rt.representation theory – Question on geometric lemma in the $p$-adic group representation

To check whether my understaning on the geometric lemma is right, I would like to ask some specific question related to it.

Let $F$ be a $p$-adic local field of characteristic zero. Let $W_n$ be a symplectic space over $F$ of dimension $2n$. Let $Sp_{n}(W_n)$ be a symplectic group and $B_n$ its standard Borel subgroup. Let $Q_{t,n-t}$ be a standard parabolic subgroup of $G$ preserving a totally isotropic subspace of $W_n$ of dimension $t$.

Let ${chi_i}_{1 le i le n}$ be unramified characters of $GL_1(F)$ such that $chi_i(omega)=1$ and let $pi$ be the normalized parabolic induced representation $text{Ind}_{Q_{3,n-3}}^{Sp_{n}}(chi_1 circ text{det}_{GL_3},chi_2,cdots,chi_{n-2})$.

I am considering $J_{Q_{2,n-2}}(pi)$, the normalized Jacquet module of $pi$ to $Q_{2,n-2}$. By the geometric lemma, there is some filtration of $J_{Q_{2,n-2}}(pi)$

$$0=tau_0 subset tau_1 subset cdots subset tau_m=J_{Q_2}(pi)$$ such that $tau_{i}backslashtau_{i+1}$ is some induced representation of the Jacquet module of the inducing data of $pi$.

I guess that such subquotient appearing in this filtration has the form $|cdot|^{-frac{1}{2}}cdot(chi_1 circ text{det}_{GL_2}) boxtimes text{Ind}_{B_{n-2}}^{Sp_{n-2}}(chi_1′,chi_2′,cdots,chi_{n-2}’)$ or
$text{Ind}_{B_2}^{GL_2}(chi_1′,chi_2′) boxtimes text{Ind}_{Q_{3,n-5}}^{Sp_{n-2}}(chi_1 circ text{det}_{GL_3},chi_4′,cdots,chi_{n-2}’)$?

(Here, ${chi_1′,cdots,chi_{n-2}’} subset {chi_1,cdots,chi_{n-2},chi_1^{-1},cdots,chi_{n-2}^{-1}}.)$

Any comments are highly appreaciated!

rt.representation theory – Frobenius reciprocity law in the $p$-adic group represenation

Let $G$ be a $p$-adic classical group and let $P_0$ be a minimal parabolic subgroup of $G$. Let $P=MN$ be a
standard parabolic subgroup containing $P_0$. Let $text{Ind}$ and $ |_M$ be the normalized parabolic induction functor and Jacquet functor, respectively.

Then for smooth representations $pi$ of $G$ and $rho$ of $M$, the Frobenius reciprocity law says that
$$text{Hom}_{G}(pi, text{Ind}(rho))simeq text{Hom}_{M}(pi|_M,rho).$$

I am wondering whether
$$text{Hom}_{G}(text{Ind}(rho),pi)simeq text{Hom}_{M}(rho, pi|_M)$$ does hold. If it does not hold in general, is there a sufficient condition for $pi$ and $rho$ which makes the above hold?

nt.number theory – Full measure properties for Zariski open subsets in $p$-adic situation

Let $F$ be a $p$-adic field and let $X$ be a smooth integral variety over $F$ (I am chiefly interested in the case when $X$ is a connected reductive group over $F$). Let $U$ be a non-empty open subset of $X$ with complement $Z$.

We can endow $X(F)$ with the Serre-Oesterle measure (e.g. as in (1,Section 2.2) or (2, Section 7.4))–this is just the standard measure coming from a top form of $X$).

My question is then whether one knows a simple proof/reference for the following:

The subset $Z(F)$ of $X(F)$ has measure zero.

I think this is proven in (1, Lemma 2.14)–but this is concerned with a more specific context which makes it non-ideal as a reference.

Any help is appreciated!

