In this thesis, we compute the mod-preduction of 2-dimensional semi-stable GQ representations. We also use the strongly divisible moduli parameterization we found to construct semi-stable deformation rings in parallel Hodge–Tate weights (0,1). We also determine the irreducible components of semistable deformation rings in parallel Hodge–Tate weights(0,1).

Therefore, we can compute the mod-preduction of semi-stable representations by computing the corresponding strongly separable moduli and Breuil moduli. We also recall the results of Kisin [4] for the existence of semi-stable deformation rings. In Section 5, we construct the required strongly separable modules, and their Breuil modules, of 2-dimensional semistable E-noncrystalline representations of GQ.

In Section 6, we construct irreducible components of semistable deformation rings over GQ.

## Witt vectors

In this section we study the ring of Witt vectors and period rings BdR,Bcris andBst to start our study of integral p-adic Hodge theory in the next section. Letx∈A/a1 and we can take a unique sequence{xn}n≥0inA/a1such thatx0=xand xn+1p =xnasA/a1is a perfect ring. To show that it is well defined, we need to show that it is independent of the choice of lift of {xn}n≥0.

Since the residual rings S/pb Sb=Fp[Xαp−∞] is perfect for characteristic p andS/pb nSb∼=S/pnS, i.e. If it is an absolutely unbranched, complete discrete valuation ring of characteristic 0 with a perfect residual field of characteristic p , then it is a strictp ring. If there exists a strict p-ringA with restringk, then there also exists a strict p-ringA′with restringk′.

We already know that Sbis is a strict p-ring with residue ringFp[Xαp−∞]and every perfect field of characteristic pis of the form a quotient of Fp[Xαp−∞].

## The ring R and its structure

Since πe=p×(unit), we can write unique. whereAn=Aand the transition map is the Frobenius map φ:A→A, a7→ap. Let K be an ap-adic field, OK its ring of integers, mK its maximum ideal with a generator π, and k its residual field perfect of characteristic p. Let Cp be the completion of Qp, Zp the integer ring of Qp, and OCp the completion of Zp.

The valuation can be extended to FrR, therefore the ring is a complete non-Archamedian perfect field with characteristic p with the ring of integers. By Theorem 2.13, the field is perfect, which has a corresponding condition: the separable closure of the field is algebraically closed. Therefore, it is sufficient to show that the field is separable closed, i.e. if a monically separable polynomial.

Let∈OCp/pOCp be the image iu∈OCp. Since θ0 is surjective, then there exists x∈Re such that θ0(x) =u, so θ0(P(x)) =0.

We can obtain the lemma asn →∞. According to comment 3 we can write xasx=. From now on, without further notice, we will adopt the following notations. Since (ξ) is a maximal ideal, B+dRis is a complete discrete valuation ring with the residual fieldCp, equipped with a continuous GK0 action. Since we can identify WL(R) withWK0(R), thenK⊆B+dR. t∈Fil1BdR and t∈/Fil2BdR, In other words: t generates the maximum ideal of B+dR. The proof of Theorem 6.22 in Fontaine-Ouyang [6]).

These representations are called the Tate turns of Zℓ, and V(r) =V⊗QpQℓ(r) are called the Tate turns of V for anyℓ-adic representationV. The Hodge–Tate weights of ap-adic representationV of GKarei∈Zsatisfying dimK(Fil−iBdR/Fil−i+1BdR⊗QpV)GK̸=0.

## Crystalline period ring B cris

Ker ¯θ= (Kerθ,p) is also a distributed power ideal. The proof of Theorem 7.5 in Fontaine-Ouyang [6]) Since p divides m!pm into Zp, it is clear.

## Semi-stable period ring B st

Furthermore, let Ccris and Cst denote the fraction fields Bcris and Bst, then both are stable under the actions GK and GK0 and the Frobenius mappingφ onto Bcris extends to Ccris. According to Lemma 7.2 in Fontaine–Ouyang [6], S is separated by a p-adic topology, then it suffices to show that for r∈N,α ∈S−pS,. Since the minimal polynomial is unique, then g(cd−1) +dη(g)t=cd−1 holds for each g∈GK0. ι:K⊗K0Bcris→BdR with λ⊗x7→λx is injective. A0cris,OK/pn⊂B+dR,K=B+dR, so consequently ι is injective.

The monodromy operator is a nilpotent operator satisfying the (1) sequence. The proof of Proposition 8.6 in Fontaine–Ouyang [6]) (1) By the definition of N, it is obvious. In this section we study semi-stable E-representations of GK, admissible filtered(φ,N)-modules and the relationship between these categories.

Semi-stable p-adic Galois representations

## Admissible filtered (φ, N)-modules

0D by the wedge product relation, and will be a filtered module (φ,N) with the image structure of ⊗dK. The ap-adic representation V is crystalline if and only if it is semistable with N=0 of Dst(V). Let V be a finite-dimensional GaloisE representation of GK, and let V∨ be the dual representation of V.

If D∗st:=Dst(V∨) is defined, then D∗st creates an anti-equivalence between categories whose quasi-inverse is given by . We can define the Hodge–Tate weights of D∗st(V) as those of the Hodge–Tate weights of a semistable representationV. Then we can give another definition of the Hodge–Tate weight for D∗st(V) as the integers such that FiljDK̸=Filj+1DK, each numbered by dimE multiple FiljDK.

Let D=Dst(V) be an admissible filtered (φ,N) module corresponding to a two-dimensional semi-stable non-crystalline p-adicE representationV of GQp with Hodge-Tate weights(0,r).

