Questions tagged [elliptic-curves]
An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.
1,503
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Why do we get a connected 2-regular graph?
In reading "PUBLIC-KEY CRYPTOSYSTEM BASED ON ISOGENIES" by Rostovtsev and Stolbunov, they claim on page 8 that the set $U=\{E_i(\mathbb{F}_p)\}$ of elliptic curves with a specific prime $l$ ...
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Elkies' family of elliptic curves of rank 19
There is a widely cited fact that Elkies had found that infinitely many curves of rank 19 in 2006, in "Z^28 in E(Q), etc. Email to the number theory mailing list
at [email protected]&...
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Reduction of elliptic curves over local fields
Let $E$ be an elliptic curve defined over a p-adic local field $K$, with $j$-invarient $j(E)\in K$. Let $\mathscr{O}_K$ be the ring of integer of $K$. If $j(E)$ does not belong to $\mathscr{O}_K$, ...
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Counting points on elliptic curves
Consider the Legendre family of elliptic curves
$$E_a: y^2=x(x-1)(x-a).$$
Let $p$ be an odd prime.
QUESTION. Is the following true? If $p\equiv 3\pmod4$ then number of solutions to $E_2$
over the ...
8
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1
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258
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A real-valued analogue of the Weierstrass $\wp$ Function
I am interested in the following function:
$$\mathcal{Q}(z) = \sum_{w \in L^*} \frac{1}{|z-w|^2} - \frac{1}{|w|^2} \, . $$
This function is analogous to the Weierstrass $\wp$ function, the only ...
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81
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Describing the hyperbolic structure of punctured torus in terms of the period lattice
Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$.
...
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115
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Finding a rational point of large height on an elliptic curve knowing a real approximation
Let $y^2=x(x^2+n)$ be an elliptic curve with $n\in\Bbb Z$ (the same question can of course
be asked for a general e.c). I know (e.g. it has rank 1) that there exists a nontrivial
rational point $(r,s)$...
2
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216
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Proof of Remark 6.14(b) of Milne's Arithmetic duality theorems'
Let $E/\mathbb{Q}$ be an elliptic curve.Let $\operatorname{Sha}(E/\Bbb{Q})$ be a Tate-Shafarevich group.
Milne's 'Arithmetic Duality Theorems' Remark 6.14(b) describe the following exact sequence.
...
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355
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Subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$
Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that
$\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
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Potential typo in "Complete Systems of Two Addition Laws for Elliptic Curves" by Bosma and Lenstra
Here is a link to the article: https://www.sciencedirect.com/science/article/pii/S0022314X85710888?ref=cra_js_challenge&fr=RR-1.
Pages 237-238 give polynomial expressions $X_3^{(2)}, Y_3^{(2)}, ...
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441
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Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
Similar to the case $x^4+y^4+z^4 = 1$ discussed in this MO post, define the system,
$$x^4+y^4+z^4+1 = (x+y+z+1)^4\tag1$$
$$\frac{x^2+x+1}{(x+y+1)(x+z+1)}=u\tag2$$
$$\frac{y^2+y+1}{(y+z+1)(y+x+1)}=v\...
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Prime splitting in the division field of an elliptic curve
Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
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Ramification of mod $\ell$ representation of elliptic curves [closed]
Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $p,\ell$ be two prime numbers.
Consider the mod $\ell$ representation $$\rho:Gal(\mathbb{\overline{Q}}/\mathbb{Q})\to Aut(E[\ell])= GL(2,\ell).$$...
7
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370
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A parametric elliptic curve for $x^4+y^4+z^4 = 1$?
Noam Elkies found that $x^4+y^4+z^4 = 1$ has infinitely many rational points $xyz \neq 0$ using an elliptic curve. We use a different approach that will produce pairs of solutions and a parametric ...
1
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1
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138
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cokernel of $H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/\operatorname{im}(\kappa_v)$
Let $F$ be a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction.
Let $\...
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1
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Can an abelian surface be bielliptic
Is an abelian surface containing an elliptic curve a bielliptic surface?
Suppose I have an abelian surface $A$ over the complex numbers that contains an elliptic curve $E$. Then
$A \to A/E$ is an ...
3
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Galois image of CM elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve with CM, with the endomorphism ring $R=\mathrm{End}_{\overline{\mathbb{Q}}}(E)$. Then for any integer $m$, we have the mod-$m$ Galois representation $\rho_m:\...
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Similar to a $d$-twist but over a cubic field
This question could be related to my old and Duality's newer questions.
I am building a $\mathbb{Z}/9\mathbb{Z}$ elliptic curve $E$ over $\mathbb{Q}$:
$$E: y^2+(t^3-3t^2+1)xy + t^3(t-1)^3y=x^2$$
For $...
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0
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77
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Elliptic curves and images of decompositions group exceptional?
