Research and Publications

Lattices in real quadratic fields and associated theta series arising from codes over ${\bf F}_4$ and ${\bf F}_2 \times {\bf F}_2$ (with Livia Betti, Fernando Gaitan, Aiyana Spear, and Japheth Varlack)
Designs, Codes and Cryptography, https://doi.org/10.1007/s10623-023-01258-w.
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Let $\mathcal{C} \subset \mathcal{R}^{n}$ be a code where $\mathcal{R}$ is ${\bf F}_4$ or ${\bf F}_2 \times {\bf F}_2$. Let $K= {\bf Q}(\sqrt{d})$ be a real quadratic field so that $\mathcal{O}_{K}/2 \mathcal{O}_{K} \cong \mathcal{R}$. One can construct a lattice $\Lambda(\mathcal{C}) \subset \mathcal{O}_{K}^{n}$ and in turn associate a theta series $\Theta_{\Lambda(\mathcal{C})}$ to the lattice. In this paper we show that if one varies $d$, the theta series agree up to a bound determined by the smaller $d$ under consideration.

Eigenform product identities of genus two Siegel modular forms of general congruence level (with Justine Dell, Hanna Noelle Griesbach, and Amanda Hernandez)
International Journal of Number Theory, 19(8), 1897-1915, (2023).
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Given two eigenforms, it is a natural question to ask if the product of the eigenforms is again an eigenform. In the case of elliptic modular forms this was answered in the full level case by Duke and Ghate and in the general level case by Johnson. In this paper we consider the case of genus two Siegel modular forms with general congruence level.

Congruence primes for automorphic forms on symplectic groups (with Huixi Li)
Glasgow Mathematical Journal, Glasgow Math. J., 63, no. 3, 660-681, (2021).
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It has been well-established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper we construct congruences for Siegel Hilbert modular forms defined over a totally real field of class number one. As an application of this general congruence, we produce congruences between paramodular Saito-Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch-Kato conjecture for elliptic newforms of square-free level and odd functional equation.

Hilbert modular forms and codes over $\mathbb{F}_{p^2}$ (with Beren Gunsolus, Jeremy Lilly, and Felice Mangeniello)
Finite Fields and Their Applications, 67, (2020).
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Let $p$ be an odd prime and consider the finite field $\mathbb{F}_{p^2}$. Given a linear code $\mathcal{C} \subset \mathbb{F}^{n}_{p^2}$, we use algebraic number theory to construct an associated lattice $\Lambda_{\mathcal{C}} \subset \mathcal{O}_{L}^{n}$ for $L$ an algebraic number field and $\mathcal{O}_{L}$ the ring of integers of $L$. We attach a theta series $\theta_{\Lambda_{\mathcal{C}}}$ to the lattice $\Lambda_{\mathcal{C}}$ and prove a relation between $\theta_{\Lambda_{\mathcal{C}}}$ and the complete weight enumerator evaluated on weight one theta series. Part of this work was done during the 2018 REU at Clemson University.

Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts (with Krzysztof Klosin)
Kyoto J. Math., Volume 60, Number 1, 179-217, (2020).
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In this paper we provide a sufficient condition for a prime to be a congruence prime for an automorphic form $f$ on the unitary group $\textrm{U}(n,n)(\mathbb{A}_F)$ for a large class of totally real fields $F$ via a divisibility of a special value of the standard $L$-function associated to $f$. We also study $p$-adic properties of the Fourier coefficients of an Ikeda lift $I_{\phi}$ (of an elliptic modular form $\phi$) on $\textrm{U}(n,n)(\mathbb{A}_{\mathbb{Q}})$ proving that they are $p$-adic integers which do not all vanish modulo $p$. Finally we combine these results to show that the condition of $p$ being a congruence prime for $I_{\phi}$ is controlled by the $p$-divisibility of a product of special values of the symmetric square $L$-function of $\phi$.

Eigenform Product Identities for Degree-Two Siegel Modular Forms (with Hugh Geller, Rico Vicente, Alexandra Walsh)
Journal of Number Theory, 204, 25-40, (2019).
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It is known via work of Duke and Ghate that there are only finiely many pairs of full level, degree one eigenforms $f$ and $g$ whose product $fg$ is also an eigenform. We prove a partial generalization of this theorem for degree two Siegel modular forms. Namely, we show that there is only one pair of eigenforms $F$ and $G$ such that $FG$ is a non-cuspidal eigenform. In the case that $FG$ is a cuspform, we provide necessary conditions for $FG$ to be an eigenform, give one example, and conjecture that is the only example. Part of this work was done during the 2018 REU at Clemson University.

