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Explicit equations for Drinfeld modular towers
Number Theory · Submitted 27 Oct 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Reducing at various primes, one obtains towers over finite fields, which are optimal, i.e., reach the Drinfeld--Vladut bound, over a quadratic extension of the finite field.
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The abstract, sentence by sentence
Elaborating on ideas of Elkies, we show how recursive equations for towers of Drinfeld modular curves $(X_0(P^n))_{n\ge 0}$ for $P\in \mathbb F_q[T]$ can be read of directly from the modular polynomial $\Phi_P(X,Y)$ and how this naturally leads to recursions of depth two. Although the modular polynomial $\Phi_T(X,Y)$ is not known in general, using generators and relations given by Schweizer, we find unreduced recursive equations over $\mathbb F_q(T)$ for the tower $(X_0(T^n))_{n\ge 2}$ and of a small variation of it (its partial Galois closure). Reducing at various primes, one obtains towers over finite fields, which are optimal, i.e., reach the Drinfeld--Vladut bound, over a quadratic extension of the finite field. We give a proof of the optimality of these towers, which is elementary and does not rely on their modular interpretation except at one point. We employ the modular interpretation to determine the splitting field of certain polynomials, which are analogues of the Deuring polynomial. For these towers, the particular case of reduction at the prime $T-1$ corresponds to towers introduced by Elkies and Garcia--Stichtenoth.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Elaborating on ideas of Elkies, we show how recursive equations for towers of Drinfeld modular curves $(X_0(P^n))_{n\ge 0}$ for $P\in \mathbb F_q[T]$ can be read of directly from the modular polynomial $\Phi_P(X,Y)$ and how this naturally leads to recursions of depth two. | contribution 0.92 | 0.94 | 0.49 | asserted 1.00 |
| s1 | Although the modular polynomial $\Phi_T(X,Y)$ is not known in general, using generators and relations given by Schweizer, we find unreduced recursive equations over $\mathbb F_q(T)$ for the tower $(X_0(T^n))_{n\ge 2}$ and of a small variation of it (its partial Galois closure). | contribution 0.62 | 0.94 | 0.10 | asserted 1.00 |
| s2 | Reducing at various primes, one obtains towers over finite fields, which are optimal, i.e., reach the Drinfeld--Vladut bound, over a quadratic extension of the finite field. | result 0.97 | 0.88 | 0.30 | asserted 1.00 |
| s3 | We give a proof of the optimality of these towers, which is elementary and does not rely on their modular interpretation except at one point. | contribution 0.54 | 0.96 | 0.25 | asserted 1.00 |
| s4 | We employ the modular interpretation to determine the splitting field of certain polynomials, which are analogues of the Deuring polynomial. | method 0.87 | 0.95 | 0.05 | asserted 1.00 |
| s5 | For these towers, the particular case of reduction at the prime $T-1$ corresponds to towers introduced by Elkies and Garcia--Stichtenoth. | result 0.42 | 0.36 | 0.05 | asserted 1.00 |
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The paper
Original abstract
Elaborating on ideas of Elkies, we show how recursive equations for towers of Drinfeld modular curves $(X_0(P^n))_{n\ge 0}$ for $P\in \mathbb F_q[T]$ can be read of directly from the modular polynomial $\Phi_P(X,Y)$ and how this naturally leads to recursions of depth two. Although the modular polynomial $\Phi_T(X,Y)$ is not known in general, using generators and relations given by Schweizer, we find unreduced recursive equations over $\mathbb F_q(T)$ for the tower $(X_0(T^n))_{n\ge 2}$ and of a small variation of it (its partial Galois closure). Reducing at various primes, one obtains towers over finite fields, which are optimal, i.e., reach the Drinfeld--Vladut bound, over a quadratic extension of the finite field. We give a proof of the optimality of these towers, which is elementary and does not rely on their modular interpretation except at one point. We employ the modular interpretation to determine the splitting field of certain polynomials, which are analogues of the Deuring polynomial. For these towers, the particular case of reduction at the prime $T-1$ corresponds to towers introduced by Elkies and Garcia--Stichtenoth.
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How the literature uses it
3 arXiv papers cite this in 22 sentences
Papers citing it · 16 sentences without a confident label
Builds on it 1 paper
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Elaborating on ideas of Elkies [ref] , Bassa and Beelen [ref] recently provided recursive equations over $_q(T)$ for the tower $X_0(T^n))_n≥2$ whose reduction at various primes (different from $T$) are optimal over a quadratic extension of the base field.
Legendre Drinfeld modules and universal supersingular polynomials · Connections to the modular tower $X_0(T^n)$ and supersingular Drinfeld modules · 0.85
Compares with it 1 paper
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(2) In Corollary 19 in [ref] , Bassa and Beelen proved the same result as the above corollary for another polynomial $p_d(s(s+1)^q-1)$.
Explicit formula of a supersingular polynomial for rank-2 Drinfeld modules and applications · An application for a supersingular polynomial · 0.99
Cites it as background 2 papers
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This was first introduced by Bassa and Beelen in [ref] .
Explicit formula of a supersingular polynomial for rank-2 Drinfeld modules and applications · An application for a supersingular polynomial · 0.90
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Several authors (see for example [ref] and the references within) addressed the nontrivial task of providing examples of curves of large genera over a finite field which are optimal in the sense that they asymptotically attain the Drinfeld-Vladut bound.
Legendre Drinfeld modules and universal supersingular polynomials · Connections to the modular tower $X_0(T^n)$ and supersingular Drinfeld modules · 0.91
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