在本論文中，我們關注兩類t模，包括德林費爾模和卡利茨模的張量冪。在第一部分，我們關注這些t模上代數點之間的線性關係。更準確地說，我們研究了在這些t模上有限多個代數點馬瑟定理[Mas88]的類比。我們估計出這些t模上代數點之間線性關係 生成元大小的上界。在第二部分，我們研究了正特徵中的多重zeta值。我們討論了一些線性獨立結果，並為某些多重zeta值所張出的向量空間建立了維度的下界。最後，我們研究了[CM21]中引入的v-進位多重zeta值。我們證明對於任何固定指標，v-進位多重zeta值對於除了有限多個素點v之外的所有素點都是v-進位整數。此外，我們證明了某些情況下古庄-山下猜想對於v-進位多重zeta值的類比。

In this dissertation, we focus on two families of t-modules including Drinfeld modules and tensor powers of the Carlitz module. In the first part, we focus on the linear relations among algebraic points on these $t$-modules. More precisely, we study an analogue of Masser's theorem [Mas88] for finitely many algebraic points on these t-modules and consequently we derive an upper bound for the generators of F_q[t]-linear relations among them. In the second part, we study multiple zeta values in positive characteristic. We discuss some linear independence results and we established a lower bound of the dimension for certain vector spaces spanned by these values. Finally, we investigate v-adic multiple zeta values introduced in [CM21]. We prove that for any fixed index v-adic multiple zeta values are v-adic integers for all but finitely many places. Furthermore, we prove an analogue of a conjecture of Furusho and Yamashita for v-adic multiple zeta values for v a finite place of degree one.

1 Introduction 1
1.1 An analogue of Masser’s theorem . . . . . . . . . . . . . . . 2
1.2 ∞-adic multiple zeta values . . . . . . . . . . . . . . . . . 5
1.3 v-adic multiple zeta values . . . . . . . . . . . . . . . . . 7
1.4 Outline of the dissertation . . . . . . . . . . . . . . . . . 9
2 Preliminaries 11
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Anderson t-modules and dual t-motives . . . . . . . . . . . . 11
3 Linear equations on t-modules 20
3.1 Siegel’s lemma for function fields and lattices over Fq[t] . .20
3.2 Linear equations on Drinfeld modules . . . . . . . . . . . . .25
3.3 Tensor powers of the Carlitz module . . . . . . . . . . . . . 42
4 Multiple zeta values in positive characteristic 50
4.1 A linear independence criterion . . . . . . . . . . . . . . . 50
4.2 On a lower bound of the dimensions of MZVs . . . . . . . . . .62
5 Integrality of v-adic multiple zeta values 66
5.1 Background of Chang and Mishiba’s v-adic multiple zeta values 67
5.2 Integrality of v-adic multiple zeta values . . . . . . . . . .73
6 Transcendence of v-adic multiple zeta values 87
6.1 A refined version of the ABP linear independence criterion . .87
6.2 Linear independence of θ-adic multiple zeta values . . . . . 100
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