The genus of projective curves discretely separates decidedly different two variable algebraic relations. So, we can focus on the connected moduli M_g of genus g curves. Yet, modern applications require a data variable (function) on such curves. The resulting spaces are versions, depending on our need from this data variable, of Hurwitz spaces. A Nielsen class is a set defined by r \ge 3 conjugacy classes C in the data variable monodromy G. It gives a striking genus analog.
Using Frattini covers of G, every Nielsen class produces a projective system of related Nielsen classes for any prime p dividing |G|. A nonempty (infinite) projective system of braid orbits in these Nielsen classes is an infinite (G,C) component (tree) branch. These correspond to projective systems of irreducible (dim r-3) components from {H(G_{p,k}(G),C)}_{k=0}^{\infty}, the (G,C,p) Modular Tower (MT). The classical modular curve towers {Y_1(p^{k+1})}_{k=0}^\infty (simplest case: G is dihedral, r=4, C are involution classes) are an avatar.
The (weak) Main Conjecture says, if G is p-perfect, there are no rational points at high levels of a component branch. When r=4, MT levels (minus their cusps) are upper half plane quotients covering the j-line. Our topics.
* Identifying component branches on a MT from g-p', p and Weigel cusp branches using the MT generalization of spin structures.
* Listing cusp branch properties that imply the (weak) Main Conjecture and extracting the small list of towers that could possibly fail the conjecture. * Formulating a (strong) Main Conjecture for higher rank MTs (with examples): almost all primes produce a modular curve-like system.
In 2000, based on his procedure for constructing explicit towers of modular curves, Elkies deduced explicit equations of rank-2 Drinfeld modular curves which coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. In 2015, Bassa, Beelen, Garcia, and Stichtenoth constructed a celebrated (recursive and good) tower (BBGS-tower for short) of curves and outlined a modular interpretation of the defining equations. Soon after that, Gekeler studied in depth the modular curves coming from sparse Drinfeld modules. In this paper, to establish a link between these …
In the 1960's Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct $χ_{5}$, the cusp form of lowest weight for the group $\operatorname{Sp}(2;Z)$. In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identifed with the orthogonal group of signature $(2,3)$ for the lattice $A_{1}$ and Igusa's form $χ_{5}$ appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three diff…
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 $Φ_P(X,Y)$ and how this naturally leads to recursions of depth two. Although the modular polynomial $Φ_T(X,Y)$ 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 fi…
In this paper, we will give an overview of known and new techniques on how one can obtain explicit equations for candidates of good towers of function fields. The techniques are founded in modular theory (both the classical modular theory and the Drinfeld modular theory). In the classical modular setup, optimal towers can be obtained, while in the Drinfeld modular setup, good towers over any non-prime field may be found. We illustrate the theory with several examples, thus explaining some known towers as well as giving new examples of good explicitly defined towers of function fields.
In this paper we investigate examples of good and optimal Drinfeld modular towers of function fields. Surprisingly, the optimality of these towers has not been investigated in full detail in the literature. We also give an algorithmic approach on how to obtain explicit defining equations for some of these towers and in particular give a new explicit example of an optimal tower over a quadratic finite field.
For a totally real number field $F$ and a nonarchimedean prime $\mathfrak{p}$ of $F$ lying above a prime number $p$ we introduce certain sheaf cohomology groups that intertwine the $\mathfrak{p}^{\infty}$-tower of a quaternionic Hilbert modular variety associated to a quaternion algebra $D$ over $F$ that is split at $\mathfrak{p}$ and a $p$-adically admissible representation of $\mbox{PGL}_2(F_{\mathfrak{p}})$. Applied to infinitesimal $p$-adic deformations of the local factor at $\mathfrak{p}$ of a cuspidal automorphic representation $π$ of $D^*(\mathbb{A})$ this yields a natural construction…
We give explicit equations for the simplest towers of Drinfeld modular curves over any finite field, and observe that they coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth.
