In this article we consider smooth projective curves $C$ of genus two described by integral equations of the form $y^2=xh(x)$, where $h(x)\in\mathbb{Z}[x]$ is monic of degree $4$. It turns out that if $h(x)$ is reducible, then the absolute discriminant of $C$ can never be an odd prime, except when $h(x)=(x-b)g(x)$ and $g(x)$ is irreducible. In this case we obtain a complete description of such genus $2$ curves. In fact, we prove that there are two one-parameter families $C_t^i$, $i=1,2$, of such curves such that if $C$ is a genus two curve with an odd prime absolute discriminant, then $C$ is $C_t^i$, for some $i$, and $t\in\mathbb{Z}$. Moreover, we show that $C_t^i$ has an odd prime absolute discriminant, $p$, if and only if a certain degree-$4$ irreducible polynomial $f^i(t)\in\mathbb{Z}[t]$ takes the value $p$ at $t$. Hence there are conjecturally infinitely many such curves. When $h(x)$ is irreducible, we give explicit examples of one-parameter families of genus $2$ curves $C_t$ such that $C_t$ has an odd prime absolute discriminant for conjecturally infinitely many integer values $t$.
In this paper we introduce a new problem called the Isogenous Embedding Problem (IEP). The existence of solutions to this problem is related to the primes of bad reduction of CM curves of genus $3$ and we can detect potentially good reduction in absence of solutions. We propose an algorithm for computing the solutions to the IEP and run the algorithm through different families of curves. We were able to prove the reduction type of some particular curves at certain primes that were open cases in [LLLR21].
We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a formula by Colmez and Obus specific to the CM case and valid when the CM field is an abelian extension of the rationals. This formula links the height and the logarithmic derivatives of an $L$-function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use results of Igusa, Liu, and Saito t…
In this paper we study general conditions to prove the infiniteness of the genus of certain towers of function fields over a perfect field. We show that many known examples of towers with infinite genus are particular cases of these conditions.
We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus two. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators…
Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We give a bound on the primes $\mathfrak{p}$ of $M$ such that the stable reduction of $C$ at $\mathfrak{p}$ contains three irreducible components of genus $1$.
Duke and the second author defined a family of linear maps from spaces of weakly holomorphic modular forms of negative integral weight and level 1 into spaces of weakly holomorphic modular forms of half integral weight and level 4 and showed that these lifts preserve the integrality of Fourier coefficients. We show that the generalization of these lifts to modular forms of genus 0 odd prime level also preserves the integrality of Fourier coefficients.
In this paper we give a passage formula between different invariants of genus 3 hyperelliptic curves: in particular between Tsuyumine and Shioda invariants. This is needed to get modular expressions for Shioda invariants, that is, for example, useful for proving the correctness of numerically computed equations of CM genus 3 hyperelliptic curves. On the other hand, we also get Shioda invariants described in terms of differences of roots of the equation defining the hyperelliptic curve, that has applications for studying the reduction type of the curve. Under certain conditions on its jacobian,…
In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the value of these modular functions at CM points of the Siegel upper-half space, which, if their…
Let $p$ be an odd prime, $N$ a square-free odd positive integer prime to $p$, $π$ a $p$-ordinary cohomological irreducible cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbb{A}_\mathbb{Q})$ of principal level $N$ and Iwahori level at $p$. Using a $p$-integral version of Rallis inner product formula and modularity theorems for $\mathrm{GSp}_{4/\mathbb{Q}}$ and $\mathrm{U}_{4/\mathbb{Q}}$, we establish an identity between the $p$-part of the critical value at $1$ of the degree $5$ $L$-function of $π$ twisted by the non-trivial quadratic Dirichlet character $ξ$ associated to the exten…
Let X be a curve over a number field K with genus g>=2, $\pp$ a prime of O_K over an unramified rational prime p>2r, J the Jacobian of X, r=rank J(K), and $\scrX$ a regular proper model of X at $\pp$. Suppose r<g. We prove that #X(K)<=#\scrX(F_{\pp})+2r, extending the refined version of the Chabauty-Coleman bound to the case of bad reduction. The new technical insight is to isolate variants of the classical rank of a divisor on a curve which are better suited for singular curves and which satisfy Clifford's theorem.
The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 defined over a function field of characteristic zero. We recover the known result for genus 2 curves and in many cases improve upon the known bound for genus 3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved.
We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof, and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.