(1) http://www.math.uni-bonn.de/people/huybrech/Magni.pdf

(2) Igusa, J.I., 2007. An introduction to the theory of local zeta functions (Vol. 14). American Mathematical Soc..

nt.number theory – P-adic distance between solutions to S-unit equation

Let $p$ be a fixed prime number and $S$ is a finite set of prime numbers which does not contain $p$. A theorem of Siegel asserts that the number of solutions to the $S$-unit equations are finite; that is, there are only finitely many $S$-unit $u$ such that $1-u$ is also an $S$-unit. Therefor for each such $S$ there exist a lower bound on $|u_1-u_2|_p$ where $u_1$ and $u_2$ are solutions to $S$-unit equations.

My question is: does there exist such a lower bound uniformly? More precisely, does there exist a lower bound for the $p$-adic distance between solutions to the $S$-unit equations that only depends on the size of $S$(and perhaps on $p$)? Here we are assuming $S$ does not contain $p$.

reference request – Parabolic inductions for p-adic reductive groups

So I wish to ask for articles/comments surveying conjectures and theorems about parabolic induction for p-adic (non-archimedean case) reductive groups, and how local Langlands behaves under such. That is:

  1. For a quasi-split group G, what we know about the sub-representations of principle series.
  2. Same setting, but what we know about the (enhanced) Langlands parameters they correspond to, both conjectures and theorems.
  3. Same questions in 1 & 2, but for parabolic induction of supercuspidal representations from Levi assumed local Langlands for that Levi is understood.

The only things I know is the articles of Bernstein-Zelevinsky that addresses part of 1 and somewhat 2 for GL_n, as well as recent work of Aubert-Moussaoui-Solleveld (and extensions of some of them with others) about a nice conjectural framework of 3.

Any comment is greatly appreciated! I am also under the impression that the above questions are largely known for classical groups given the local Langlands established by Arthur and many others, but love to know a reference good for quick reading. If more is known in the local function field case, it will be fantastic to learn too. Thank you very much!!!

arithmetic geometry – Generalized Hodge-Tate weights of $p$-adic Galois representation

Let $K$ be an finite extension of $mathbb{Q}_p$, and $E$ be a Galois extension of $K$ whose Galois group is a open subgroup of $text{SL}_2(mathbb{Z}_p)$, then we have a $2$-dim $p$-adic Galois representation given by $$G(bar{K}/K)rightarrow G(E/K)rightarrowtext{SL}_2(mathbb{Z}_p)rightarrowtext{GL}_2(mathbb{Q}_p),$$then I want to know why the sum of the two Sen weights of this representation is $0$. Note that by definition, Sen weights or generalized Hodge-Tate weights are the eigenvalues of Sen operator.

Thanks!

number theory – Integral of a Chebyshev polynomial with respect to this special measure (Plancherel p-adic measure for GL2 (Q_p))

I'm trying to show that the integral $ int _ {- 2} ^ 2 U_n left ( frac {x} {2} right) frac {p + 1} { pi} frac { sqrt {1- frac {x ^ 2 } {4}}} { left ( sqrt {p} + frac {1} { sqrt {p}} right) ^ 2 – x ^ 2} dx $ It does not matter $ p ^ – n / 2} $, when $ n $ is an even integer.
Here, $ U_n (x) $ is he $ n $-th Chebyshev polynomial of the second type, defined as $ U_n (x) = frac {sin ((n + 1) cos ^ {- 1} x)} { sqrt {1-x ^ 2}} $; $ p $ is cousin $ n geq 0 $.

The corresponding integral when $ n $ is odd is zero since the integrand is an odd function. However, I cannot solve this integral for the even case. The integrand is simplified a bit once the definition of $ U_n $ used but I can't deal with it $ left ( sqrt {p} + frac {1} { sqrt {p}} right) ^ 2 – x ^ 2 $ finished. Some help?

