I'm trying to see that the Hecke algebra defined as $\mathbb{Q}[\text{GL}_2(\mathbb{Z}_p)\backslash \text{GL}_2(\mathbb{Q}_p)/\text{GL}_2(\mathbb{Z}_p)]$ maps to the ring of correspondences $\text{Corr}_\sim^0(M_n,M_n)$ where $M_n$ is the modular curve of elliptic curves with full $n$-torsion...

I am self studying Algebraic Geometry from the Clader and Ross Algebraic geometry textbook: Beginnings in Algebraic Geometry. This proof is given on page 275 of the textbook and I am quite confused about it. Please help me. Theorem 9.53 (Projective Nullstellensatz) Assume that $K$ is...

For instance we have the projective space itself is a manifold, and we often talk about zero set of polynomials over the projective space in algebraic geometry. So, how do the non trivial manifold properties /algebraic topology play into the study of algebraic geometry?

It is well known that for a finite field $K$ of characteristic $p$ the ring of $n$-truncated Witt vectors $W_n(K)$ is used to classify field extensions of $K$ of degree $p^n$; for details see e.g. Bosch's Algebra, chapter 4.10 on general Kummer theory. Basically the upshot is, cyclic subgroups...

I. Manifolds A non-singular homogeneous polynomial of degree $n+2$ with $n+2$ variables is a compact Calabi-Yau manifold, some of which important in string theory. For $n=2,3,5$ dimensions, we have, $$x_1^4+x_2^4+x_3^4+x_4^4 = 0$$ $$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5 = 0$$...

This is different from asking whether the Prime Number Theorem implies the asymptotic. I am specifically asking whether this asymptotic has an explicit published reference (journal article, book, or monograph), rather than whether it follows from standard methods. I am looking for a literature...

Let $b\ge 2$. Partition $[0,1)$ into the $b$ equal half-open intervals $$ I_j=\left[\frac{j}{b},\frac{j+1}{b}\right), \qquad 0\le j\le b-1. $$ Define the same-bin indicator $$ H_b(x,y)= \begin{cases} 1, & x,y\text{ lie in the same }I_j,\\ 0, & \text{otherwise}, \end{cases} $$ and the centered...

Let $a(b,n)$ be the number of integer tuples $(x_1, x_2, ..., x_{k+1})$ where $0 \leq x_i \leq b-1$, such that $|x_i - x_{i+1}| = d_i$ for all $i$, where $(d_1, d_2, ..., d_k)$ are digits of $n$ in base $b$. Related patterns in this specific sequence are discussed here and here. Now consider the...

For the Fermat curve $x^4+y^4=z^4$, what does its L-function look like? I know its zeta function over prime $p$ should have the form $\frac{P_p(t)}{(1-t)(1-pt)}$, with $P_p$ a polynomial of degree $6$. The L-function should be $\prod_p P_p(t)^{-1}$, right? I think the only possible bad primes...

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(31i)}{\eta(i)}$ that is missing. Can someone help me solve in radical form the following equation, whose solution is the value of Dedekind's modular...

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(43i)}{\eta(i)}$ that is missing. Can someone help me solve /in radical form) the following equation, whose solution is the value of Dedekind's modular...

Let $S\subset \mathbb R^3$ be a fixed sphere and let $E\subset \mathbb R^3$ be a fixed ellipsoid containing $S$. Consider tetrahedra $$A_1A_2A_3A_4$$ such that $$A_i\in E$$ and each face is tangent to $S$. Let $D_i\in S$ be the tangency point of the face opposite $A_i$. Poncelet’s closure...

Let $R = k[x_1, \dots, x_n]$ be a polynomial ring over a field $k$ equipped with a standard grading, and let $<$ be a monomial order on $R$. Let $I$ be a homogenous ideal of $R$, and let $\operatorname{in}_<(I)$ denote the initial ideal of $I$ with respect to $>$. My question is If $f\notin I$...

Let $Q_N$ denote the cubical cell complex given by the poset of faces of the $N$-cube, and let $$ V_N=\{\pm 1\}^N $$ be its set of $0$-cells. Let $$ A_N:=V_N/\{\pm 1\} $$ be the set of antipodal pairs of $0$-cells. We have $$ |A_N|=2^{N-1} $$ For each antipodal pair $$ \alpha=\{v,-v\}\in A_N $$...

It seems that for some integer $N$, namely any $N\equiv4\pmod {16}$, then they can be expressed as sum of $4$th powers of $4$ positive rational numbers. For example: $$15236 =\left(\frac{1875}{251}\right)^4+\left(\frac{11767}{3263}\right)^4+...

I have designed a new variant of the Collatz conjecture based on modulo 4 remainders, and I am looking for computational data or heuristic analysis regarding its convergence. Definition of the Mapping Let $n$ be a positive integer. We define the transition function $g(n)$ as follows based on $n...

The curve $y^2=P(x)$ over the field of complex numbers, where $P(x)$ is a polynomial of degree $4$ without repeating roots, can be transformed with a birational transformation into $Y^2=Q(X)$ with $Q(x)$ of degree $3$ without repeating roots. That is, an elliptic curve. However, if $P(x)$ does...

Consider the recurrence $$ x_{n+1}=x_{n-1}+2\log x_n, $$ with positive initial values chosen so that the sequence remains positive and increasing. Since this may be rewritten as $$ \frac{x_{n+1}-x_{n-1}}{2}=\log x_n, $$ it resembles the centred-difference discretisation of the differential...

Suppose $a(n)$ is the minimum integer $k$ such that $k^{11}$ can be expressed as the sum of $n$ distinct positive 11th powers. Q: Find $a(n)$ for $10\le n\le 19$. For example, $a(20)=119$ because $199^{11}$ is the sum of 20 terms of $11$th powers,...