Given an elliptic curve $E$ and its mod $p$ Galois representation $\bar{\rho}_{E,p}$, I am wondering what are the possibilities for $\bar{\rho}_{E,p}(G_l)$, where $G_l:=$Gal($\overline{\mathbb{Q}_l}/\...
2
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0
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61
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Average rank of elliptic curves over one-parameter family
Let $E_t:y^2=x^3+f(t)x+g(t)$ be an one parameter family of elliptic curves with $f,g\in \mathbb{Z}[t]$. I found one Silverman's result https://www.degruyter.com/document/doi/10.1515/crll.1998.109/pdf ...
7
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357
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Relationship between Serre-Tate coordinates of ordinary elliptic curves and Tate curves
Let $K$ be a complete extension of $\mathbb{Q}_{p}$ with valuation $v$ over $p$, valuation ring $R$, maximal ideal $\mathfrak{m}$ and residue field $k$. It is well known that if $E/K$ is an elliptic ...
2
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120
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On the elliptic curve $X^3+6d^2X-7d^3 = Y^2$ and the ellipse $p^2+3q^2-d = 0$?
From the ellipse $p^2+3q^2 - d = 0$ we can find a solution to the equation,
$$a^3+b^3+c^3 = (c+m)^3$$
if we solve the elliptic curve,
$$E:=X^3+6d^2X-7d^3 = Y^2$$
More details can be found in this MSE ...
3
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245
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When I know the two points on an elliptic curve, and the two points satisfy the relationship: $Q=e \cdot P$, is it possible for me to solve for e [closed]
When I know the two points on an elliptic curve, and the two points satisfy the relationship: $Q=e \cdot P$, is it possible for me to solve for $e$.
The equation of the curve is: $y^2 = x^3 + ax + b \...
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80
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Using the trace map to find rational points on elliptic curves
Let $K$ be a number field and let $E/K$ be an elliptic curve. Let $L/K$ be a finite extension. Consider the trace map
$$
\operatorname{Tr}_{L/K}:E(L)\longrightarrow E(K),\qquad \operatorname{Tr}_{L/K}(...
6
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194
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Ranks of elliptic curves over cubic fields
We are writing a paper on the ranks of elliptic curves over cubic fields. The curves of different torsion subgroups are created by the formulas in Jeon et al. and by our new parametrizations.
D. Jeon,...
7
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175
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Explicit equations for the universal vector extension of an elliptic curve
The universal vector extension $E$ of an abelian variety $A$ is an algebraic group, an extension of $A$ by a vector group $0 \to V \to E \to A \to 0$, such that any other extension of $A$ by a vector ...
2
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Torsion of an elliptic curve injects under reduction - question
Let $E/K$ be an elliptic curve over a number field. I am interested in the folowing statement: the map $E(K)[m]\rightarrow \tilde E_v(\tilde k_v)$ is injective for any place of $K$ provided there is ...
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80
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Finiteness of elliptic curves with trivial conductor over function fields
Let $K=\mathbb{F}_q(C)$ be the function field of a smooth projective curve $C$ over a finite field $\mathbb{F}_q$ with $\text{cha}(K)>3$ and let $E$ be an elliptic curve over $E$. To $E$ we may ...
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135
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Étaleness of Isom scheme $\operatorname{Isom}_S(X,Y)$
Let $S$ be a quasi-projective scheme over base field $k$ and $X, Y$ two finite étale schemes over $S$ and assume we are in situation we know that the isom space $\operatorname{Isom}_S(X,Y)$ exists as ...
1
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Criterion for an etale cover $E[\ell]\to \mathbb{G}_m$ to be tamely ramified in $0, \infty$
Suppose $E\to \mathbb{G}_m/k$ is an elliptic curve with $k$ field of characteristic $p>0$ and $E[m]$ it $m$-torsion group with $(m,p)=1$.
Consider the induced finite etale cover $E[\ell]\to \mathbb{...
11
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2
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606
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Does the number of roots of the modular form associated to an elliptic curve, on the positive imaginary axis, equal the analytic rank?
Recently I've been playing around with elliptic curves and have seemingly come up with a conjecture that I could not find elsewhere:
Let $E$ be an elliptic curve, and $f(q)$ its associated modular ...
2
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0
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166
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Is the Weil restriction of an elliptic curve self-dual?
$\DeclareMathOperator\res{res}$Let $K=\mathbb{Q}(\sqrt{-3})$, and let $$p\equiv 1\pmod 3$$
be a prime split in $K$. Assume that
$$p=\omega*\overline\omega,\quad\text{where}\quad\omega\equiv 1\pmod 3.$$...
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118
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Trivializing covers of $\ell$- torsion of elliptic curve
Let $E \to \mathbb{G}_m/k$ an elliptic curve over $ \mathbb{G}_m$ ($k$ field of char $p>0$) and $E[\ell]$ for $(\ell,p)=1$ the $\ell$-torsion group.