The context is in analytic number theory, particularly the results in vertical Sato-Tate. The measure here is actually $ mu_p (x) = frac {p + 1} { pi} frac { sqrt {1-x ^ 2}} { left ( sqrt {p} + frac {1} { sqrt { p}} right) ^ 2 – x ^ 2} dx $, which is the measure of Plancherel in GL$ _2 ( mathbb {Q} _p) $. However, I don't think this should be relevant to the integral, I think that the integral can be solved using simple calculation techniques.

arithmetic geometry – function L in p-adic spaces

I've been learning more about different $ p $-Adicic geometries, namely, Berkovich spaces, Huber Adic spaces and ridiculous analytical spaces. In arithmetic geometry, it is often very interesting to associate and study the function L of a space. My question is whether this has been studied for such $ p $-the adicos spaces. A naive way of waiting for such L functions to be obtained is to take the L function associated with the cohomology of said space and study it. However, I could not find a reference for this.

The problem is that Google searches lead directly to $ p $-adic L-functions, which as far as I can tell is not what I want.

cv.complex variables – Asymptotic analysis using the p-adic Mellin Transform?

In the ordinary analysis, given a nice enough $ f: left (0, infty right) rightarrow mathbb {C} $, if we can calculate the Mellin transformation: $$ mathscr {M} left {f right } left (s right) = int_ {0} ^ { infty} x ^ {s-1} f left (x right) dx $$ in closed form, the calculation of the inverse transformation of Mellin using the residue theorem provides us with formulas (usually asymptotic, but sometimes exact) for the behavior of $ f left (x right) $ how $ x $ decreases to $ 0 $ I eat $ x $ increases to $ infty $. Former:

$$ int_ {0} ^ { infty} x ^ {s-1} sum_ {n = 0} ^ { infty} e ^ {- 2 ^ {n} x} dx = frac { Gamma left (s right)} {1-2 ^ {- s}} $$ it implies: $$ sum_ {n = 0} ^ { infty} e ^ {- 2 ^ {n} x} = frac {1} {2} – frac { gamma + ln x} { ln2} + frac {1} { ln2} sum_ {k in mathbb {Z} ^ { times}} Gamma left ( frac {2k pi i} { ln2} right) x ^ {- frac {2k pi i} { ln2}} – sum_ {n = 1} ^ { infty} frac { left (-1 right) ^ {n}} {2 ^ {n} -1 } frac {x ^ {n}} {n!} $$ an identity (emotionally satisfactory!) that is fulfilled for all x> 0.

But, now consider the p-adic analog: $$ mathscr {M} _ {p} left {f right } left (s right) = int _ { mathbb {Z} _ {p}} left | mathfrak {y} right | _ {p} ^ {s-1} f left ( mathfrak {y} right) d mathfrak {y} $$ where $ d mathfrak {y} $ is the measure of probability of hair in $ mathbb {Z} p Y $ f: mathbb {Z} _ {p} rightarrow mathbb {C} $ is it nice enough is there a similar interpretation for the inverse transformation of Mellin of $ mathscr {M} _ {p} left {f right } $ in terms of the asymptotes of $ f $? Answers or references will be appreciated.

Test using p-adic assessment

$$ text {Prove that} sum_ {i = 1} ^ n frac {1} {i} text {is not an integer
for} n ge 2 $$

Here is the solution:

Realise $$ sum_ {i = 1} ^ n frac {1} {i} = sum_ {i = 1} ^ n frac { frac {n!} {i}} {n!} $$ We consider $ v_2 left ( sum_ {i = 1} ^ n frac {n!} {i} right) $. We know $$ v_2 left ( frac {n!} {2i-1} + frac {n!} {2i} right) = v_2 left ( frac {n!} {2i} right) $$Then we get $ v_2 left ( frac {n!} {4i-2} + frac {n!} {4i} right) = v_2 left ( frac {n!} {4i} right) $ and repeating to summarize the factorial in this way we come to $$ v_2 left ( sum_ {i = 1} ^ n frac {n!} {i} right) = v_2 left ( frac {n!} {2 ^ { lfloor log_2 n rfloor} } right) tag {1} $$

I do not understand the part: "and repeating to summarize the factorial in this way we arrive at …", where the equation was made $ (1) $ comes from? Can anyone help? Thank you!