Each finite $p$-perfect group $G$ ($p$ a prime) has a universal central $p$-extension. For a perfect group these central extensions come from its {\sl Schur multiplier}. Serre gave a Stiefel-Whitney class approach to analyzing spin covers of alternating groups ($p=2$) aimed at geometric covering space problems. This included the regular version of the Inverse Galois Problem. Every finite simple group with order divisible by $p$ has an infinite string of perfect centerless group covers exhibiting nontrivial Schur multipliers for the prime $p$. Sequences of moduli spaces of curves attached to $G…
We give a general recipe for explicitly constructing asymptotically optimal towers of modular curves such as {X_0(l^n): n=1,2,3,...}. We illustrate the method by giving equations for eight towers with various geometric features. We conclude by observing that such towers are all of a specific recursive form, and speculate that perhaps every tower of this form that attains the Drinfeld-Vladut bound is modular.
Let $k$ denote an algebraically closed field. We revisit a construction of the author of families of elliptic curves over the rational function field $k(t)$. Combining a combinatorial analysis with a rank formula of Ulmer we prove that, for all but finitely many families of these curves, the Mordell-Weil groups over $k(t^{1/d})$ have rank zero, as $d$ ranges over positive integers prime to the characteristic of $k$.
We formulate and for the most part prove a conjecture in the style of Mazur-Greenberg for the nonvanishing of central values of Rankin-Selberg $L$-functions attached to elliptic curves in abelian extensions of imaginary quadratic fields. This in particular generalizes the theorem of Rohrlich on $L$-functions of elliptic curves in cyclotomic towers to the setting of abelian extensions of imaginary quadratic fields, corresponding to families of degree-four $L$-functions given by $\operatorname{GL}(2)\times\operatorname{GL}(2)$ Rankin-Selberg $L$-functions. It also generalizes the theorems of Roh…
We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$ over a finite field of characteristic $p$, and consider the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower defined by $\bar f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb{A}^1$, whose Galois group is canonically isomorphic to $\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified extension of $\mathbb{Z}_p$, which is abstractly isomorphic to $(\mathbb{Z}_p)^\ell$ as a topological group. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton …
Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an elliptic curve in the ${\bf{Z}}_p^2$-extension is finitely generated modulo Heegner points. The novelty of the approach here is to use the existence of a suitable $p$-adic $L$-function to reduce the problem to a minimal nonvanishing criterion, which should be applicable to a broader class of problems than considered here.
We study the $Λ$-module structure of the ordinary parts of the arithmetic cohomology groups of modular Jacobians made out of various towers of modular curves. We prove that the ordinary parts of $Λ$-adic Selmer groups coming from two different towers have "almost same" $Λ$-module structures. We also prove the cotorsionness of $Λ$-adic Tate-Shafarevich group under mild assumptions.
Let $E$ be an elliptic curve defined over a number field $F$ with good ordinary reduction at all primes above $p$, and let $F_\infty$ be a finitely ramified uniform pro-$p$ extension of $F$ containing the cyclotomic $\mathbb{Z}_p$-extension $F_{cyc}$. Set $F^{(n)}$ be the $n$-th layer of the tower, and $F^{(n)}_{cyc}$ the cyclotomic $\mathbb{Z}_p$-extension of $F^{(n)}$. We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $λ$-invariant of the Selmer group over $F^{(n)}_{cyc}$ as $n\rightarrow \infty$. This method has its origins in work of A.Cuoco, who studied $\math…
We study the variation of Mordell-Weil ranks in the Jacobians of curves in a pro-p tower over a fixed number field. In particular, we show that under mild conditions the Mordell-Weil rank of a Jacobian in the tower is bounded above by a constant multiple of its dimension. In the case of the tower of Fermat curves, we show that the constant can be taken arbitrarily close to 1. The main result is used in the forthcoming paper of Guillermo Mantilla-Soler on the Mordell-Weil rank of the modular Jacobian J(Np^m).
Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.
Let $k$ be a field of characteristic zero and let $K=k(t)$ be the rational function field over $k$. In this paper we combine a formula of Ulmer for ranks of certain Jacobians over $K$ with strong upper bounds on endomorphisms of Jacobians due to Zarhin to give many examples of higher dimensional, absolutely simple Jacobians over $k(t)$ with bounded rank in towers $k(t^{1/p^r})$. In many cases we are able to compute the rank at every layer of the tower.
This paper concerns towers of curves over a finite field with many rational points, following Garcia--Stichtenoth and Elkies. We present a new method to produce such towers. A key ingredient is the study of algebraic solutions to Fuchsian differential equations modulo $p$. We apply our results to towers of modular curves, and find new asymptotically good towers.
To extend Iwasawa's classical theorem from ${\mathbb Z}_p$-towers to ${\mathbb Z}_p^d$-towers, Greenberg conjectured that the exponent of $p$ in the $n$-th class number in a ${\mathbb Z}_p^d$-tower of a global field $K$ ramified at finitely many primes is given by a polynomial in $p^n$ and $n$ of total degree at most $d$ for sufficiently large $n$. This conjecture remains open for $d\geq 2$. In this paper, we prove that this conjecture is true in the function field case. Further, we propose a series of general conjectures on $p$-adic stability of zeta functions in a $p$-adic Lie tower of funct…
We classify all complex quadratic number fields with 2-class group of type (2,2^m) whose Hilbert 2-class fields have class groups of 2-rank equal to 2. These fields all have 2-class field tower of length 2. We still don't know examples of fields with 2-class field tower of length 3, but the smallest candidate is the field with discriminant -1015.
Let $C$ be a smooth projective curve defined over a number field $k$, $X/k(C)$ a smooth projective curve of positive genus, $J_X$ the Jacobian variety of $X$ and $(τ,B)$ the $k(C)/k$-trace of $J_X$. We estimate how the rank of $J_X(k(C))/τB(k)$ varies when we take an unramified abelian cover $π:C'\to C$ defined over $k$.
Let p$\ge$2 be a given prime number. We prove, for any number field kappa and any integer e$\ge$1, the p-rank $ε$-conjecture, on the p-class groups Cl\_F, for the family F\_kappa^p^e of towers F/kappa built as successive degree p cyclic extensions (without any other Galois conditions) such that F/kappa be of degree p^e, namely: #(Cl\_F[p])<<\_{kappa,p^e,$ε$}($\sqrt$D\_F)^$ε$, where D\_F is the absolute value of the discriminant (Theorem 3.6) and, more generally, #(Cl\_F[p^r])<<\_{kappa,p^e,$ε$}($\sqrt$D\_F)^$ε$, for any r$\ge$1 fixed. This Note generalizes the case of the family F\_Q^p (Genus …
Inspired by the Odlyzko root discriminant and Golod--Shafarevich $p$-group bounds, Martinet (1978) asked whether an imaginary quadratic number field $K/\mathbb{Q}$ must always have an infinite Hilbert $2$-class field tower when the class group of $K$ has $2$-rank $4$, or equivalently when the discriminant of $K$ has $5$ prime factors. No negative results are known. Benjamin (2001, 2002) and Sueyoshi (2004, 2009, 2010) systematically established infinite $2$-towers for many $K$ in question, by casework on the associated Rédei matrices. Others, notably Mouhib (2010), have also made progress, but…
We prove an Equivariant Main Conjecture in Iwasawa Theory along any rank one, sign-normalized Drinfeld modular, split at infinity Iwasawa tower of a general function field of characteristic p, for the Iwasawa modules recently considered by Greither and Popescu, in their proof of the classical Equivariant Main Conjecture along the (arithmetic) cyclotomic Iwasawa tower.
