arXiv subjects

Padmavathi Srinivasan

Publications and source records attributed to Padmavathi Srinivasan.

18 recordsLinked to original sources

arXiv2026

Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers I

We prove conductor-discriminant inequalities for all $\mathbb{Z}/n$-covers of $\mathbb{P}^1$ defined over discretely valued fields $K$ with excellent valuation ring $\mathcal{O}_K$ and perfect residue field of characteristic not dividing $n$, modulo some calculations appearing in work of the third author (arXiv:2609.20585). Specifically, when such a curve $X$ is given by $y^n = f(x)$ with $f(x) \in\mathcal{O}_K[x]$ and $n\mid\text{deg}(f)$, and if $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $\mathcal{X}$ is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$. This is a direct generalization of previous work of the first two authors on hyperelliptic curves, which in turn generalized work of Ogg, Saito, Liu, and the second author. When $f$ is monic, this strengthens a result of Kohls stating that the conductor exponent of the Jacobian of such a curve is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$.

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arXiv2026

Covers of curves, Ceresa cycles, and Unlikely intersections

Fix a smooth, projective, geometrically integral curve $C$ of genus $g \geq 2$ over a characteristic zero field. We prove that the Ceresa cycle $\mathrm{Cer}(\widetilde{C})$ of a very general ramified cover $\widetilde{C}$ of $C$ is nontorsion in the Chow group of its Jacobian. We also show that there exist infinitely many families of ramified covers of a varying family of curves where a general point of these families corresponds to a curve with nontorsion Ceresa cycle. To illustrate this, we write down two explicit $1$-dimensional and $2$-dimensional families of genus $6$ curves where the locus of curves with torsion Ceresa cycle is Zariski closed and has positive codimension. Our strategy is to reduce the question of whether the Ceresa cycle is torsion to the question of whether a related point on the Jacobian of the curve is torsion. For this, we use the ``relative canonical shadow" of the Ceresa cycle, which is a point in the Jacobian of the curve obtained by intersecting the Ceresa cycle with a natural correspondence arising from the covering map. We combine this with ideas from unlikely intersection theory (namely the relative Manin--Mumford theorem) to study the locus where the relative canonical shadows of the Ceresa cycle become torsion.

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arXiv2026

p-adic adelic metrics and Quadratic Chabauty I

We give a new construction of $p$-adic heights on varieties over number fields using $p$-adic Arakelov theory. In analogy with Zhang's construction of real-valued heights in terms of adelic metrics, these heights are given in terms of $p$-adic adelic metrics on line bundles. In particular, we describe a construction of canonical $p$-adic heights on abelian varieties and we show that we recover the canonical Mazur--Tate height and, for Jacobians, the height constructed by Coleman and Gross. Our main application is a new and simplified approach to the Quadratic Chabauty method for the computation of rational points on certain curves over the rationals, by pulling back the canonical height on the Jacobian with respect to a carefully chosen line bundle. We show that our construction allows us to reprove, without using $p$-adic Hodge theory or arithmetic fundamental groups, several results due to Balakrishnan and Dogra. Our method also extends to primes $p$ of bad reduction. One consequence of our work is that for any canonical height ($p$-adic or $\mathbb{R}$-valued) on an abelian variety (and hence on pull-backs to other varieties), the local contribution at a finite prime $q$ can be constructed using $q$-analytic methods.

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arXiv2025

Minimal regular normal crossings models of superelliptic curves

Let $K$ be a complete discretely valued field with perfect residue field $k$. If $X \to \mathbb{P}^1_K$ is a $\mathbb{Z}/d$-cover with $\text{char } k \nmid d$, we compute the minimal regular normal crossings model $\mathcal{X}$ of $X$ as the normalization of an explicit normal model $\mathcal{Y}$ of $\mathbb{P}^1_K$ in $K(X)$. The model $\mathcal{Y}$ is given using Mac Lane's description of discrete valuations on the rational function field $K(\mathbb{P}^1)$.

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arXiv2025

Curve equations from expansions of 1-forms at a nonrational point

We exhibit an algorithm to compute equations of an algebraic curve over a computable characteristic 0 field from the power series expansions of its regular 1-forms at a nonrational point of the curve, extending a 2005 algorithm of Baker, González-Jiménez, González, and Poonen for expansions at a rational point. If the curve is hyperelliptic, the equations present it as an explicit double cover of a smooth plane conic, or as a double cover of the projective line when possible. If the curve is nonhyperelliptic, the equations cut out the canonical model. The algorithm has been used to compute equations over $\mathbb{Q}$ for many hyperelliptic modular curves without a rational cusp in the L-functions and Modular Forms Database.

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arXiv2024

Certifying nontriviality of Ceresa classes of curves

The Ceresa cycle is a canonical algebraic $1$-cycle on the Jacobian of an algebraic curve. We construct an algorithm which, given a curve over a number field, often provides a certificate that the Ceresa cycle is non-torsion, without relying on the presence of any additional symmetries of the curve. Under the hypothesis that the Sato--Tate group is the whole of $\operatorname*{GSp}$, we prove that if the Ceresa class (the image of the Ceresa cycle in étale cohomology) is non-torsion, then the algorithm will eventually terminate with a certificate attesting to this fact.

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arXiv2024

Quadratic enrichment of the logarithmic derivative of the zeta function

We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck--Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure.

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arXiv2023

Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic

We prove an inequality between the conductor and the discriminant for all hyperelliptic curves defined over discretely valued fields $K$ with perfect residue field of characteristic not 2. Specifically, if such a curve is given by $y^2 = f(x)$ with $f(x) \in \mathcal{O}_K[x]$, and if $X$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $X$ (and thus also the number of irreducible components of the special fiber of $X$) is bounded above by the valuation of disc$(f)$. There are no restrictions on genus of the curve or on the ramification of the splitting field of $f$. This generalizes earlier work of Ogg, Saito, Liu, and the second author.

