2025/2026 SEMINARS |
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---|---|---|---|
Math 208 - Algebraic Geometry |
Oprea, Dragos |
Oprea, Dragos |
Oprea, Dragos |
Math 209 - Number Theory |
Bucur, Alina |
Bucur, Alina |
Bucur, Alina |
Math 211A - Algebra |
Golsefidy, Alireza |
Golsefidy, Alireza |
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Math 211B - Group Actions |
Frisch, Joshua |
Frisch, Joshua |
Frisch, Joshua |
Math 218 - Biological Systems |
Miller, Pearson |
Miller, Pearson |
Miller, Pearson |
Math 243 - Functional Analysis |
Ganesan, Priyanga & Vigdorovich, Itamar |
Ganesan, Priyanga & Vigdorovich, Itamar |
Vigdorovich, Itamar |
Math 248 - Real Analysis |
Bejenaru, Ioan |
Bejenaru, Ioan |
Bejenaru, Ioan |
Math 258 - Differential Geometry |
Spolaor, Ioan |
Spolaor, Ioan |
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Math 268 - Logic |
TBD |
TBD |
TBD |
Math 269 - Combinatorics |
Rhoades, Brendon & Warnke, Lutz |
Rhoades, Brendon & Warnke, Lutz |
Rhoades, Brendon & Warnke, Lutz |
Math 278A - CCOM |
Cheng, Li-Tien |
Cheng, Li-Tien |
Cheng, Li-Tien |
Math 278B - Math of Info, Data |
Cloninger, Alexander |
Cloninger, Alexander |
Cloninger, Alexander |
Math 278C - Optimization |
Nie, Jiawang |
Nie, Jiawang |
Nie, Jiawang |
Math 288A - Probability |
Peca-Medlin, John |
Peca-Medlin, John |
Peca-Medlin, John |
Math 288B - Statistics |
TBD |
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Math 292 - Topology Seminar |
Chow, Bennett |
Chow, Bennett |
Chow, Bennett |
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10:00 am
Xie Wu
Iwasawa theory of Taelman class modules of t-modules
Thesis Defense
APM 6402
AbstractIn 1974, Drinfeld published his great paper Elliptic Modules in which he introduced what we now call arbitrary rank Drinfeld modules. Let $k$ be the field of rational functions on a smooth, projective, geometrically connected curve $X$ defined over the finite field $\mathbb{F}_q$, and $K$ be a finite separable extension of $k$. Let $\infty$ be a closed point of $X$, and $A\subset k$ be the ring of functions regular away from $\infty$. A Drinfeld module defined over $K$ is an $\mathbb{F}_q$-algebra morphism from $A$ to a twisted polynomial ring with coefficients in $K$, satisfying some leading term condition. The notion of a Drinfeld module has been historically proven to be highly powerful. In 1986, Anderson generalized this notion to the higher-dimensional $t$-modules. If Drinfeld modules are analogous to elliptic curves, then $t$-modules are analogous to general abelian varieties.
On the other hand, given a Drinfeld module $E$ defined over $K$, Taelman defined an associated class module $H(E/A)$. He proved a class number formula which states that the special value at $s=0$ of the $L$-function of the Drinfeld module $E$ is equal to the product of the unique monic generator of the Fitting ideal $\mathrm{Fitt}_A^0(H(E/A))$ of $H(E/A)$ and a regulator term, in the case that $A=\mathbb{F}_q[t]$. Later, Popescu and his collaborators proved an equivariant Tamagawa number formula (ETNF) for the special value at $s=0$ of a Goss-type $L$-function, equivariant with respect to a Galois group $G$, and associated to a Drinfeld module defined over a finite, integral extension of $\mathbb{F}_q[t]$. Then Popescu and his collaborators generalize their result to the $t$-module case. In this case, if $K/F$ is a finite abelian extension of function fields of Galois group $G$ and $E$ is a $t$-module defined over the integral closure $O_F$ of $A=\mathbb{F}_q[t]$ in $F$, then the special value at $s=0$ of the $G$-equivariant $L$-function associated to $E$ is related to $\mathrm{Fitt}^0_{A[G]}(H(E/O_K))$.
The main theme of this talk is the theory of abelian $t$-modules of arbitrary dimension. In the first part, we talk about an Iwasawa theory for their Taelman class modules, extending work of Higgins from the 1-dimensional case to the arbitrary dimensional case. The second part deals with the theory of formal $t$-modules and extends the results of Michael Rosen from the 1-dimensional case to the case of arbitrary dimension.