Sect 1 introduces Nielsen classes attached to (G,C), where C is r conjugacy classes in a finite group G, and a braid action on them. These give reduced Hurwitz spaces, denoted H(G,C)^rd. The section concludes with a braid formula for the genus of these spaces when r = 4. If there is at least one prime l for which G is divisible by l, but has no Z/l quotient, then there is a canonical tower of reduced Hurwitz spaces over H(G,C)^rd, using the Universal Frattini cover, ~G, of G, and ~G_ab, its abelianized version. The towers are nonempty assuming C are l' classes satisfying a cohomological condit…
Let E --> C be an elliptic surface defined over a number field K. For each finite covering C' --> C defined over K, let E' --> C' be the pullback. We give a strong upper bound for the rank of E'(C'/K) in the case that C' --> C is an unramified abelian covering and under the assumption that the Tate conjecture is true for the surface E'/K. In the case that C is an elliptic curve and the map C'=C --> C is the multiplication-by-n map, the rank of E'(C'/K) is O(n^e) for every e > 0, which may be compared with the elementary bound of O(n^2).
We calculate the effect of simple Hecke operators on u-expansions of higher rank Drinfeld modular forms, the eigenvalue for the Drinfeld discriminant function $Δ_t$ and show that a certain natural class of Hecke operators is completely multiplicative.
This is the first of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank r. In the present part, we develop the analytic theory. Most of the work goes into defining and studying the u-expansion of a weak Drinfeld modular form, whose coefficients are weak Drinfeld modular forms of rank r-1. Based on that we give a precise definition of when a weak Drinfeld modular form is holomorphic at infinity and thus a Drinfeld modular form in the proper sense.
We use the adelic language to show that any homomorphism between Jacobians of modular curves arises from a linear combination of Hecke modular correspondences. The proof is based on a study of the actions of $\mathrm{GL}_2$ and Galois on the étale cohomology of the tower of modular curves. We also make this result explicit for Ribet's twisting operators on modular abelian varieties.
In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian plus an associated space of Chow-Heegner points. We then apply this condition to prove the finiteness of this set for any modular curves $X_{\mathrm{ns} }^+ (N)$ and $X_0 ^+ (N)$ of genus at least 2 with N prime. The proof relies on the existence of a quotient of their Jacobians whose Mordell-Weil rank is equal to its dimension (and at least 2), which is prove…
We investigate the modular properties of a new partition rank, the $M_d$-rank of overpartitions. In fact this is an infinite family of ranks, indexed by the positive integer $d$, that gives both the Dyson rank of overpartitions and the overpartition $M_2$-rank as special cases. The $M_d$-rank of overpartitions is the holomorphic part of a certain harmonic Maass form of weight $\frac{1}{2}$. We give the exact transformation of this harmonic Maass form along with a few identities for the $M_d$-rank.
In this paper, we determine all modular Jacobian varieties $J_1(M,MN)$ over the number field $\mathbb{Q}(ζ_M)$ with the Mordell-Weil rank zero following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown.
We use Maeda's Conjecture to prove that the Rankin-Cohen bracket of an eigenform and any modular form is only an eigenform when forced to be because of the dimensions of the underlying spaces. We further determine when the Rankin-Cohen bracket of an eigenform and modular form is not forced to produce an eigenform and when it is determined by the injectivity of the operator itself. This can also be interpreted as using the Rankin-Cohen bracket operator of eigenforms to create evidence for Maeda's Conjecture.
We present an algorithm to compute the action of the Hecke operators on the top dimensional integral cohomology of certain torsion-free arithmetic subgroups of algebraic groups of Q-rank one. This generalizes the modular symbol algorithm to a setting including Bianchi groups and Hilbert modular groups. In addition, we generalize some results of Voronoi for real positive-definite quadratic forms to self-adjoint homogeneous cones of arbitrary Q-rank.
