(I'm sorry for my English.) Hello. I have been reading B.Mazur's article "An introduction to the deformation theory of Galois representations". I'm at the proof of proposition1 in §30, where he discusses I-ordinary deformation. Let me write down the settings : $A$ : a Noetherian local ring which...

I am self learning Algebraic Geometry from Daniel Perrin's Algebraic Geometry textbook. I have a question on last paragraph of page $41$. Let $D(f)$ be the set of points where the function doesn't vanish and $\Gamma(V)= k[X_1,...,X_n]/I(V)$ where $k$ is a commutative field. Let $r$ denote the...

I am reading up on some algebraic geometry and came across the above definition of sheaves. Some things confuse me. When the author says in 2) of Definition 4.1, "For each inclusion of open sets $V \subset U$..", does he mean that $V$ is an open set in $X$ or in the induced topology on $U$? In...

Problem Statement Let $f: X \to Y$ be a surjective morphism of finite type between affine Noetherian schemes, where $X = \operatorname{Spec} B$ and $Y = \operatorname{Spec} A$. Suppose that for every closed point $y \in Y$ and for every $x \in X_y$ that is closed in $X_y$, the stalk map...

Let $k$ be an infinite field. Definition 1. ( Affine Hilbert function ). Let $R := k[x_1, \dots ,x_n]$ be a polynomial ring which can be viewed as a vector space over $k$. Let $R_{\le s} := k[x_1, \dots, x_n]_{\le s} $ denote the set of polynomials of total degree $\le s$ in $R$. Note that...

This question is from Proposition $6.3$ of the textbook Algebraic geometry by Daniel Perrin( Page 32). Here $k$ is a commutative field.Let $V$ be a projective algebraic set and consider a homogeneous element $f \in \Gamma_h(V)= k[X_0,...,X_n]/I_p(V)$ of degree>0. $I_p(V)$ is the ideal of...

Let $s \ge 4$ be a composite integer with $s \ne 0 \pmod3$. Computationally, for every such $s \le 2000$, I can find a prime $p$ and integer $k \ge 1$ such that $p^k - s$ is prime (usually with $p \in \{2,3\}$ and small $k$). Heuristically this seems unsurprising: for fixed small $p$, the values...

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 $f\notin I$...