Draft:Shankar Sen

• Comment: Resubmitting without addressing the issues is considered disruptive. Citing only Shankar Sen's publications is not enough to meet the criteria so do take the time actually read the criteria. If resubmitted without substantial improvement, the draft may be rejected meaning it will no longer be considered. S0091 (talk) 20:38, 8 April 2025 (UTC)

• Comment: Is there any sources that discuss Sen or support his information? McMatter ^(talk)/_(contrib) 19:22, 8 April 2025 (UTC)

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Shankar Sen
Born	Shankar Sen
Alma mater	Harvard University
Known for	Sen Theory Sen's Filtration Theorem Sen's Operator Tate-Sen Conditions Ax-Sen-Tate Theorems
Scientific career
Fields	Mathematics
Institutions	Cornell University
Thesis	An Automorphism of Local Fields (Published in Annals of Mathematics)  (1967)
Doctoral advisor	John Tate (mathematician)
Doctoral students	James Barker Coykendall IV Charles Helou Antonio José de Oliveira Machiavelo Alicia Norma Sevilla
Website	math.cornell.edu/shankar-sen

Shankar Sen is an Indian-American Mathematician who is currently a professor at Cornell University. His research primarily concerns with invariants associated with representations of Galois groups of p-adic fields and algebraic number fields.

Education

Shankar Sen obtained his PhD from Harvard University in 1967 under the supervision of John Tate (mathematician). His thesis titled "An Automorphism of Local Fields" concerns important results and is published in the pretigious journal Annals of Mathematics.

Sen Theory

He is known for his most important contribution in the field of p-adic Galois representation of the absolute galois , now known as Sen Theory. He applied techniques of p-adic analytic function theory and functional analysis in analysing of these representations. His theory was mainly made to overcome the following situation: Given a $p$-adic field $k$ and an infinitely ramified $\mathbb {Z} _{p}$-extension, $k_{\infty }=\cup _{n}k_{n}$ (for example $k_{n}=k(\mu _{p^{n}})$) one wants to descend a $\mathbb {C} _{p}$-representation of $G_{k}$ to a finite level, i.e over $k_{n}$ over some $n$. However we have $\mathbb {C} _{k}^{G_{k_{\infty }}}={\widehat {k_{\infty }}}$ and hence taking $G_{k_{\infty }}$-invariants we get a functor {{center|${\text{Rep}}_{\mathbb {C} _{k}}(G_{k})\longrightarrow {\text{Rep}}_{\widehat {k_{\infty }}}({\text{Gal}}(k_{\infty }/k))$ The classical Sen theory constructs a functor

$D_{\text{Sen}}:{\text{Rep}}_{\mathbb {C} _{k}}(G_{k})\longrightarrow {\text{Rep}}_{k_{\infty }}({\text{Gal}}(k_{\infty }/k))$

which is quasi inverse to $W\mapsto \mathbb {C} _{k}\otimes _{k_{\infty }}W$. This argument is called decompletion argument.

Advances in Sen Theory and further directions

This decompletion argument of Sen was further generalised by Pierre Colmez who gave the generalised Tate-Sen conditions, and applied it to deduce results primarily known as overconvergence of p-adic representations.

Sen's theory is also used to investigate properties of overconvergence of $(\varphi ,\Gamma )$-modules. Later Kiran Kedlaya and his student Ruochuan Liu formulated generalised decompletion systems.

Notable papers

• Sen, Shankar (1969), "On automorphisms of local fields", Annals of Mathematics, 90 (1): 33–36, doi:10.2307/1970680, JSTOR 1970680
• Sen, Shankar (1993), "Galois cohomology and Galois representations", Inventiones Mathematicae, 112, Springer: 639–656, Bibcode:1993InMat.112..639S, doi:10.1007/BF01232450