# Tagged Questions

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### What is the arithmetic Nullstellensatz?

The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same ...
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### Rational points techniques on curves not using their Jacobian

Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...
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### Which of the Mochizuki's works are the most closely related to elliptic curves?

I'm very much interest about algebraic geometry and number theory along with cryptography, but I have a special interest about the elliptic curves. I have heard a lot of interesting things about ...
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### Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)

Suppose X/K is a curve over a field K, which we want to think of as non-algebraically closed, and let x be a point of X(K). The Ceresa cycle is defined as follows; you can embed X in Jac(X) by sending ...
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### Applications of anabelian geometry to Galois representations?

One aspect of anabelian geometry is the study of the action of the absolute Galois group of a field $K$ on the etale fundamental group $\pi_1(X_\overline K)$, where $X$ is a (anabelian) variety and ...
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### Obstruction and 1st order infinitesimal deformations of Generalized Elliptic Curves (Deligne-Rapoport)

We consider the deformation theory of a generalized elliptic curve $(C_0,+)$ over a field $k$. Let $D$ be the deformation functor. And now we only consider the case that $C_0$ is irreducible as in ...
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### Tate-Shafarevich groups over finitely generated fields

Let $G$ be an algebraic group over a number field $k$. One defines the Tate-Shafarevich set of $G$ to be $$Ш(k,G) = \ker\left(H^1(k,G) \to \prod_{v} H^1(k_v,G)\right),$$ where the product is over all ...
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### Examples of subspaces singled out by modular forms

I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature. To be more clear let me explain the example I have in mind. Let ...
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### The minimum genus of a family of degree $12$ algebraic curves which comes from the resultant of two quartic polynomials

Let $f(t)$ be a rational normal cubic curve in $\mathbb{P}^3$ (it is not contained in any plane) and also we assume that this cubic curve passes through two points $(0,0,0)$ and $(1,0,0)$. By an easy ...
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### modular forms, invertible sheaves, and quotients

I'm very confused about some contradicatory statements, and I hope someone can help me clarify this. Let $\Gamma$ be a congruence subgroup. It is well known that modular forms of weight $k$ for ...
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### Finding a divisor on a curve in a given linear equivalence class made with points over a “small” field

Suppose I have a curve $C$ defined over $\mathbb Q$ and a line bundle $\mathcal L$ on it, also defined over $\mathbb Q$. I am interested in trying to find a (non-effective) divisor $D=\sum_i a_ip_i$ ...
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### Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve

Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us ...
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### K3 surfaces that correspond to rational points of elliptic curves

In his work on mirror symmetry (http://arxiv.org/pdf/alg-geom/9502005v2.pdf) Igor Dolgachev has considered families of K3 surfaces of Picard rank at least 19 with the base given by $X_0(n)^+$, the ...
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### Whats the prime-to-$p$ Etale fundamental group of the affine line minus two points over $\mathbb{F}_p$?

Background: I've often heard that the fundamental group of $\mathbb{P}^1/\mathbb{Q}-\{0,1,\infty\}$ is extremely hard to understand. First of all, it has a surjective map to the galois group ...
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### Is the ordinary locus affine?

Let $p$ be a prime number and let $Y$ over $\mathbb F_p$ be a Siegel modular variety, with minimal compactification $X$. It is well known that $X^{\operatorname{ord}}$, the ordinary locus of $X$ is ...
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### Conjugate surfaces: informations about the orbits

Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex ...
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### Rank 2^n quadratic form with non trivial invariant e_n

I advise you to have a look this question of mine first Rank four quadratic Form with non trivial discriminant in I(k) From quadratic form theory its well known that for a field $k$ and the ...
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### Motivic L-function vs motivic zeta function

Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of $$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$ where $Fr_p$ is a ...
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### Newton polygons of modular polynomials

This is pretty much straightforward curiosity. Is there anything known about Newton polygons of classical modular polynomials (polynomial relations between $j(\tau)$ and $j(n\tau)$)? I understand that ...
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### Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$

Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Then $SL_2(\mathcal{O}_K)$ acts on $\mathbb{H}^3$ ...
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### Weil's Riemann Hypothesis for dummies?

Weil's Riemann Hypothesis is a deep result that I don't fully understand, but it has understandable corollaries which interest me. For example: (a) For any projective curve $X$ satisfying certain ...
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### A letter from J. P. Serre

Which is the letter where J. P. Serre present "Analogues Kählériens de certaines conjectures de Weil" to Weil?
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### Time-line until the publicaton of Weil of “Numbers of solutions of equations in finite fields”

In "On the history of the Weil Conjectures" Dieudonné says: "Appropriately enough, the story, as with so many problems in number theory, begins with Gauss...". C. F. Gauss, Disquisitiones ...
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### What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?

I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that ...
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### Connection of Galois representation and arithmetic geometry

This is might be a dumb question. There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that ...
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### Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure

This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...
### $\zeta(n)$ as a mixed Tate motive
I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that $M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$ and $\zeta(n)$, ...