This is a sequel to the paper [F. Breuer, H.-G. Rück, Drinfeld modular polynomials in higher rank, J. Number Theory 129 (2009), 59-83.], in which we introduced Drinfeld modular polynomials of higher rank, using an analytic construction. These polynomials relate the isomorphism invariants of Drinfeld F_q[T]-modules of rank r\geq 2 linked by isogenies of a specified type. In the current paper, we give an algebraic construction of greater generality, and prove a generalization of the Kronecker congruences relations, which describe what happens when modular polynomials associated to P-isogenies ar…
We construct and study a natural compactification $\overline{M}^r(N)$ of the moduli scheme $M^r(N)$ for rank-$r$ Drinfeld $\F_q[T]$-modules with a structure of level $N \in \F_q[T]$. Namely, $\overline{M}^r(N) = {\rm Proj}\,{\bf Eis}(N)$, the projective variety associated with the graded ring ${\bf Eis}(N)$ generated by the Eisenstein series of rank $r$ and level $N$. We use this to define the ring ${\bf Mod}(N)$ of all modular forms of rank $r$ and level $N$. It equals the integral closure of ${\bf Eis}(N)$ in their common quotient field $\widetilde{\MF}_r(N)$. Modular forms are characterized…
In this paper we make a systematic study of certain motivic cohomology classes ("Rankin-Eisenstein classes") attached to the Rankin--Selberg convolution of two modular forms of weight $\ge 2$. The main result is the computation of the $p$-adic syntomic regulators of these classes. As a consequence we prove many cases of the Perrin-Riou conjecture for Rankin--Selberg convolutions of cusp forms.
The modular invariant of rank 1 Drinfeld modules is introduced and used to formulate and prove an exact analog of the Weber-Fueter theorem for global function fields. The main ingredient in the proof is a version of Shimura's Main Theorem of Complex Multiplication for global function fields, which is also proved here.
This is the third part of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present article we construct and study some examples of Drinfeld modular forms. In particular we define Eisenstein series, as well as the action of Hecke operators upon them, coefficient forms and discriminant forms. In the special case A=F_q[t] we show that all modular forms for GL_r(Γ(t)) are generated by certain weight one Eisenstein series, and all modular forms for GL_r(A) and SL_r(A) are generated by certain coefficient forms and discriminant forms. We …
This paper studies modular forms of rank four and level one. There are two possiblities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we describe when each case occurs for general choices of exponents for the T -matrix. In the remaining sections we describe how to write down corresponding differential equations satisfied by minimal weight forms, and how to use these minimal weight forms to describe the entire graded module of holomorphic modular forms. Unfortunately the differential equations that…
Continuing the work of \cite{7} and \cite{8}, we derive an analogue of the classical "$k/12$-formula" for Drinfeld modular forms of rank $r \geq 2$. Here the vanishing order $ν_ω(f)$ of one modular form at some point $ω$ of the complex upper half-plane is replaced by the intersection multiplicity $ν_{\bo}(f_1,\ldots,f_{r-1})$ of $r-1$ independent Drinfeld modular forms at some point $\bo$ of the Drinfeld symmetric space $\OM^r$. We apply the formula to determine the common zeroes of $r-1$ consecutive Eisenstein series $E_{q^{i}-1}$, where $n-r<i<n$ for some $n \geq r$.
We show that the absolute value $|f|$ of an invertible holomorphic function $f$ on the Drinfeld symmetric space $\OM^r$ $(r \geq 2)$ is constant on fibers of the building map to the Bruhat-Tits building $\MB\MT$. Its logarithm $\log|f|$ is an affine map on the realization of $\MB\MT$. These results are used to study the vanishing loci of modular forms (coefficient forms, Eisenstein series, para-Eisenstein series) and to determine their images in $\MB\MT$.
In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate that the resulting elements have high order. Between the two constructions, we are able to generate high order elements in every characteristic. Despite the use of the modular recursions of Elkies, our methods are quite elementary and require no knowledge of modular curves. We compare our results to…
