc. math. France, vol.90, p.164, 1962.

P. G. Goerss and M. J. Hopkins, Moduli spaces of commutative ring spectra, Structured ring spectra, vol.315, p.22, 2004.

G. Paul, H. Goerss, and . Henn, The Brown-Comenetz dual of the K(2)-local sphere at the prime 3, Adv. Math, vol.288, pp.648-678, 2016.

P. Goerss, H. Henn, and M. Mahowald, The homotopy of L 2 V (1) for the prime 3, Categorical decomposition techniques in algebraic topology, vol.215, pp.125-151, 2001.
URL : https://hal.archives-ouvertes.fr/hal-00129678

P. Goerss, H. Henn, M. Mahowald, and C. Rezk, A resolution of the K(2)-local sphere at the prime 3, Ann. of Math, vol.162, issue.2, pp.777-822, 2005.

P. Goerss, H. Henn, M. Mahowald, and C. Rezk, On Hopkins' Picard groups for the prime 3 and chromatic level 2, J. Topol, vol.8, issue.1, pp.267-294, 2015.

P. G. Goerss, Quasi-coherent sheaves on the moduli stack of formal groups, p.22, 2008.

H. Henn, On finite resolutions of K(n)-local spheres, Elliptic cohomology, vol.342, pp.122-169, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00129702

M. J. Hopkins and B. H. Gross, The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc, vol.30, issue.1, pp.76-86, 1994.

N. Hans-werner-henn, M. Karamanov, and . Mahowald, The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited, Math. Z, vol.275, issue.3-4, p.32, 2013.

M. Hill and T. Lawson, Topological modular forms with level structure, Invent. Math, vol.203, issue.2, p.29, 2016.

D. Heard, G. Li, and X. Shi, Picard groups and duality for real morava e-theories, vol.124, 2018.

M. J. Hopkins, M. Mahowald, and H. Sadofsky, Constructions of elements in Picard groups, Topology and representation theory, vol.158, pp.89-126, 1992.

M. Hovey, Operations and co-operations in Morava E-theory

, Homology Homotopy Appl, vol.6, issue.1, p.158, 2004.

J. Michael, J. H. Hopkins, and . Smith, Nilpotence and stable homotopy theory, II. Ann. of Math, vol.148, issue.2, pp.1-49, 1998.

M. Hovey and N. P. Strickland, Morava K-theories and localisation, Mem. Amer. Math. Soc, vol.139, issue.666, p.167, 1999.

O. Stanley and . Kochman, Stable homotopy groups of spheres, Lecture Notes in Mathematics, vol.1423, p.73, 1990.

M. Lazard, Sur les groupes de Lie formels à un paramètre, Bull. Soc. Math. France, vol.83, pp.251-274, 1955.

M. Lazard, Groupes analytiques p-adiques, Inst. Hautes Études Sci. Publ. Math, issue.26, pp.389-603, 1965.

J. Lubin and J. Tate, Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France, vol.94, p.22, 1966.

A. Mathew, The homology of tmf, Homology Homotopy Appl, vol.18, issue.2, p.28, 2016.

J. Milnor, The Steenrod algebra and its dual, Ann. of Math, vol.67, issue.2, pp.150-171, 1958.

S. A. Mitchell, Finite complexes with A(n)-free cohomology, Topology, vol.24, issue.2, pp.227-246, 1985.

R. and M. F. Moss, Secondary compositions and the Adams spectral sequence, Math. Z, vol.115, p.73, 1970.

C. Douglas and . Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math, vol.106, issue.2, p.35, 1984.

C. Douglas and . Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol.121, 1986.

C. Douglas and . Ravenel, Nilpotence and periodicity in stable homotopy theory, vol.128

C. Rezk, Notes on the Hopkins-Miller theorem, Homotopy theory via algebraic geometry and group representations, vol.220, p.22, 1997.

J. Rognes, The adams spectral sequence

K. Shimomura, The homotopy groups of the L 2 -localized Toda-Smith spectrum V (1) at the prime 3, Trans. Amer. Math. Soc, vol.349, issue.5, pp.1821-1850, 1997.

K. Shimomura, The homotopy groups of the L 2 -localized mod 3 Moore spectrum, J. Math. Soc. Japan, vol.52, issue.1, pp.65-90, 2000.

J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol.106, 2009.

V. Stojanoska, Calculating descent for 2-primary topological modular forms. In An alpine expedition through algebraic topology, Contemp. Math, vol.617, p.27, 2014.

N. P. Strickland, Gross-Hopkins duality, Topology, vol.39, issue.5, pp.1021-1033, 2000.

P. Symonds and T. Weigel, Cohomology of p-adic analytic groups, New horizons in pro-p groups, vol.184, p.23, 2000.

K. Shimomura and X. Wang, The homotopy groups ? * (L 2 S 0 ) at the prime 3, Topology, vol.41, issue.6, pp.1183-1198, 2002.

K. Shimomura and A. Yabe, The homotopy groups ? * (L 2 S 0 ), Topology, vol.34, issue.2, pp.261-289, 1995.

C. Martin and . Tangora, On the cohomology of the Steenrod algebra, Math. Z, vol.116, p.152, 1970.

H. Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, issue.49, p.73, 1962.