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2021/08/31

Observations:

In a recollement, the derived category of the open can be the left or right full subcategory in the corresponding semiorthogonal decomposition. For left, use \(j_!\) to embed, and for right, use \(j_*\) to embed.

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2020/09/04

Reading:

Mallory, “Hilbert schemes of points”

for any projective smooth geometrically connected variety \(X\) over a field, write \(X^{[n]}\) for its Hilbert scheme of \(n\) points. we have the Hilbert–Chow morphism \(X^{[n]}_\mathrm{red}\rightarrow X^n/\mathfrak{S}_n\)

when \(X\) has dimension \(1\), this is an isomorphism (and \(X^{[n]}\) is reduced already)

when \(X\) has dimension \(2\), the Hilbert scheme is smooth and geometrically connected, so looking at the locus where the points are distinct shows that the Hilbert–Chow morphism is a birational equivalence

when \(X\) has dimension at least \(3\), the Hilbert scheme is generally neither smooth nor irreducible

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2020/08/09

Reading:

Loeffler, “What kind of non-cuspidal automorphic representation are not isobaric sums?”

let \(B\) be the standard Borel subgroup of \(\mathrm{GL}_2\), and work globally. note that \(I_B^{\mathrm{GL}_2}(|\cdot|^{1/2}\otimes|\cdot|^{-1/2})\) has infinite length, and after taking smooth vectors, the composition factors are given by products of the trivial representation at cofinitely places and the Steinberg at all others. Eisenstein series imply that they are all automorphic, but clearly they don’t satisfy strong multiplicity

thus the notion of isobaric is meant to pick out the canonical subquotient, which here is the trivial representation

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2019/11/26

Reading:

Gelbart, “Lectures on the Arthur–Selberg Trace Formula”

on page 9, for \(\mu=\mathrm{diag}(\alpha,\beta)\) in \(M(\mathbb{A})\), let \(t=-(\log\lvert\alpha^{-1}\beta\rvert)/(2[F:\mathbb{Q}])\), \(z=\beta e^t\), and \(a=\alpha\beta^{-1}e^{-2t}\). then \(\mu=zh_tm\)

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2019/11/21

Reading:

Wikipedia, “Picard group”

the Picard group of the line with two origins in \(\mathbb{Z}\)

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2019/11/18

Observations:

let \(K\) be a local field, write \(\mathcal{O}\) for its ring of integers, and write \(\kappa\) for its residue field. let \(p\) be a prime number. if \(p\mid\#\kappa-1\), we can form \(C_{p^2}\)-extensions of \(K\) whose inertia subgroup is \(C_p\) as follows: consider the quotient \(\widehat{\mathbb{Z}}\times\mathcal{O}^\times\rightarrow(\mathbb{Z}/p^2\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})\). then further quotient by \((\alpha\cdot p,-1)\), where \(\alpha\) is any unit in \(\mathbb{Z}/p^2\mathbb{Z}\).

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2019/09/29

Reading:

Darmon, “Algebraic Modular Forms”

by Weierstrass uniformization, the set of lattices \(\Lambda\subset\mathbb{C}\) is in bijection with the set of pairs \(E,\omega\), where \(E\) is an elliptic curve over \(\mathbb{C}\) and \(\omega\) is a generator of \(\Omega^1_{E/\mathbb{C}}\approx\mathscr{O}_E\) via \(\Lambda\mapsto(\mathbb{C}/\Lambda,\mathrm{d}z)\) and \((E,\omega)\mapsto\left\{\int_\gamma\omega\mid\gamma\in H_1(E(\mathbb{C}),\mathbb{Z})\right\}\)

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2019/09/22

Observations:

let \(N\) be a positive integer, and let \(d\) be in \((\mathbb{Z}/N\mathbb{Z})^\times\). the (contravariant) diamond operator \(\langle d\rangle\) sends cusp forms \(f(\tau)\) in \(S_k(\Gamma_1(N))\) to \((c\gamma^{-1}(\tau)+d)^2f(\gamma^{-1}(\tau))\), where \(\gamma=\begin{bmatrix} a & b \ c & d\end{bmatrix}\) lies in \(\Gamma_0(N)\).

let \(a\) be a positive integer, let \(H\) be a subgroup of \((\mathbb{Z}/aN\mathbb{Z})^\times\), and write \(H’\) for the image of \(H\) in \((\mathbb{Z}/N\mathbb{Z})^\times\). then for any \(f(\tau)\) in \(S_2(\Gamma_{H’}(N))\), the function \(f(a\tau)\) lies in \(S_2(\Gamma_H(aN))\)

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2019/09/03

Reading:

Wikipedia, “Simplicial complex”

the star of a set \(S\) of simplices is the set of simplices touching \(S\), i.e. the set of simplices where one of the faces is in \(S\)

the link of a set \(S\) of simplices is \(\mathrm{Cl}\,\mathrm{St}\,S\smallsetminus\mathrm{St}\,\mathrm{Cl}\,S\)

Observations:

let \(K\) be a local field, write \(\kappa\) for its residue field, and let \(G\) be a connected split semisimple group over \(K\). then the link of any vertex of the affine building of \(G\) over \(K\) is isomorphic to the spherical building of \(G\) over \(\kappa\)

