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1. Counting points on smooth plane quartics

   https://doi.org/10.1007/s40993-022-00397-8  

   Costa, Edgar; Harvey, David; Sutherland, Andrew V. (March 2023, Research in Number Theory)

   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. 

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2. Overconvergent modular forms are highest-weight vectors in the Hodge-Tate weight zero part of completed cohomology

   https://doi.org/10.1017/fms.2021.16  

   Howe, Sean (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract We construct a $$(\mathfrak {gl}_2, B(\mathbb {Q}_p))$$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $$0$$ of a sheaf on $$\mathbb {P}^1$$ , landing in the compactly supported completed $$\mathbb {C}_p$$ -cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest-weight vector for any overconvergent modular form of infinitesimal weight not equal to $$1$$ . For classical weight $$k\geq 2$$ , the Verma has an algebraic quotient $$H^1(\mathbb {P}^1, \mathcal {O}(-k))$$ , and on classical forms, the pairing factors through this quotient, giving a geometric description of ‘half’ of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $$\mathbb {P}^1$$ . Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan, but the perspective here is different, and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology. 

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3. Differential operators and families of automorphic forms on Unitary groups of arbitrary signature

   Eischen, Ellen; Fintzen, Jessica; Mantovan, Elena; Varma, Ila (June 2018, Documenta mathematica)

   In the 1970's, Serre exploited congruences between q-expansion coefficients of Eisenstein series to produce p-adic families of Eisenstein series and, in turn, p-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to p-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on q-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack q-expansions when the signature is of the form (a,b), a≠b. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a p-adic measure taking values in the space of p-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit p-adic family of Eisenstein series. One application of our results is to the recently completed construction of p-adic L-functions for unitary groups by the first named author, Harris, Li, and Skinner. 

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4. Differential operators and families of automorphic forms on Unitary groups of arbitrary signature

   https://doi.org/10.25537/dm.2018v23.445-495  

   Eischen, Ellen; Fintzen, Jessica; Mantovan, Elena; Varma, Ila (June 2018, Documenta mathematica)

   In the 1970's, Serre exploited congruences between q-expansion coefficients of Eisenstein series to produce p-adic families of Eisenstein series and, in turn, p-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to p-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on q-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack q-expansions when the signature is of the form (a,b), a≠b. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a p-adic measure taking values in the space of p-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit p-adic family of Eisenstein series. One application of our results is to the recently completed construction of p-adic L-functions for unitary groups by the first named author, Harris, Li, and Skinner. 

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5. Refined Chabauty–Kim computations for the thrice-punctured line over $$\mathbb {Z}[1/6]$$

   https://doi.org/10.1007/s40993-024-00597-4  

   Lüdtke, Martin (March 2025, Research in Number Theory)

   Abstract The Chabauty–Kim method and its refined variant by Betts and Dogra aim to cut out theS-integral points$$X(\mathbb {Z}_S)$$ $X\left(Z_S\right)$on a curve inside thep-adic points$$X(\mathbb {Z}_p)$$ $X\left(Z_p\right)$by producing enough Coleman functions vanishing on them. We derive new functions in the case of the thrice-punctured line whenScontains two primes. We describe an algorithm for computing refined Chabauty–Kim loci and verify Kim’s Conjecture over$$\mathbb {Z}[1/6]$$ $Z[1/6]$for all choices of auxiliary prime $$p < 10{,}000$$ $p<10,000$. 

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