On the action of the $U_p$ operator on Siegel modular forms (with Krzysztof Klosin)
Ramanujan Journal, 44(3), 597-615, (2017).
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In this article we study the action of the $U_p$ Hecke operator on the normalized spherical vector $\phi$ in the representation of $\textrm{GSp}_4(\mathbb{Q}_p)$ induced from a character on the Borel subgroup. We compute the Petersson norm of $U_p \phi$ in terms of certain local $L$-values associated with $\phi$.

Amicable pairs and aliquot cycles for elliptic curves over number fields (with David Heras, Kevin James, Rodney Keaton, and Andrew Qian)
Rocky Mountain Journal of Mathematics, 46(6), 1853-1866, (2016).
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This paper is a result of one of the projects at the 2012 REU at Clemson University. The notion of amicable pairs and aliquot cycles on elliptic curves was introduced by Silverman and Stange. They provided a very detailed analysis of these concepts for elliptic curves over the rational numbers. In this paper we consider amicable pairs and aliquot cycles over general number fields. We focus on the existence of aliquot cycles in various cases and explore some of the sublteties of dealing with primes of different degrees.

Counting tamely ramified extensions of local fields up to isomorphism (with Robert Cass, Kevin James, Rodney Keaton, Salvatore Parenti, and Daniel Shankman)
Integers, 16 #A53, 1-12, (2016).
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This paper is a result of one of the projects at the 2013 REU at Clemson University. Let $p$ be a prime number and let $K$ be a local field of residue characteristic $p$. In this paper we give a formula that counts the number of degree $n$ tamely ramified extensions of $K$ in the case $p$ is of order 2 modulo $n$. This result is achieved via elementary counting methods and simple group theory.

Mixed level Saito-Kurokawa liftings (with Dania Zantout)
The Ramanujan Journal, 39 , 247-257, (2016).
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In a 2007 paper R. Schmidt constructed the congruence level and paramodular level Saito-Kurokawa lifts via representation theoretic methods. We use these methods to construct Saito-Kurokawa lifts of more general levels. In particular, we recover a mixed level Saito-Kurokawa lift that was claimed by M. Manickam and B. Ramakrishnan.

Saito-Kurokawa lifts of odd square-free level (with Mahesh Agarwal)
Kyoto Journal of Mathematics, 55(3), 641-662 (2015).
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In the paper On the Bloch-Kato conjecture for elliptic modular forms of square-free level the Saito-Kurokawa lift of square-free level and its arithmetic properties are heavily used. However, until Ibukiyama provided a construction of the Maass lifting with level in 2012 there was no correct classical construction of this Saito-Kurokawa lift. In this paper we put together the classical and automorphic construction in one article as well as give the necessary arithmetic properties. We also calculate the norm of the Saito-Kurokawa lift.

Congruence Primes for Ikeda Lifts and the Ikeda ideal (with Rodney Keaton)
Pacific J. Math., 274(1), 27-52 (2015).
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Let $f$ be a newform of level 1 and weight $2k-n$ for $k$ and $n$ positive even integers. In this paper we study congruence primes for the Ikeda lift of $f$. In particular, we consider a conjecture of Katsurada stating that primes dividing certain $L$-values of $f$ are congruence primes for the Ikeda lift. Instead of focusing on a congruence to a single eigenform, we deduce a lower bound on the number of all congruences between the Ikeda lift and forms not lying in the space spanned by Ikeda lifts by considering the same primes and slightly relaxed hypotheses. In particular, we define the Ikeda ideal and show how this can be used to study all congruences instead of focusing on a single congruence.

Degree 14 extensions of $\mathbb{Q}_7$ (with Robert Cass, Rodney Keaton, Salvatore Parenti, and Daniel Shankman)
Int. J. of Pure and Appl. Math., 100(2), 337-345 (2015).
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This paper is a result of one of the projects at the 2013 REU at Clemson University. We calculate all degree 14 extensions of $\mathbb{Q}_7$ up to isomorphism. We give the Galois group of each extension along with enough information about each Galois group so that one can distinguish between them. The data that accompanies this paper can be found at data.

Classifying extensions of the field of formal Laurent series over $\mathbb{F}_p$ (with Alfeen Hasmani, Lindsey Hiltner, Angela Kraft, Daniel Scofield, and Kirsti Wash)
Rocky Mountain Journal of Mathematics, 45(1), 115-130 (2015).
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This paper is a result of one of the projects at the 2012 REU at Clemson university. In previous works, Jones-Roberts and Pauli-Roblot have studied finite exten- sions of the p-adic numbers $\mathbb{Q}_p$. This paper focuses on results for local fields of characteristic $p$. In particular we are able to produce analogous results to Jones-Roberts in the case that the characteristic does not divide the degree of the field extension. Also in this case, following from the work of Pauli-Roblot, we prove that the defning polynomials of these extensions can be written in a simple form amenable to computation. Finally, if $p$ is the degree of the extension, we show there are infinitely many extensions of this degree and thus these cannot be classified in the same manner.