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2019/08/21

Observations:

let \(G\) be an algebraic group over a field \(k\), and let \(A\) and \(B\) be closed subgroups of \(G\). then \((A\cap B)^\circ\subseteq A^\circ\cap B^\circ\), so in particular \(\mathrm{dim}(A\cap B)\leq\mathrm{dim}(A^\circ\cap B^\circ)\)

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2019/08/13

Reading:

Morris, Introduction to Arithmetic Groups

let \(G\) be a connected semisimple group over \(\mathbb{Q}\). then arithmetic subgroups of \(G\) are cocompact in \(G(\mathbb{R})\)

conversely, let \(G\) be a connected semisimple subgroup of \(\mathrm{SL}_n\) over \(\mathbb{R}\). if \(G\) has no \(\mathbb{R}\)-quasisimple component \(H\) with \(H(\mathbb{R})\) compact and \(G(\mathbb{R})\cap\mathrm{SL}_n(\mathbb{Z})\) is cocompact in \(G(\mathbb{R})\), then \(G\) is defined over \(\mathbb{Q}\)

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2019/08/10

Reading:

Wikipedia, “Approximation in algebraic groups”

let \(F\) be a global field, let \(G\) be a non-solvable algebraic group over \(F\), and let \(S\) be a nonempty set of places of \(F\). then \(G\) satisfies strong approximation if and only if \(R(G)\) is unipotent, \(G/R(G)\) is simply connected, and every \(F\)-quasisimple component \(H\) of \(G/R(G)\) has noncompact \(H(F_S)\)

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2019/07/22

Notes:

write \(R\) for the right adjoint of the functor \(M\mapsto\mathbb{Z}[M]\) from absorbing monoids to \(\Lambda\)-rings. then for all rings \(A\), the Teichmuller map is given by applying the unit of the forgetful-Witt adjunction to \(W(A)\), and then taking \(R\)

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2019/06/28

Reading:

Wikipedia, “Conductor-discriminant formula”

for any finite Galois extension \(L/K\) of local or global fields, the relative discriminant equals \(\Delta_{L/K} = \prod_\rho \mathfrak{f}(\rho)^{\dim\rho}\), where the product ranges over all irreducible complex representations \(\rho\) of \(\mathrm{Gal}(L/K)\), and \(\mathfrak{f}(\rho)\) denotes the Artin conductor of \(\rho\)

ProofWiki, “Completed Riemann Zeta Function has Order One”

the completed entire Riemann zeta function has order 1

Wikipedia, “Multiplicative group of integers modulo n”

for \(k\geq3\), the group \((\mathbb{Z}/2^k\mathbb{Z})^\times\) is isomorphic to \(C_2\times C_{2^{k-2}}\) via letting \(-1\) be the first generator and \(3\) be the second generator. the \(2\)-torsion is \(\{\pm1,2^{k-1}\pm1\}\)

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2019/06/27

Reading:

Wikipedia, “Conductor”

let \(L/K\) be a finite separable extension of local fields, and let \(E\) be the maximal abelian subextension of \(L\). then \(\mathrm{Nm}_{L/K}(L^\times)=\mathrm{Nm}_{E/K}(E^\times)\)

in particular, the conductor of \(L/K\) depends only on \(E\)

let \(L/K\) be a finite Galois extension of local fields. as an ideal, the conductor of \(L/K\) is the least common multiple of the conductors of one-dimensional representations of \(\mathrm{Gal}(L/K)\)

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2019/06/05

Reading:

Wikipedia, “Ramification group”

in the lower numbering, the \(s\)-th ramification subgroup of \(\mathrm{Gal}(\mathbb{Q}_p(\mu_{p^n})/\mathbb{Q}_p)\) is \(\mathrm{Gal}(\mathbb{Q}_p(\mu_{p^n})/\mathbb{Q}_p(\mu_{p^e}))\), where \(e=\lfloor\log_ps\rfloor+1\)

the lower numbering is compatible with subgroups, while the upper numbering is compatible with quotient groups and subgroups 

Notes:

using the Grothendieck–Lefschetz trace formula for \(\mathrm{Bun}_{\mathrm{SL}_n}\), one can show that \(\lvert\zeta_{X_{\mathbb{F}_{q^n}}}(-m)\rvert\sim(q^{3(2m+1)(g-1)})^n\) as a function of \(n\), where \(X\) is a geometrically connected smooth proper curve over \(\mathbb{F}_q\) of genus \(g\), and \(\zeta\) denotes the Hasse–Weil zeta function

by the adjoint functor theorem, the functor \(M\mapsto\mathbb{Z}[M]\) from absorbing monoids to \(\Lambda\)-rings has a right adjoint \(R\). by comparing with the forgetful-Witt adjunction to rings, we see \(R\) takes \(W(A)\) to the absorbing monoid underlying \(A\) for any ring \(A\)

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2019/06/04 — Today’s Notes

Reading:

nLab, “model structure on simplicial sets”

Kan equivalences can be characterized as the smallest weakly saturated set of morphisms satisfying the 2-out-of-3 property and containing the maps \(\Delta^n\rightarrow\Delta^0\). in particular, we can characterize them purely in terms of simplicial sets

Quillen’s original proof of the Kan model structure is purely combinatorial