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Abstract We reduce the classification of finite extensions of function fields (of curves over finite fields) with the same class number to a finite computation; complete this computation in all cases except when both curves have base field$$\mathbb {F}_2$$ $F_2$and genus$$>1$$ $>1$; and give a conjectural answer in the remaining cases. The conjecture will be resolved in subsequent papers. 

Abstract We complete the computation of all$$\mathbb {Q}$$ $Q$-rational points on all the 64 maximal Atkin-Lehner quotients$$X_0(N)^*$$ $X_0{\left(N\right)}^{\ast }$such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levelsN, we classify all$$\mathbb {Q}$$ $Q$-rational points as cusps, CM points (including their CM field andj-invariants) and exceptional ones. We further indicate how to use this to compute the$$\mathbb {Q}$$ $Q$-rational points on all of their modular coverings. 

Abstract In this paper we prove that there are finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we show that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a modular curve of Shimura type admits a smooth plane model of degree 8 we show that it must be a twist of one of four curves. 

Abstract We present a new elementary algorithm that takes $$ \textrm{time} \ \ O_\epsilon \left( x^{\frac{3}{5}} (\log x)^{\frac{8}{5}+\epsilon } \right) \ \ \textrm{and} \ \textrm{space} \ \ O\left( x^{\frac{3}{10}} (\log x)^{\frac{13}{10}} \right) $$ time O ϵ x 3 5 ( log x ) 8 5 + ϵ and space O x 3 10 ( log x ) 13 10 (measured bitwise) for computing $$M(x) = \sum _{n \le x} \mu (n),$$ M ( x ) = ∑ n ≤ x μ ( n ) , where $$\mu (n)$$ μ ( n ) is the Möbius function. This is the first improvement in the exponent of x for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to $$O(x^{1/5} (\log x)^{5/3})$$ O ( x 1 / 5 ( log x ) 5 / 3 ) by the use of (Helfgott in: Math Comput 89:333–350, 2020), at the cost of letting time rise to the order of $$x^{3/5} (\log x)^2 \log \log x$$ x 3 / 5 ( log x ) 2 log log x . 

Abstract Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy $$\text {PGL}_2(\mathbb {F}_q)$$ PGL 2 ( F q ) or $$\text {PSL}_2(\mathbb {F}_q)$$ PSL 2 ( F q ) . We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus. 

Abstract We present efficient algorithms for counting points on a smooth plane quartic curve X modulo a prime p . We address both the case where X is defined over  $${\mathbb {F}}_p$$ F p and the case where X is defined over $${\mathbb {Q}}$$ Q and p is a prime of good reduction. We consider two approaches for computing $$\#X({\mathbb {F}}_p)$$ # X ( F p ) , one which runs in $$O(p\log p\log \log p)$$ O ( p log p log log p ) time using $$O(\log p)$$ O ( log p ) space and one which runs in $$O(p^{1/2}\log ^2p)$$ O ( p 1 / 2 log 2 p ) time using $$O(p^{1/2}\log p)$$ O ( p 1 / 2 log p ) space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $$X/{\mathbb {Q}}$$ X / Q that compute $$\#X({\mathbb {F}}_p)$$ # X ( F p ) for good primes $$p\leqslant N$$ p ⩽ N in $$O(N\log ^3 N)$$ O ( N log 3 N ) time using O ( N ) space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $${\mathbb {P}}^1$$ P 1 , which in combination with previous results addresses all curves of genus $$g\leqslant 3$$ g ⩽ 3 . Our algorithms also compute Cartier–Manin/Hasse–Witt matrices that may be of independent interest. 

Abstract We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we investigate endoscopy using theta series and a theorem of Rallis. Along the way, we exhibit many examples and pose several conjectures. As a first application, we express counts of Kneser neighbours in terms of coefficients of classical or Siegel modular forms, complementing work of Chenevier–Lannes. As a second application, we prove new instances of Eisenstein congruences of Ramanujan and Kurokawa–Mizumoto type. 

Abstract We present a new method for computing the zeta function of an algebraic curve over a finite field. The algorithm relies on a trace formula of Harvey to count points on a plane model of the curve. The zeta function of the curve is then obtained by making corrections at singular points. We report on an implementation and provide some examples in MAGMA which demonstrate an improvement over Tuitman’s algorithm. 

Abstract We use the invariant theory of binary quartics to give a new formula for the Cassels–Tate pairing on the 2-Selmer group of an elliptic curve. Unlike earlier methods, our formula does not require us to solve any conics. An important role in our construction is played by a certain K 3 surface defined by a (2, 2, 2)-form.