Let C/Q be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of Q, but may not have a hyperelliptic model of the usual form over Q. We describe an algorithm that computes the local zeta functions of C at all odd primes of good reduction up to a prescribed bound N. The algorithm relies on an adaptation of the "accumulating remainder tree" to matrices with entries in a quadratic field. We report on an implementation, and compare its performance to previous algorithms for the ordinary hyperelliptic case.
This paper provides an algorithm enumerating superspecial trigonal curves of genus $5$ over finite fields. Executing the algorithm over a computer algebra system Magma, we enumerate them over finite fields $\mathbb{F}_{p^a}$ for any natural number $a$ if $p \leq 7$ and for odd $a$ if $p \leq 13$.
The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier's work on elliptic curves. Experimentally, for genus 2 curves with CM by a quartic CM field K, it appears that primes dividing the denominators of the discriminants of the Igusa class polynomials all have the property 1) that they are bounded by d, the absolute value of the discriminant of K, and 2) that they divide d-x^2, for some integer x whose square is less than d. A slightly stronger condition is given in Section 3. Such primes are primes of bad reduction for the genus 2 curve and primes of …
Let $p$ be an odd prime number and be an integer coprime to $p$. We survey an algorithm for computing explicit rational representations of $(\ell,...,\ell)$-isogenies between Jacobians of hyperelliptic curves of arbitrary genus over an extension $K$ of the field of $p$-adic numbers $\mathbb{Q}\_p$. The algorithm has a quasi-linear complexity in $\ell$ as well as in the genus of the curves.
We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X_0(N), for N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q-coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves.
The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points on curves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32 \le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves and trigonal curves over $\mathbb{F}_3$ which have many $\mathbb{F}_9$-rational points (and $\mathbb{F}_3$-rational points), especially the maximal number of $\mathbb{F}_9$-rational points of those curves is $30$. Kudo-Harashita studied the nonhyperelliptic and nontrigonal case,where they found a new example of curves (over $\mathbb{F}_3$) of genus five which attains $32$ and…
One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus $g$? The same question can be asked of various subclasses of curves. In this article we classify the possibilities for the number of points on genus 4 hyperelliptic supersingular curves over finite fields of order $2^n$, $n$ odd.
Let p be an odd prime number and g $\ge$ 2 be an integer. We present an algorithm for computing explicit rational representations of isogenies between Jacobians of hyperelliptic curves of genus g over an extension K of the field of p-adic numbers Qp. It relies on an efficient resolution, with a logarithmic loss of p-adic precision, of a first order system of differential equations.
We compute rational points on genus $3$ odd degree hyperelliptic curves $C$ over $\mathbb{Q}$ that have Jacobians of Mordell-Weil rank $0$. The computation applies the Chabauty-Coleman method to find the zero set of a certain system of $p$-adic integrals, which is known to be finite and include the set of rational points $C(\mathbb{Q})$. We implemented an algorithm in Sage to carry out the Chabauty-Coleman method on a database of $5870$ curves.
A hyperelliptic curve over $\mathbb Q$ is called "locally soluble" if it has a point over every completion of $\mathbb Q$. In this paper, we prove that a positive proportion of hyperelliptic curves over $\mathbb Q$ of genus $g\geq 1$ are locally soluble but have no points over any odd degree extension of $\mathbb Q$. We also obtain a number of related results. For example, we prove that for any fixed odd integer $k > 0$, the proportion of locally soluble hyperelliptic curves over $\mathbb Q$ of genus $g$ having no points over any odd degree extension of $\mathbb Q$ of degree at most $k$ tends …
We consider the proportion of genus one curves over $\mathbb{Q}$ of the form $z^2=f(x,y)$ where $f(x,y)\in\mathbb{Z}[x,y]$ is a binary quartic form (or more generally of the form $z^2+h(x,y)z=f(x,y)$ where also $h(x,y)\in\mathbb{Z}[x,y]$ is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove that the local density at a prime $p$ is given by a fixed degree-$9$ rational function of $p$ for all odd $p$ (and for the generalised equation, the sa…
We prove that if q is a power of an odd prime then there is no genus-2 curve over F_q whose Jacobian has characteristic polynomial of Frobenius equal to x^4 + (2-2q)x^2 + q^2. Our proof uses the Brauer relations in a biquadratic extension of Q to show that every principally polarized abelian surface over F_q with the given characteristic polynomial splits over F_{q^2} as a product of polarized elliptic curves.