On the Bloch-Kato conjecture for elliptic modular forms of square-free level (with Mahesh Agarwal)
Mathematische Zeitschrift, 276 (3), 889-924 (2014).
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This paper contains generalizations of the results of On the cuspidality of pullbacks of Siegel Eisenstein series and applications to the case of square-free level. It also contains a discussion of the relation of the bounds on the Selmer group to the Bloch-Kato conjecture. This discussion is absent in On the cuspidality of pullbacks of Siegel Eisenstein series and applications. The computational evidence will be removed from the final version, but will remain available here: Computational Evidence. One should note that in the main theorem we require the conductor of the Dirichlet character be divisible by the level of the modular form in question. Such an assumption is not necessary; one can take an auxilliary level divisible by the conductor and the level of the modular form and work in that level. That should make it more feasible to expand the computational data provided.

Special values of $L$-functions for Saito-Kurokawa lifts (with Ameya Pitale)
Mathematical Proceedings of the Cambridge Philosophical Society, 155 (2), 237-255 (2013).
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In this paper we obtain special value results for $L$-functions associated to classical and paramodular Saito-Kurokawa lifts. In particular, we construct standard L-functions associated to Saito-Kurokawa lifts as well as degree eight L-functions obtained by twisting with an automorphic form defined on $\textrm{GL}(2)$. The results are obtained by combining classical and representation theoretic arguments. One should note here there is a typo in Theorem 4.5 (replace $m-1$ with $\varphi(m)$) and in Corollary 4.6 (replace $m-1$ with $\varphi(m)$ and remove the $m+1$ in $C_{k,m}$.)

Level stripping for Siegel modular forms with reducible Galois representations (with Rodney Keaton)
Journal of Number Theory, 133 (5), 1492-1501 (2013).
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In this paper we consider level stripping for genus 2 cuspidal Siegel eigenforms. In particular, we show that it is possible to strip primes from the level of weak endoscopic lifts as well as from Saito-Kurokawa lifts that arise as theta lifts with a mild restriction on the associated character.

Pullbacks of Siegel Eisenstein series and weighted averages of critical $L$-values (with Nadine Amersi, Jeff Beyerl, Allison Proffer, and Larry Rolen)
The Ramanujan Journal, 27(2), 151-162 (2012).
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This paper is the result of an REU project I directed at the Clemson REU on Combinatorics and Computational Number Theory during the summer of 2010. We study the pullback of a Siegel Eisenstein series of weight $k$ and full level from $\textrm{Sp}(6)$ to $\textrm{Sp}(4) \times \textrm{Sp}(2)$. By explicitly working out the constants in this pullback formula we are able to produce a weighted average formula for the special values $D(k-1,f)$ where $f$ runs over an orthogonal basis of $S_{k}(\textrm{SL}_2(\mathbb{Z})$.

On the cuspidality of pullbacks of Siegel Eisenstein series and applications
International Mathematics Research Notices, 7, 1706-1756 (2011).
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This paper is essentially a follow-up paper to Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture. (One should see the description of that paper for the set-up of this one.) Here we remove some of the ad-hoc arguments used to produce a congruence between a Saito-Kurokawa lift and a cuspidal Siegel eigenform with irreducible Galois representation. The results are phrased in terms of the CAP-ideal (an ideal analogous to the Eisenstein ideal). Some results of E. Urban are generalized which allow us to strengthen the results from earlier work to give essentially one inclusion of the Bloch-Kato conjecture for the k-th twist of the Galois representation associated to f (up to our technical hypotheses needed to produce the congruence). This paper does not include an explicit description of the relation of the main result stated and the Bloch-Kato conjecture. Such a description is included in a forthcoming paper with Mahesh Agarwal that generalizes these results to include newforms of square-free level.

On the cuspidality of pullbacks of Siegel Eisenstein series to $\textrm{Sp}(2m) \times \textrm{Sp}(2n)$
Journal of Number Theory, 131, 106-119 (2011).
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This paper studies the conditions under which one can conclude that the pullback of a Siegel Eisenstein series from $\textrm{Sp}(2m) \times \textrm{Sp}(2n)$ is cuspidal in the smaller variable. It was shown by Garrett that if $n=m$, for a certain choice of section the pullback of the associated Eisenstein series is cuspidal in each variable. Here we generalize this to show that if $m$ is not equal to $n$, the pullback of the Eisenstein series is cuspidal in the smaller variable.