Let $f:T \to \mathbb{G}_m$ an finite etale ...
2
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195
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Construction refuting the existence of nonisotrivial elliptic curve over $\mathbb{G}_m$
I have some troubles to understand the construction in detail presented here by Daniel Litt used to show that there cannot exist an elliptic curve over $\mathbb{G}_m/k$, $k$ of characteristic $p >3$...
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Tate-Shafarevich group and its twist such that $\text{Sha}(E_D/\Bbb{Q})=0$ or some constant
Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$.
Let $D\in \Bbb{Z}$ be a square free integer and $E_D/\Bbb{Q}$ be its quadratic twist.
It is widely known that for all $E/\Bbb{Q}$: elliptic ...
0
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1
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281
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Tate–Shafarevich group and $\sigma \phi(C)=-\phi \sigma(C)$ for all $C \in \operatorname{Sha}(E/L)$
$\DeclareMathOperator\Sha{Sha}\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a quadratic extension of number field $K$.
Let $\sigma$ be a generator of $\Gal(L/K)$.
Let $E/K$ be an elliptic curve defined ...
2
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0
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100
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elliptic curves on general 3-folds of degree 7
Do there exist elliptic curves on a general 3-fold hypersurface $X_7 \subset \mathbb{P}^4$ of degree $7$?
Clemens proved that for $d \ge 8$ there are no elliptic curves on the general hypersurface $...
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How fast can elliptic curve rank grow in towers of number fields?
Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2\subset\cdots$ how fast can $\operatorname{rank}(E(K_n))$ grow in ...
7
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402
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Cubic twist of elliptic curves and its rank
Let $E/\mathbb{Q}$ be an elliptic curve defined by $E: y^2 = x^3 + b$ (where $b \in \mathbb{Q}$).
Let $E_D$ be an elliptic curve defined by $E_D: y^2 = x^3 + bD^2$.
$E$ and $E_D$ are isomorphic over $\...
1
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0
answers
86
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Select random point on elliptic curve
If I have an elliptic curve $E$ over some finite field $F_p$ what is a step by step algorithm to pick a random point that lays on this curve? There is definitely a naive approach to brute force all ...
4
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83
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Elliptic integral as quantity associated with Riemann surface?
There are many elliptic integrals, so to show my point let me
just pick one of them (complete elliptic integral of the first
kind [1]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
2
votes
1
answer
266
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On Iwasawa theory of elliptic curves in $\mathrm{PGL}_2(\mathbb{Z}_p)$-extension
Let $E$ be an elliptic curve over the rationals $\mathbb{Q}$. We consider the Galois representation attached to $E$ by acting on its $p$-adic Tate module $T_p(E)$,
$$
\rho_E: G_{K} \rightarrow \mathrm{...
8
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0
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167
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Do there exist Calabi-Yau 3-folds that contain a finite number of elliptic curves?
The moduli space $M_1(X, e)$ of degree $e$ elliptic curves on $X$ has virtual dimension zero if $X$ is a Calabi-Yau 3-fold. I am wondering if there is an example of such an $X$ so that each $M_1(X, e)$...
0
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0
answers
218
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Reference book on the relation between modular forms and elliptic curves
What is a modern reference book to understand the relation between modular forms and elliptic curves after the proof of the Taniyama–Shimura theorem?
3
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1
answer
235
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Global duality theorem for 2-part
$\DeclareMathOperator\coker{coker}\DeclareMathOperator\Sha{Sha}$Let $K$ be a number field.
Let $E/K$ be an elliptic curve over $K$.
Suppose finiteness of $\Sha(E/K)$.
According to Global duality ...
5
votes
2
answers
435
views
When are two elliptic curves with zero j invariant isogenous?
Consider elliptic curves of the form $E_B\colon y^2=x^3+B$ for $B\in\mathbb Q$. These are exactly the elliptic curves with zero $j$-invariant. I would like to know when are two elliptic curves $E_B$ ...
0
votes
0
answers
69
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Is there an $\mathbb{F}_{\!q}$-curve of geometric genus 3 and $\mathbb{F}_{\!q^3}$-cover to an elliptic $\mathbb{F}_{\!q}$-curve of $j$-invariant 0?
Let $E$ be an elliptic curve $y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $3 \mid (q-1)$. Is there an absolutely irreducible $\mathbb{F}_{\!q}$-curve $C$ of ...
2
votes
2
answers
315
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Upper bound on number of integral solutions of elliptic curves
I was studying M. Bhargava Et al's seminal paper titled "Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves"
And came across a very fascinating ...
6
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2
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1k
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Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD
I am posting this question on MO since I haven't received any answers on MSE.
Below is my (very elementary) attempt. Feel free to post a solution using facts in algebraic geometry and facts about ...
6
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0
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136
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Fourier transform and Hodge-$*$ operator
Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says
$$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$
...