Let $K$ be a number field, and $g \geq 2$ a positive integer. We define $c_K(g)$ as the smallest integer $n$ such that there exist infinitely many $\overline{K}$-isomorphism classes of genus $g$ hyperelliptic curves $C/K$ with all Weierstrass points in $K$ having potentially good reduction outside $n$ primes in $K$. We show that $c_K(g) > π_{K, \textrm{odd}}(2g) + 1$, where $π_{K, \textrm{odd}}(n)$ denotes the number of odd primes in $K$ with norm no greater than $n$, as well as present a summary of various conditional and unconditional results on upper bounds for $c_K(g)$.
Poonen and Stoll have shown that the reduced Shafarevich-Tate group of a principally polarized abelian variety over a global field can have order twice a square (the odd case) as well as a square (the even case). For a curve over a global field, they give a local diophantine criterion for its jacobian to be even or odd. In this note we use the Cherednik-Drinfeld p-adic uniformization to study the local points and divisors on certain Atkin-Lehner quotients of Shimura curves. We exhibit an explicit family of such quotient curves over the rationals with odd jacobians, with genus going to infinity…
In this paper we first of all determine all possible genera of (odd and even) definite unimodular lattices over an imaginary-quadratic field. The main questions are whether the partial class numbers of lattices with given Steinitz class within one genus are equal for all occuring Steinitz classes and whether the partial masses of those partial genera are equal. We show that the answer to the first question in general is "no" by giving a counter example, while the answer to the second question is "yes" by proving a mass formula for partial masses. Finally, we determine a list of all single-clas…
For integers $N\geq 3$ and $g\geq 1$, we study bounds on the cardinality of the set of points of order dividing $N$ lying on a hyperelliptic curve of genus $g$ embedded in its jacobian using a Weierstrass point as base point. This leads us to revisit division polynomials introduced by Cantor in 1995 and strengthen a divisibility result proved by him. Several examples are discussed.
We describe how the quadratic Chabauty method may be applied to explicitly determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell--Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or nontrivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin--Lehner quotients $X_0^+(N…
We find a closed formula for the number $\operatorname{hyp}(g)$ of hyperelliptic curves of genus $g$ over a finite field $k=\mathbb{F}_q$ of odd characteristic. These numbers $\operatorname{hyp}(g)$ are expressed as a polynomial in $q$ with integer coefficients that depend on the set of divisors of $q-1$ and $q+1$. As a by-product we obtain a closed formula for the number of self-dual curves of genus $g$. A hyperelliptic curve is self-dual if it is $k$-isomorphic to its own hyperelliptic twist.
We compute the $L$-functions of a large class of algebraic curves, and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local $L$-factor and the conductor exponent at the primes of bad reduction. Most of our examples are hyperelliptic curves of genus $g\geq 2$ defined over $\mathbb{Q}$ which have semistable reduction at every prime $p$. We also treat a few more general examples of superelliptic curves.
We study a mass formula for Artin--Schreier curves of genus $g$ defined over a finite field $k$ of characteristic $p$. For an odd prime $p$ and for small $g$, we determine the number of $k$-isomorphism classes of Artin-Schreier curves of genus $g$, weighted by the order of the centralizer of the Artin-Schreier involution in the automorphism group. This extends earlier results by several authors in characteristic $p=2$. Keywords: Artin-Schreier curve, finite field, automorphism, Mass formula, moduli space, arithmetic statistics
In this note, we give a so-called representative classification for the strata by automorphism group of smooth $\bar{k}$-plane curves of genus $6$, where $\bar{k}$ is a fixed separable closure of a field $k$ of characteristic $p = 0$ or $p > 13$. We start with a classification already obtained by the first author and we use standard techniques. Interestingly, in the way to get these families for the different strata, we find two remarkable phenomenons that did not appear before. One is the existence of a non $0$-dimensional final stratum of plane curves. At a first sight it may sound odd, but …
We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using $p$-adic cohomology.
We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p > 2. Using this, we prove that the Z/\ell-monodromy of every irreducible component of the stratum H_g^f of hyperelliptic curves of genus g and p-rank f is the symplectic group Sp_{2g}(Z/\ell) if g > 2, f > 0 and \ell is an odd prime distinct from p. These results yield applications about the generic behavior of hyperelliptic curves of given genus and p-rank. The first application is that a generic hyperelliptic curve of genus g > 2 and p-rank 0 is not su…