## Strongly divisible modules

S⊗W(k)D is an equivalence of categories, where FM+K,E(φ,N) is a category whose objects are filtered (φ,N)-modules K,E with positive Hodge–Tate weights. For a strongly divisible module Mof weight r, there exists a unique admissible filtered (φ,N)-moduleD with Hodge–Tate weights lying in [0,r] such that. Let D be an admissible filtered (φ,N)-module K=Qpof of the second rank with Hodge–Tate weights (0,1), i.e. there exists a basee= (e1,e2) such that D=E·(e1,e2) together with .

E GIs the category of GK-stableOE lattices in semi-stable E-representations of GKwith Hodge–Tate weights lying in[0,r], provided 0≤r≤p−2, and where. Therefore all adjuncts of H in GK are also open, and the adjuncts of H form an open cover of G.

## Breuil modules

There exists a natural bisection between the set of neighbors of [Λ] and the set of proper non-trivial F-subspaces of Λ/pΛ. Then the remaining representationρΛ is irreducible over Fif and only ifΛ is the only stable lattice up to homothety. If the remaining representationρΛ is nonsplit reducible over F, thenΛ is a unique stable lattice up to homothety.

Potentially semi-stable representations

Galois groups

## Deformation rings

A representation ρ0:G→ GLn(F) is called reducible if the representation space Fn has a proper subspace that is invariant under the action of G. Finally, it is called absolutely irreducible if there is no extension F′/F such thatρ0⊗FF′ is reducible. Artinian ring (sometimes Artin ring) is a ring such that there is no infinite descending sequence of ideals.

Let C0(resp.CΛ0) be the full subcategory of C(resp.CΛ) whose objects are artin local rings (resp.Λ-algebra)With residual fieldF. Let pu:G→GLn(Rρ0) be the universal deformation for Dρ0, and letπ:Rρ0 ↠RP be the quotient map.

## Weil-Deligne representations

0,OE→Qp results in a potentially semistable representation with Hodge–Tate weights and Galois typeγ if and only if the x-factors throughπst. In this section, we review the result of Dousmanis [1] to find all admissible filtered (φ,N)-modules corresponding to all 2-dimensional semi-stable representations of GQp with a parallel denoted by Hodge–.

## Admissible filtered (φ, N)-modules

Since Dis is free of rank two overEf, the componentsDτi are equal-dimensional overE, each of dimension two. The labeled Hodge–Tate weights of Dis defined(Wi)τi, where Wi is the set containing the labeled Hodge–. Let D be a ranked two-filteredφ-module with an ordered basis η of D with respect to [φ]η =diag(pλ·⃗1,λ·⃗1)for someλ∈E×.

Έστω D μια φιλτραρισμένη(φ,N)-μονάδα της κατάταξης δύο με διατεταγμένη βάσηηofD σε σχέση με [φ]η =diag(pλ·⃗1,λ·⃗1)για μερικάλ ∈E×και[N]η.

Another admissible filtered (φ ,N)-modules

Strongly divisible module (Case 1)

Breuil module (Case 1)

Strongly divisible module (Case 2)

## Breuil module (Case 2)

In this section we find the irreducible components of the deformation ring in parallel Hodge–Tate weights(0,1) of a nonsplit residual representationρ0:GQf →GL2(F). We define the category MDrR of strongly divisible SR modules as follows: an object in MDrR consists of quadruple (M,FilrM,φr,N)where. Moreover, the condition (7) is equivalent toNφ=pφNonM. 2) When R=OE, the above definition is equivalent to the definition of a strongly divisible module in Section 3.3. 3) M/(mR,FilpSR) obviously has a structure of Breuil module over(k⊗FpF)[u]/(op).

Let R be an object in C, I be an ideal containing mnR for somen>0, and M be a strongly divisible module of weightroverSR.

## Case 1

M+(X0, · · ·,Xf−1) becomes a highly divisible module if and only if specializationXiinmg leads to a highly divisible module on state-owned enterprises. Since OE[[X0,· · ·,Xf−1]]isp torsion-free and has a reduced generic fiber, there exists an OE-algebra morphismq•:R(0,1),ψρ. Note that the characteristic 0 closed points of these irreducible components exhaust all two-dimensional semi-stable lifts of ρ0 with parallel Hodge-Tate weights (0,1).

## Case 2

Dousmanis, "Rank two filtered(φ,n) modules with Galois descent data and coefficients," Transactions of the American Mathematical Society, vol. Liu, "On lattices in semi-stable representations: a proof of a Breuil conjecture," Compositio Mathematica, vol. Mézard, "Multiplicityes modulaires et représentations de GL2(Zp) et de Gal(Qp/Qp)enl=p,"Duke Mathematical Journal, vol. 2022) Theory of p-adic galois representations.

Available: http://staff.ustc.edu.cn/~yiouyang/galoisrep.pdf. 2001) p-adic mixture theory, deformations and local longlands. First of all, I would like to express my sincere appreciation to my advisor, Chol Park, for everything I learned from him. He was the best advisor I could ever meet and the best mentor for study and research in mathematics.

In addition to my advisor, I would like to thank the other members of the thesis committee: Hae-sang Sun and Peter Jaehyun Cho for their insightful comments and questions. They were good mathematics lecturers and good assistants in preparing for my high school graduation and studying abroad. I also thank the UNIST number theory group, especially the graduate students in the arithmetic geometry lab: Jeonghyo Park, Seongjae Han, and Euntaek Lee, for the fun we had.

Not only at UNIST, but also at Pusan National University, where I studied as an undergraduate, there are people to whom I must be grateful. I was able to get interested in Algebra and Number Theory through Mitsugu Hirasaka and Donghi Lee. I was able to get interested in math because they taught me math straight through high school.

I also want to thank my younger brother, Hyeonwoo Cheon, who became my best friend in my life. She always made me laugh, helped me not to get tired and prayed for me.