$L$-functions on $\textrm{GSp}(4) \times \textrm{GL}(2)$ and the Bloch-Kato conjecture
International Journal of Number Theory, 6(8), 1901-1926 (2010).
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Here we use a pullback formula of B. Heim that calculates the inner product of the pullback of a Siegel Eisenstein series on $\textrm{GSp}(10)$ to $\textrm{GSp}(4) \times \textrm{GSp}(4) \times \textrm{GL}(2)$ with the Saito-Kurokawa lift of a newform $f$ in each of the $\textrm{GSp}(4)$ variables and a newform $g$, allowed to vary, in the $\textrm{GL}(2)$ variable. We show how this can be used to give results towards the Bloch-Kato conjecture for $f$. In particular, this gives a different flavor of result than Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture as the freedom in the technical hypotheses there are in varying a character and here we are allowed to vary a modular form.

On the congruence primes of Saito-Kurokawa lifts of odd square-free level
Mathematical Research Letters, 17(5), 977-991 (2010).
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In this paper we generalize a conjecture of H. Katsurada about the congruence primes of Saito-Kurokawa lifts to the case of odd square-free level. We also provide evidence for this new conjecture.

The first negative Hecke eigenvalue of genus 2 Siegel cuspforms with level $N \geq 1$
International Journal of Number Theory, 6(4), 857-867 (2010).
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In this short note we extend results of W. Kohnen and J. Sengupta on the sign of eigenvalues of Siegel cuspforms. We show that their bound for the first negative Hecke eigenvalue of a genus 2 Siegel cuspform of level 1 extends to the case of level $N \geq 1$. We also discuss the signs of Hecke eigenvalues for CAP forms.

Level lowering for half-integral weight modular forms (with Yingkun Li)
Proceedings of the American Mathematical Society, 138, 1171-1173 (2010).
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This is one of two papers that resulted from the Summer Undergraduate Research Fellowship (SURF) of Yingkun Li that I directed at Caltech during the summer of 2008. Here we provide a level stripping result for half-integral weight modular forms that we originally thought we would need in the work contained in Distribution of powers of the partition function modulo $\ell^{j}$.

Distribution of powers of the partition function modulo $\ell^{j}$ (with Yingkun Li)
Journal of Number Theory, 129, 2557-2568 (2009).
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This is one of two papers that resulted from the Summer Undergraduate Research Fellowship (SURF) of Yingkun Li that I directed at Caltech during the summer of 2008. In this paper we study Newman's conjecture for powers of the partition for exceptional primes. We settle this conjecture in many cases for small powers of the partition function by generalizing results of Ono and Ahlgren.

Residually reducible representations of algebras over local Artinian rings
Proceedings of the American Mathematical Society, 136(10), 3409-3414 (2008).
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In this paper we generalize a result of E. Urban on the structure of residually reducible representations on local Artinian rings from the case the semi-simplification of the residual representation splits into 2 absolutely irreducible representations to the case where it splits into $m >2$ absolutely irreducible representations. In particular, the case of $m=3$ is needed in On the cuspidality of pullbacks of Siegel Eisenstein series and applications.

An inner product relation on Saito-Kurokawa lifts
The Ramanujan Journal, 14(1), 89-105 (2007).
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This paper consists of the calculation of the Petersson norm of a Saito-Kurokawa lift of square-free level. One should note this calculation is based on a construction of the Saito-Kurokawa lift that was later shown to be incorrect. The correct construction and calculation can be found in On the Bloch-Kato conjecture for elliptic modular forms of square-free level with Mahesh Agarwal. The paper also has an error in section 7 due to a mistake in factoring the $L$-function in Proposition 7.5.

Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture
Compositio Mathematica, 143(2), 290-322 (2007).
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Let $f$ be a normalized eigenform of weight $2k-2$ and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypotheses if $p$ divides the algebraic part of $L(k,f)$, the $p$ divides an appropriate Selmer group. We demonstrate this by establishing a congruence between the Saito-Kurokawa lift of $f$ and a cuspidal Siegel eigenform with irreducible Galois representation. The method here is essentially due to Ribet and his proof of the converse of Herbrand's theorem.

The argument given in the proof of Theorem 4.4 is not complete. This also impacts several of the papers above. Please see the following short note that fixes the gap in the proof: fixes.

Saito-Kurokawa lifts and applications to arithmetic
Conference Proceedings of the 9th Autumn Conference on Number Theory, Hakuba Japan, 1-11 (2007)
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These are notes from my plenary address at the 9th Autumn Conference on Number Theory in Hakuba, Japan. The topic of the conference was automorphic forms on $\textrm{GSp}(4)$.

Variation of Hodge Structure (with Kirsten Eisentrager, Krzysztof Klosin, Jorge Pineiro, Mak Trifkovic, and Oliver Watson)
Arizona Winter School on Periods 2002
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This is the short write-up from the project supervised by Johan de Jong during the 2002 Arizona Winter School.