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arXiv2023

Computing nonsurjective primes associated to Galois representations of genus $2$ curves

For a genus $2$ curve $C$ over $\mathbb{Q}$ whose Jacobian $A$ admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes $\ell$ for which the Galois action on $\ell$-torsion points of $A$ is not maximal. Building on work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are Mitchell's classification of maximal subgroups of $\mathrm{PSp_4}(\mathbb{F}_\ell)$, sampling of the characteristic polynomials of Frobenius, and the Khare--Wintenberger modularity theorem. The algorithm has been submitted for integration into Sage, executed on all of the genus~$2$ curves with trivial endomorphism ring in the LMFDB, and the results incorporated into the homepage of each such curve.

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arXiv2023

Surface bundles and the section conjecture

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of this conjecture. In so doing we produce many examples of curves satisfying the section conjecture over fields of geometric interest, and then over p-adic fields and number fields via a Chebotarev argument. We construct two Galois cohomology classes o_1 and o_2, which obstruct the existence of pi_1-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curves M_g studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to produce curves over p-adic fields and number fields where each class obstructs pi_1-sections and hence rational points. Among our geometric results are a new proof of the section conjecture for the generic curve of genus g>2, and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).

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arXiv2023

Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class

Let l be a prime and G a pro-l group with torsion-free abelianization. We produce group-theoretic analogues of the Johnson/Morita cocycle for G -- in the case of surface groups, these cocycles appear to refine existing constructions when l=2. We apply this to the pro-l etale fundamental groups of smooth curves to obtain Galois-cohomological analogues, and discuss their relationship to work of Hain and Matsumoto in the case the curve is proper. We analyze many of the fundamental properties of these classes and use them to give an example of a non-hyperelliptic curve whose Ceresa class has torsion image under the l-adic Abel-Jacobi map.

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arXiv2022

Explicit minimal embedded resolutions of divisors on models of the projective line

Let $K$ be a discretely valued field with ring of integers $\mathcal{O}_K$ with perfect residue field. Let $K(x)$ be the rational function field in one variable. Let $\mathbb{P}^1_{\mathcal{O}_K}$ be the standard smooth model of $\mathbb{P}^1_K$ with coordinate $x$ on irreducible special fiber. Let $f(x) \in \mathcal{O}_K[x]$ be a monic irreducible polynomial with corresponding divisor of zeroes $\text{div}_0(f)$ on $\mathbb{P}^1_{\mathcal{O}_K}$. We give an explicit description of the minimal embedded resolution $\mathcal{Y}$ of the pair $(\mathbb{P}^1_{\mathcal{O}_K}, \text{div}_0(f))$ by using Mac Lane's theory to write down the discrete valuations on $K(x)$ corresponding to the irreducible components of the special fiber of $\mathcal{Y}$.

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arXiv2021

Finiteness of reductions of Hecke orbits

We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension $g$ only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga-Satake construction, we also show that only finitely many supersingular $K3$-surfaces admit CM lifts. Our tools include $p$-adic Hodge theory and group theoretic techniques.

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arXiv2021

Conductors and minimal discriminants of hyperelliptic curves: A comparison in the tame case

Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is perfect and that $\mathrm{char}\ k > 2g+1$. Let $S = \mathrm{Spec}\ R$. Let $X$ be the minimal proper regular model of $C$ over $S$. Let $\mathrm{Art}\ (C/K)$ denote the Artin conductor of the $S$-scheme $X$ and let $ν(Δ_C)$ denote the minimal discriminant of $C$. We prove that $-\mathrm{Art}\ (C/K) \leq ν(Δ_C)$. The key ingredients are a combinatorial refinement of the discriminant introduced in this paper (called the metric tree) and a recent refinement of Abhyankar's inversion formula for studying plane curve singularities. We also prove combinatorial restrictions for $-\mathrm{Art}\ (C/K) = ν(Δ_C)$.

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arXiv2020

An arithmetic count of the lines meeting four lines in P^3

We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field $k$, this enrichment counts the number of lines meeting four lines defined over $k$ in $\mathbb{P}^3_k$, with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in $\mathbb{A}^1$-homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms. In the appendix, the condition that the four lines each be defined over $k$ is relaxed to the condition that the set of four lines being defined over $k$.

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arXiv2018

A virtually ample field that is not ample

A field $K$ is called ample if for every geometrically integral $K$-variety $V$ with a smooth $K$-point, $V(K)$ is Zariski-dense in $V$. A field $K$ is virtually ample if some finite extension of $K$ is ample. We prove that there exists a virtually ample field that is not ample.

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arXiv2015

Conductors and minimal discriminants of hyperelliptic curves with rational Weierstrass points

Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is perfect and that $\mathop{\textrm{char}} k \neq 2$. Assume that the Weierstrass points of $C$ are $K$-rational. Let $S = \mathop{\textrm{Spec}} R$. Let $\mathcal{X}$ be the minimal proper regular model of $C$ over $S$. Let $\mathop{\textrm{Art}} (\mathcal{X}/S)$ denote the Artin conductor of the $S$-scheme $\mathcal{X}$ and let $ν(Δ)$ denote the minimal discriminant of $C$. We prove that $-\mathop{\textrm{Art}} (\mathcal{X}/S) \leq ν(Δ)$. As a corollary, we obtain that the number of components of the special fiber of $\mathcal{X}$ is bounded above by $ν(Δ)+1$.

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arXiv2015

Zeta functions of a class of Artin-Schreier curves with many automorphisms

This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of the curves in the class over the field of definition of all automorphisms in the subgroup. As a consequence, we obtain new examples of maximal curves.

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