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

## 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.

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